MATH 121 -Calculus Interactive Notes Fall 2014math121/Notes/notes_fall_math121.pdf · MATH 121...

MATH 121 - Calculus

Interactive Notes

Fall 2014

Unit #1 : Transformation of Functions, Exponentials and Loga-rithms

Goals:

• Review core function families and mathematical transformations.

Textbook reading for Unit #1: Read Sections 1.1–1.4

Familiar Functions - 1

Example: The graphs of ex, ln(x), x2 and x12 are shown below. Identify

each function’s graph.

x

y


Comment on the properties of the graphs of

• inverse functions -

• exponentials -

• logarithms -

• powers of x -


Knowing the graphs and properties of essential families of functions is crucial foreffective mathematical modeling.Name other families of functions.


Give examples of members of each family, and state some of their commonproperties.

Transforming Functions - 1

The core families of functions can be made even more versatile by being trans-formed.Example: Sketch the graph of y = x2, over the interval x ∈ [−4, 4].

On the same axes, sketch the graph of y = 4− 12(x + 1)2.


Review the four common types of function transformations.Type Form Example


Type Form Example

Modeling With Transformations - 1

Example: Consider the data shown below, showing the concentration of achemical produced in a reaction vessel, over time.

0 20 40 60 80 100

05

1015

Time (hours)

Con

cent

ratio

n (p

pm)

What family of functions would best describe this graph? Point out specificfeatures of the graph that make the choice a reasonable one.


Give a general mathematical form for the func-tion, based on the shape of the graph.e.g. C(t) = ...

0 20 40 60 80 100

05

1015

Time (hours)

Con

cent

ratio

n (p

pm)


Determine as many of the numerical values in theformula C(t) = ... as you can, given the graph.Sketching related graphs along the way might behelpful.

0 20 40 60 80 100

05

1015

Time (hours)

Con

cent

ratio

n (p

pm)


Looking closely at the graph, you see that after30 hours, the concentration has reached almostexactly 12 ppm. Determine the value for the fi-nal missing parameter in your concentration func-tion.

0 20 40 60 80 100

05

1015

Time (hours)

Con

cent

ratio

n (p

pm)

Logarithm Definition - 1

Logarithm Review

Most students are quite comfortable with exponential functions, but many findlogarithms less familiar. To address this we will do a more comprehensive reviewof the logarithmic function and its use in transforming equations.

Log/Exponential Equivalency

ac = x means loga x = c

Simplify loga(a7).

Simplify aloga(25).


Without using a calculator, find log10(10, 000), and log10(1/100).


These problems suggest the following equations, which also follow from the factthat ax and loga(x) are inverse functions.

loga(ax) = x and aloga x = x

Rules for Computing with Logarithms

1. loga(AB) = logaA + logaB

2. loga(A/B) = logaA− logaB

3. loga(AP ) = P logaA

Changing Log Bases - 1

Changing logarithmic bases

The functions ax and loga are not provided on calculators unless a = 10 or a = e(see next section of these notes). For other values of a, ax and loga can be expressedin terms of 10x and log10. To calculate loga x, we use the following formula:

Conversion of Log Bases

loga x =log10 x

log10 aor

loge x

loge a

Prove the above formula, using the Rules for Computing Logarithms and thefact that loga x = c means x = ac.

Changing Log Bases - 2

Example: Without an exact calculation, determine which of log10 1000 andlog2 1000 is the larger numeric value.

Compute the numeric value of both the log values above, using your calculatorif necessary.

Note: since the logarithm in base 10 is commonly used in science, we define log x(no subscript) to mean log10 x, for brevity.For the natural logarithm (base e), we use ln instead of loge.

Graphs of Logarithmic Functions - 1

Graphs of Logarithmic Functions

The graph of loga x may be obtained from the graph of its corresponding expo-nential function by reversing the axes (that is, by reflecting the graph in the liney = x). (If drawing the graph of inverse functions is unfamiliar, please read Section1.3 in the text.)E.g. for y = log10 x and y = 10x,

10x

log10(x)

Graphs of Logarithmic Functions - 2

What is the domain of log x? What is the range of log x?

Sketch the logarithm function for the bases e and 2.

Continuous Growth With Exponentials - 1

Classic Applications of Exponentials and Logarithms

Example: A cup of coffee contains 100 mg of caffeine, which leaves thebody at a continuous rate of 17% per hour.

Sketch the graph of caffeine level over time, after drinking one cup of coffee.


There are two natural interpretations of the question statement which lead totwo different formulae for A(t). Write down both formulae.

Compare the predicted caffeine level after 10 hours, using each model. Basedon those values, how similar are these two models in practice?


The key phrase continuous rate has a special meaning in mathematics and science,and it associated with the natural exponential form ert. It is typically associatedwith processes like chemical reactions, population growth, and continuously com-pounded interest.Common alternative statements about percentage growth or decay, where the rateis assumed to be measured at the end of one time period (hour, day year), areusually of the form (1± r)t.


Write out an appropriate mathematical model for the following scenarios:

• Infant mortality is being reduced at a rate of 10% per year.

• My $10,000 investment is growing at 5% per year.

• A savings account offers daily compound interest, at a 4% annual rate.

• Bacteria are reproducing at a continuous rate of 125% per hour.


We now return to our earlier modeling problem.Example: A cup of coffee contains 100 mg of caffeine, which leaves thebody at a continuous rate of 17% per hour. Write the formula for A(t).

What is the caffeine level at t = 4 hours?

At what time does the caffeine level reach A = 10 mg?


Find the half-life of caffeine in the body.

Unit #2 : Limits, Continuity, and the Derivative

Goals:

• Study and define continuity

• Review limits

• Introduce the derivative as the limit of a difference quotient

• Discuss the derivative as a rate of change

Textbook reading for Unit #2: Study sections 1.7, 1.8, and 2.1-2.2

Intro to Continuity - 1

We spent the previous unit discussing various families of functions, and how wecould transform them to obtain functions that suit a particular purpose.In this unit we will study properties and tools for all functions: continuity andlimits, and the idea of the derivative.

Continuity

To prepare for this topic, you should read Section 1.7 in the textbook.

Consider this statement: You were once exactly 1 meter tall.

A. True

B. False


Below is a graph for drawing your height over time.

Put your birth height, and your current height on the graph.

Put the line h = 1 meter on the graph.

Why must the graph of your height cross the h = 1 line?


The more formal way to state the property we used is through the IntermediateValue Theorem.

Intermediate Value TheoremSuppose f is continuous on a closed interval [a, b]. If k is any number betweenf (a) and f (b), then there is at least one number c in [a, b] such that f (c) = k.

This introduces another question, though: how do we define a continuous function?What characterizes the graph of a continuous function?


Give examples of functions that are continuous everywhere.

Give example of functions with discontinuities.

Is the function y = |x| continuous at x = 0?

A. Yes

B. No

Continuity From Formulas - 1

Continuity and Formulas

The continuity of a function (or lack thereof) is usually obvious if we have a graphof a function. It can be less clear if we just have a formula.

Example: Where must the function y =sin(x)

xbe continuous? Where might

it be discontinuous?


Use sample points to sketch the graph of y =sin(x)

xon the interval x ∈

[−1, 1].

What is the value of y at x = 0?


What value does y approach as x approaches 0?

How do we write the last question mathematically?

Intuition about Limits - 1

Intuitive Definition of the LimitThe limit lim

x→cf (x) is the number L if we can make f (x) as close to L as we want,

for all the values of x close enough to c.It is possible that such a number L may not exist; if that is the case, we would saythe limit does not exist.

Note: The limit never depends on the value of the function at the limiting point,or even if that point is defined or not.


Example: Explain why the limit limx→2

f (x) is not 6 for the graph below.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

02

46

8

x

f(x)


0.0 0.5 1.0 1.5 2.0 2.5 3.0

02

46

8

x

f(x)

Describe the real value and limits for x = 2 of the graph of f (x) above.


For each of the graphs below, explain why the limit as x→ 0 does not exist.

x

f(x)


x

f(x)

The Definition of Continuity - 1

We can use our definition of the limit to help us define continuity.

Definition of ContinuityA function is continuous at a point x = c if

• f (c) is defined

• limx→c

f (x) is defined

• both these values are equal

A function is continuous on an interval x ∈ [a, b] if it is continuous at all the pointson the interval.


Use this definition to state why y =sin(x)

xis not continuous at x = 0.

Use the consequences of this definition to compute limx→4

(x2 + x− 3).


Use the terms ‘limit’ and ‘continuous’ to describe the situation around x = 2on the graph below.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

02

46

8x

f(x)

Computing Limits - 1

Computing limits

Example: Consider the function g(x) =x2 − 4

x + 2.

Find the points where the function is discontinuous.

Evaluate the limit of g(x) as you approach the discontinuity.


One- and Two-Sided Limits

Example: Consider the piecewise function

h(x) =

{−x + 1 x ≤ 1

x2 − 1 x > 1

On what intervals is the function clearly continuous?

At what point(s) might the function be discontinuous?


Deciding whether a function is continuous or not, we realize that in our previouslimit questions, we (implicitly or explicitly) considered points on both sides of thelimiting point. In this new case, we have different behaviours on either side, so weneed a way to handle this special case.

Two-Sided Definition of the LimitThe limit lim

x→cf (x) exists and equals L if and only if

• the right-hand limit limx→c+ exists and equals L, and

• the left-hand limit limx→c− also exists and equals L.


h(x) =

{−x + 1 x ≤ 1

x2 − 1 x > 1

Does the limit limx→1 h(x) exist, and if so what is its value?


Analyze the continuity of h(x) at x = 1.


Example: Sketch examples where both one-sided limits exist, but the overalllimit does not, and where one or both of the one sided limits do not exist.

Limits at Infinity - 1

Limits at Infinity

Aside from studying questions of continuity, limits can help to analyze the be-haviour of the functions when x is very large. We say that the limit of f (x) asx approaches infinity is equal to L (written limx→∞ f (x) = L) if f (x) becomesarbitrarily close to L when x is arbitrarily large. We make the similar definitionfor x→ −∞ when x is negative.

Use limits at infinity to express the horizontal asymptote of ex.

Find limt→∞

15− 10e−.04t

Limits at Infinity - 2

Determine limx→∞

x2 + 1

2x2 − 1

Average Rates and the Derivative - 1

Average Rates and The Derivative

To prepare for this topic, you should read Sections 2.1 and 2.2 in the textbook.

Consider the graph below, and assume it represents the position of a person overtime; t is in seconds, and f (t) is in meters.

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

t

f(t)

Compute the average speed of the person between t = 1 and t = 4 seconds.


0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

t

f(t)

How can you show the value of the average speed by using the points t = 1and t = 4 on the graph?


0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

t

f(t)

On the graph, sketch the line that would reflect the instantaneous speed att = 1.


Based on the earlier question about average speed, how would you computethe instantaneous speed?

What is the difficulty with that approach?


Our earlier work with limits was not tied to any particular function or any particularapplication. Now we use limits as a tool to help answer a completely separatequestion.If we cannot use rise-over-run or distance-over-time directly to compute theinstantaneous speed, write how we could use limits to help us.


Sketch on the graph what this limit represents.

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

t

f(t)

Definition of the Derivative at a Point - 1

Definition of the Derivative at a Point

By formalizing the intuitive idea of an “instantaneous” slope, or speed, or rate ofchange, we have created a formal way to compute slopes at points. We call thevalue of the slope at a point the derivative of the function at that point.


The DerivativeThe derivative of a function f (x) at a point x = c is defined as

limh→0

f (c + h)− f (c)

h

Sometimes this is written with ∆x instead of h, to make the relationship to thegraph more intuitive:

lim∆x→0

f (c + ∆x)− f (c)

∆xA third way is to write as if we were moving two separate points together, ratherthan using one reference point (c) and a distance (∆x or h)

limx→c

f (x)− f (c)

x− cNote that all these forms share the common “rise over run” ratio, and “limit asdenominator goes to zero”. You may use any of the forms, as they are all equivalent.


Example: Find the slope of the function f (x) = x2 at the point x = 2, usingthe definition above.


The graph of x2 around x = 2 is shown below. Confirm your answer graphi-cally by sketching the tangent line with the slope you just computed.

1.0 1.5 2.0 2.5 3.0

12

34

56

78

x

y

Unit #3 : Differentiability, Computing Derivatives, Trig Review

Goals:

• Determine when a function is differentiable at a point

• Relate the derivative graph to the the graph of an original function

• Compute derivative functions of powers, exponentials, logarithms, and trig func-tions.

• Compute derivatives using the product rule, quotient rule, and chain rule forderivatives.

• Review trigonometric and inverse trigonometric functions.

Textbook reading for Unit #3 : Study Sections 1.5, 2.3, 2.4, 2.6 and 3.1–3.6.

Secants vs. Derivatives - 1

In the previous unit we introduced the definition of the derivative. In thisunit we will use and compute the derivative more efficiently. As a lead in, though,let us review how we arrived at the derivative concept.

Interpretations of Secants vs Derivatives

From Section 2.4The slope of a secant line gives

• the average rate of change of f (x) over some interval ∆x.

• the average velocity over an interval, if f (t) represents position.

• the average acceleration over an interval, if f (t) represents velocity.

Give the units of the slope of a secant line.

Secants vs. Derivatives - 2

The derivative gives

• the limit of the average slope as the interval ∆x approaches zero.

• a formula for slopes for the tangent lines to f (x).

• the instantaneous rate of change of f (x).

• the velocity, if f (t) represents position.

• the acceleration, if f (t) represents velocity.

Give the units of the derivative.

Differentiability - 1

Differentiability

From Section 2.6Recall the definition of the derivative.

f ′(x) =d

dxf =

df

dx= lim

∆x→0

∆f

∆x= lim

h→0

f (x + h)− f (x)

h

A function f is differentiable at a given point a if it has a derivative at a, orthe limit above exists. There is also a graphical interpretation differentiability: ifthe graph has a unique and finite slope at a point. Since the slope inquestion is automatically the slope of the tangent line, we could also say that

f is differentiable at a if its graph has a (non-vertical)tangent at (a, f (a)).

For functions of the form y = f (x), we do not consider points with vertical tangentlines to have a real-valued derivative, because a vertical line does not have a finiteslope.


Here are the ways in which a function can fail to be differentiable at a point a:

1. The function is not continuous at a.

2. The function has a corner (or a cusp) at a.

3. The function has a vertical tangent at (a, f (a)).

Sketch an example graph of each possible case.


Investigate the limits, continuity and differentiability of f (x) = |x| at x = 0graphically.


Use the definition of the derivative to confirm your graphical analysis.


We have seen at a point that a function can have, or fail to have, the followingdescriptors:

• continuous;

• limit exists;

• is differentiable.

Put these properties in decreasing order of stringency, and sketch relevantillustrations.


Reminder: Differentiability is CommonYou will notice that, despite our concern about some functions not being differen-tiable, most of our standard functions (polynomials, rationals, exponentials, loga-rithms, roots) are differentiable at most points. Therefore we should investigatewhat all these possible derivative/slope values could tell us.

Interpreting the Derivative - 1

Interpreting the DerivativeFrom Section 2.4

•Where f ′(x) > 0, or the derivative is positive, f (x) is increasing.

•Where f ′(x) < 0, or the derivative is negative, f (x) is decreasing.

•Where f ′(x) = 0, or the tangent line to the graph is horizontal, f (x)has a critical point.


Example: Consider the graph of f (x) shown below.

A

B

C

D E

FG

On what intervals is f ′(x) > 0?


A

B

C

D E

FG

Where does f ′(x) takes on its largestnegative value?

Graphs of the Derivative - 1

Graphs, and Graphs of their Derivatives

Example: Consider the same graph again, and the graph of its derivative.Identify important features that associate the two.

A

B

C

D E

FG

A B C D E F G


Question: Consider the graph of f (x) shown:

−1 0 1 2

−1

1

2

Which of the following graphs is the graph of the derivative of f (x)?


−1 0 1 2

−1

1

2

−1 0 1 2

−1

1

2

−1 0 1 2

−1

1

2

A B

−1 0 1 2

−1

1

2

−1 0 1 2

−1

1

2

C D

Computing Derivatives - Basic Formulas - 1

Computing Derivatives

Note: The standard formulas for derivatives are covered in the Grade 12 Ontariocurriculum. While they will be reviewed here, students who are not familiarwith them should begin both textbook reading and the assignmentproblems for this unit as soon as possible.From Sections 3.1-3.6Beyond the graphical interpretation of derivatives, there are all the algebraic rules.All of these rules are based on the definition of the derivative,

f ′(x) =d

dxf =

df

dx= lim

∆x→0

∆f

∆x= lim

h→0

f (x + h)− f (x)

h

However, by finding common patterns in the derivatives of certain families of func-tions, we can compute derivatives much more quickly than by using the definition.


Sums, Powers, and Differences

Constant Functions:d

dxk = 0

Power rule:d

dxxp = pxp−1

Sums :d

dxf (x) + g(x) =

(d

dxf (x)

)+

(d

dxg(x)

)

Differences:d

dxf (x)− g(x) =

(d

dxf (x)

)−(d

dxg(x)

)

Constant Multiplier:d

dx[kf (x)] = k

(d

dxf (x)

), so long as k is a constant


Example: Evaluate the following derivatives:d

dx

(x4 + 3x2

)

d

dx

(2.6√x− πx3 + 4

)


Question: The derivative of −3x2 − 1

x2is

1. −6x3 + 21

x3

2. −6x + 21

x3

3. −6x− 21

x3

4. −x3 + 21

x


Exponentials and Logs

e as a base:d

dxex = ex

Other bases:d

dxax = ax(ln(a))

Natural Log:d

dxln(x) =

1

x

Other Logs:d

dxloga(x) =

1

x

1

ln(a)



dx

(4 · 10x + 10 · x4

)

d

dx(ex + log10(x))

(Exponential and log derivatives are relatively straightforward, until we mix in theproduct, quotient, and chain rules.)

Computing Derivatives - Product and Quotient Rules - 1

Product and Quotient Rules

Products:d

dxf (x) · g(x) = f ′(x)g(x) + f (x)g′(x)

Quotients:d

dx

f (x)

g(x)=f ′(x)g(x)− f (x)g′(x)

(g(x))2


dx

(4x2ex

)


d

dx(x ln(x))

d

dx

(5x2

ln(x)

)


Question: The derivative of10x

x3is

1.10x

ln(10)x−3 + 10x(−3x−4)

2.10x ln(10)x3 − 10x(3x2)

x6

3.10x 1

ln(10)x3 − 10x(3x2)

x6

4. ln(10)10xx−3 + 10x(−3x−4)

Computing Derivatives - Chain Rule - 1

Chain Rule

Nested Functions:d

dx[f (g(x))] = f ′(g(x)) · g′(x)

Liebnitz formd

dxf (g(x)) =

df

dg

dg

dx



dxex

2


d

dxln(x4)


d

dx

(1

1 + x3

)


d

dx

(x4 + 103x

)


Question: The derivative of e√x is

1.1

2e

1√x

2. e√x(√

x)

3.1

2e√x

(1√x

)

4.1

2e√x(√

x)

Trigonometry Review - 1

Trigonometry Review

From Section 1.5In our earlier discussion of functions, we skipped over the trigonometric functions.We return to them now to discuss both their properties and their derivative rules.

The trigonometric functions are usually defined for students first using triangles(recall the mnemonic device, “SOHCAHTOA”).


Use the 45/45 and 60/30 triangles to compute the sine and cosine of thesecommon angles.


Extending Trigonometric DomainsOne difficulty with limiting ourselves to the triangle ratio definition of the trigfunctions is that the possible angles are limited to the range θ ∈ [0, π2 ] radians orθ ∈ [0, 90] degrees.To remove this limitation, mathematicians extended the definition of the trigono-metric functions to a wider domain via the unit circle.

θ


How does the circle definition lead to the trigonometric identity sin2(θ) +cos2(θ) = 1?


Show how the circle and triangle definitions define the same values in the firstquadrant of the unit circle.

It is useful to understand both definitions of trig functions (circle and triangle) assometimes one is more helpful than the other for a particular task.

Sine and Cosine as Functions - 1

Sine and Cosine as Oscillating Functions

Despite the geometric source of the trigonometric functions, they are used morecommonly in biology and many other sciences as because their periodicity andoscillatory shapes. For many cyclic behaviours in nature, trigonometric func-tions are a natural first choice for modeling.


Question The graph of y = 10 + 4 cos(x) is shown in which of the followingdiagrams?

−6

−4

−2

2

4

6

8

10

12

14

2

4

6

8

10

12

14

A B

2

4

6

8

10

12

2

4

6

8

10

12

C D

Show the amplitude and the average on the correct graph.


Period and Phase

How can you find the period of the function cos(Ax)?


How can you reliably determine where the function cos(Ax + B) ‘starts’ onthe graph? (For a cosine graph, where the ‘start’ represents a maximum, thestarting time or x value is sometimes called the “phase” of the function.)


Consider the graph of the function y = 5 + 8 cos(π(x − 1)). What are thefollowing properties of the function:

• amplitude

• period

• average

• phase


Sketch the graph on the axes below. Include at least one full period of thefunction.

Non-Constant Amplitudes - 1

More complicated amplitudes

In the form y = A + B cos(Cx + D), the B factor sets the amplitude. In manyinteresting cases, however, that amplitude need not be constant.Sketch the graph of |y| = 5, and the graph of y = 5 cos(x) on the axes below.


Sketch the graph of |y| = x, and the graph of y = x cos(πx) on the axes below.Use only x ≥ 0


Use your intuition to sketch the graph of y = ex cos(πx) on the axes below.

Derivatives of Trigonometric Functions - 1

Derivatives of Trigonometric Functions

From Section 3.5Having covered the graphs and properties of trigonometric functions, we can nowreview the derivative formulae for those same functions.The derivation of the formulas for the derivatives of sin and cos are an interestingstudy in both limits and trigonometric identities. For those who are interested,many such derivations can be found on the web1. However, it is in some ways moreuseful to derive the formula in a graphical manner.

1For example, http://www.math.com/tables/derivatives/more/trig.htm#sin


Below is a graph of sin(x). Use the graph to sketch the graph of its derivative.

−3 π /2 −π −π/2 0 π/2 π 3 π/2

−1

1

−3 π /2 −π −π/2 0 π/2 π 3 π/2

−1

1


From this sketch, we have evidence (though not a proof) that

Theoremd

dxsinx =


Most students will also be familiar with the other derivative rules for trig functions:

d

dxcos(x) = − sin(x)

d

dxtan(x) = sec2(x)

d

dxsec(x) = sec(x) tan(x)

d

dxcsc(x) = − csc(x) cot(x)

d

dxcot(x) = − csc2(x)


Prove the secant derivative rule, using the definition sec(x) =1

cos(x)and the

other derivative rules.


Question: Find the derivative of 4 + 6 cos(πx2 + 1)

1. 4− 6 sin(πx2 + 1) · (2πx)

2. −6 cos(πx2 + 1) · (2πx)

3. −6 sin(πx2 + 1) · (2πx)

4. −6 sin(πx2 + 1) · (πx2 + 1)

5. 6 sin(2πx)

Inverse Trig Functions - 1

Inverse Trig Functions

From Section 1.5, 3.6In addition to the 6 trig functions just seen, there are 6 inverse functions as well,though the inverses of sine, cosine, and tangent are the most commonly used.Sketch the graph of sin(x) on the axes below

On the same axes, sketch the graph of arcsin(x), or sin−1 x, or the inverse ofsin(x).

Inverse Trig Functions - 2

What is the domain of arcsin(x)?

What is the range of arcsin(x)?

Derivative of arcsin - 1

Derivative of arcsin

Simplify sin(arcsinx)

Differentiate both sides of this equation, using the chain rule on the left. You

should end up with an equation involvingd

dxarcsinx.

Derivative of arcsin - 2

Solve ford

dxarcsinx, and simplify the resulting expression by means of the

formula

cos θ =√

1− sin2 θ,

which is valid if θ ∈ [−π2,π

2].

d

dxarcsinx =

Unit #4 : Interpreting Derivatives, Local Linearity, Newton’sMethod

Goals:

• Develop natural language interpretations of the derivative

• Create and use linearization/tangent line formulas

• Investigate Newton’s Method as a tool for solving non-linear equations that arenot solvable by hand

Textbook reading for Unit #4 : Study Sections 3.1–3.6 (from last week), 3.9,Appendix C

Quick Derivative Review - 1

Quick Derivative Review

Example: On the graph of y = 2 sin(5x) + 3, what is the slope at x = π/4?

Example: On the graph of x = 14e−t − 3t, at what rate is x changing att = 2?


Interpreting the Derivative

Example: Consider the statement “I am walking at 1.2 m/s.”How far will you travel in the next second?

How far will you travel in the two seconds?

How far will you travel in the next1

3of a second?

How far will you travel in the next 10 minutes?


Note that all the values computed above are estimates or predictions. Whichof the estimates you just calculated will be the most accurate?

What assumptions are you using to reach your answers?


Example: Let R = f (A) be the monthly revenue for a company, given ad-vertising spending of A per month. Both variables are measured in thousandsof dollars.

Interpret f ′(200) = 1.8 in words.


If A = 200 currently, and you increased advertising spending by 2 thousanddollars, what would you expect your revenue increase to be?

If A = 200 currently, and you increased advertising spending by 1 milliondollars, what would you expect your revenue increase to be?

If f ′(200) = 0.8, and you are currently spending 200 thousand on advertising,should you spend more or less next month?


Question: A chemical reaction consumes reactant at a rate given by f (c), wherec is the amount (mg) of catalyst present. f (c) is given in moles per second.The units of the derivative, f ′(c), are

(a) mg/s

(b) moles/s

(c) moles/(s mg)

(d) (mg moles)/s


Question: If f ′(10) = −0.2,

(a) Adding more catalyst to the 10 mg present will speed up the reaction.

(b) Adding more catalyst to the 10 mg present will slow down the reaction.

(c) Removing catalyst, from 10 mg present, will speed up the reaction.

(d) Removing catalyst, from 10 mg present, will slow down the reaction.

Local Linearity - 1

Local Linearity

In all these estimates we have been making, we have been relying on the locallinearity of a differentiable function.

If a function is differentiable at a point, then it behaves like a linear function for xsufficiently close to that point.

Another interpretation of differentiability is that if we “zoom in” sufficiently on apoint, the graph will eventually look like a straight line.

Local Linearity - 2

Consider the graph of y = sin(x) at different scales, around the point x = 0.4:

−8 −6 −4 −2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−1.5

−1

−0.5

0

0.5

1

1.5

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

Local Linearity - 3

And the more exotic y = sin(1/x) at different scales, around x = 0.1:

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−1.5

−1

−0.5

0

0.5

1

1.5

−0.05 0 0.05 0.1 0.15 0.2 0.25−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.09 0.095 0.1 0.105 0.11−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.099 0.0992 0.0994 0.0996 0.0998 0.1 0.1002 0.1004 0.1006 0.1008 0.101−0.64

−0.62

−0.6

−0.58

−0.56

−0.54

−0.52

−0.5

−0.48

−0.46

−0.44

Local Linearity - 4

Sketch a graph of a locally linear function f (x). Add on the tangent line, anduse the derivative to estimate ∆y for a given change in x.

Local Linearity - 5

Derivative as Approximation of Change

f ′(x) =dy

dx≈ ∆y

∆xso given a value of ∆x,

∆y ≈ f ′(x)∆x

assuming that ∆x is “sufficiently” small.The larger ∆x is, the worse the approximation will generally be.

Local Linearity - 6

Let’s return for a minute to an earlier example, and see how we can formalize ourprevious work.Example: Let R = f (A) be the monthly revenue for a company, given ad-vertising spending of A per month. Both variables are measured in thousandsof dollars.If A = 200 currently, and you increased advertising spending by 2 thousanddollars, what would you expect your revenue increase to be?

If A = 200 currently, and you increased advertising spending by 1 milliondollars, what would you expect your revenue increase to be?

The Tangent Line, or Local Linearization - 1

The Tangent Line, or Local Linearization

In the last few examples, we focused on the change in y (or f , or revenue, etc.),based on a set change in the input. Note that all these changes were relative to agiven starting value. (A = 200, c = 10, etc.)We can take the ideas one step further and create a linear function that approxi-mates our given (usually non-linear) function.


Example: Let us return to the advertising problem, where R = f (A) rep-resents the revenue of a company, given the amount A spent on advertising.Suppose f (200) = 1500, and f ′(200) = 1.8.

State the interpretation of both relationships in words.


Recall the point/slope form for a linear function:

y = m(x− a) + c

Sketch out the graph of this function, indicating the effect of the parametersm, a and c on the graph.


Use the point/slope formula, and the information about f (200) and f ′(200), tobuild a local linear approximation for the revenue function R for advertisingbudgets A around 200.


What revenue would we expect if we reduced advertising to 190 thousand dol-lars?


We can construct a linear approximation of a function, given a reference pointx = a, using

f (x) ≈ f ′(a)(x− a) + f (a)

This approximation is good assuming that the x values used are “sufficiently” closeto the reference point x = a.The larger (x− a) (or ∆x) is, the worse the approximation will generally be.


Show that f (x) ≈ f ′(a)(x−a)+f (a) is equivalent to our earlier approximation

f ′(a) ≈ ∆y

∆x

Some Linearization Examples - 1

Example: Build a local linear approximation formula for the populationof Canada, given it is currently 33 million, and the population is currentlyincreasing at a rate 300,000 people per year.

Use your approximation to estimate the Canadian population two years fromnow.


Question: Given that the Canadian population is growing exponentially (around1% per year), will your previous population estimate above an underestimate oran overestimate of the real population in that year?

(a) Overestimate

(b) Underestimate


Support your answer with a sketch of the population curve, and the linearapproximation.


We can also construct interesting geometric questions using tangent lines.

Example: Find the equations of all the lines through the origin that arealso tangent to the parabola (From H-H Section 3.1 #57.)

y = x2 − 2x + 4


Continued y = x2 − 2x + 4


Sketch the parabola and the lines you found.

Linearity to Help Solve Nonlinear Equations - 1

Solving Nonlinear Equations

Example: Solve the equation x2 + 3x− 4 = 0.


Example: Solve the equation 10e−x = 7


Example: Solve the equation 10e−x + x = 7


Perhaps surprisingly to some students, there are many relatively simple equationsthat cannot be solved by hand. We look now at a classical numerical methodthat lets us approximate the solution.Note: It is never better to use numerical methods instead of solving by hand, if aby-hand solution is available.

• Numerical solutions are always approximations, not exact.

• By-hand solutions can often be generalized, while numerical solutions have tobe re-calculated if anything changes.


Re-arrange the equation 10e−x + x = 7 so the RHS is zero.

Call the LHS f (x), and plot a few points of its graph, between x = 0 andx = 4.


Where could a solution to f (x) = 0 be, based on the points you plotted?

Bonus: what property of f (x) did you use to find the region of the solution?


Here are a few more points on the graph. f (x) = 10e−x + x− 7

0 1 2 3 4

−4

−3

−2

−1

1

2

3

4

For maybe not-so-obvious reasons, compute the derivative of f (x) at x = 0.5,a point which is close to a root/solution.


Use the derivative information to sketch the tangent line at x = 0.5 on thezoomed-in graph below.

0 0.1 0.2 0.3 0.4 0.5

−4

−3

−2

−1

1

2

3

4

Sketch the curve on the same graph (lightly, since we don’t know its exactshape).

Would the root of the tangent line be close to the root of the real (curved)function? Why?

Newton’s Method - 1

Newton’s Method

1. Convert an equation like g(x) = h(x) into a function on the left hand side:f (x) = g(x)− h(x) = 0

2. Select a starting value of x, x0, near a root of f (x).

3. Use the formula xn+1 = xn−f (xn)

f ′(xn)to find the root of the tangent line at xn.

4. Repeat Step 3 until the xn+1 estimate is sufficiently close to a root.


Rationale for Step 3 of Newton’s Method: For an arbitrary function,f (x), and a point x = xn, find where a tangent line to f (x) at xn would reachy = 0.


Apply Step 3 of Newton’s method twice to improve our estimate of the solutionto our original equation 10e−x + x− 7 = 0.


Evaluate the quality of the x estimate you found.


Sketch the values we computed on the axes below.


It can be shown that, under certain common conditions, and a “sufficiently close”initial estimate of the root, Newton’s method will converge very quickly towards anearby root. It will always give just an estimate, though, not an exact answer; asa result, you always have to trade off the amount of work you are willing to do formore steps/increased accuracy.

Newton’s Method Example - 1

Example: Try to find a solution to sin(x) =x

3by hand.

Example: Sketch both functions to identify roughly what x values might besolutions.


Use three iterations of Newton’s method to find an approximate non-zero so-

lution to sin(x) =x

3.


Confirm your approximate solution by subbing it in to the equation sin(x) =x

3,

and checking that the LHS and RHS are (very close to) equal.

Unit #5 : Implicit Differentiation, Related Rates

Goals:

• Introduce implicit differentiation.

• Study problems involving related rates.

Textbook reading for Unit #5 : Study Sections 3.7, 4.6

Tangent Lines to Relations - Implicit Differentiation - 1

Implicit Differentiation

If we define a graph by the relationship y = f (x), then we have a formula fortangent lines.

Question: What function gives the tangent line to y = e−x at x = 0?

(a) y = −x + 1

(b) y = −(x− 1) + 1

(c) y = −x + e

(d) y = x + 1

(e) y = −(x− 1)− 1


Question: Can we define a function in the form y = f (x) that describes thepoints on the circle below?

(a) Yes.

(b) No.

(c) Maybe.

−4 −3 −2 −1 0 1 2 3 4

−4

−3

−2

−1

1

2

3

4


−4 −3 −2 −1 0 1 2 3 4

−4

−3

−2

−1

1

2

3

4

Write out a formula for the points in the circle shown.


Question: Can we, in principle, define a tangent line to the circle at the point(2,√

12)?

(a) Yes.

(b) No.

(c) Maybe.

−4 −3 −2 −1 0 1 2 3 4

−4

−3

−2

−1

1

2

3

4


To find the slope at this point, we introduce a technique called implicit differen-tiation, because we will differentiate a relationship that defines a graph implicitly,rather than differentiating a function.Find the slopes of tangent lines for points on the circle x2 + y2 = 16.


Find the slope at the point (2,√

12), and sketch it on the graph.

−4 −3 −2 −1 0 1 2 3 4

−4

−3

−2

−1

1

2

3

4


What is different about the derivative found by implicit differentiation, com-pared to the derivative of a function?

Implicit Differentiation Examples - 1

Example: Sketch the curve defined by the relationship x = y2.

Find the equation of the tangent line to this graph at the point (4, 2).


(Continued)


Note that implicit derivatives of even complicated relationships are straightforwardto compute.

Example: Use implicit differentiation to calculatedy

dxwhen

x3 + 2x2y + sin(xy) = 1.


x3 + 2x2y + sin(xy) = 1

Is the point (1, 0) on this curve?


x3 + 2x2y + sin(xy) = 1

Sketch the curve locally around the point (1, 0).


x3 + 2x2y + sin(xy) = 1

Estimate the y location of the nearby point on the graph at x = 0.95.


Here is the graph of the relation, shown at two different zoom levels.

−6 −4 −2 0 2 4 6 8

−3

−2

−1

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sketch the tangent line found in the previous question.

Related Rates - Car Example - 1

Related Rates

To prepare for this topic, you should read Section 4.6 in the textbook.

We can use the Chain Rule and Implicit Differentiation to solve problems involv-ing related rates. As the name suggests, we use the rate of change (i.e., thederivative) of one function to calculate the rate of change (derivative) of a secondfunction.


Example: A car starts driving north at a point 150 meters east of anobserver at point A. The car is traveling at a constant speed of 20 meters persecond. How fast is the distance between the observer and car changing after10 seconds?

A 150 m

Car

Start

20 m/s


(Continued)

A 150 m

Car

Start

20 m/s

Related Rates - Water Tank - 1

By using implicit differentiation, we can solve these problems even if we do not havean explicit formula for the function in terms of the independent variable (usuallytime).Example: A conical water tank with a top radius of 2 meters and height4 meters is leaking water at 0.5 liters per second. How fast is the height ofthe water in the tank changing when the remaining water in the tank is at aheight of 2 meters?


(Continued)


Question: When the water height is above 2 m, is the water height changingmore quickly or more slowly than the value we just found?

(a) more quickly

(b) more slowly

Related Rates - Ladder - 1

General method for Related Rates problems

• Draw a picture (if possible) of the situation, and label all relevant variables.

• Identify which of the variables and their rate of change are known.

• Identify which rate you are trying to determine.

•Write an equation involving the changing variables, including the function whoserate you are trying to find.

• Apply implicit differentiation to the equation, before substituting any knownvariables.

• Substitute known variables and rates.

• Solve for desired rate.


No calculus course is complete without a related rates ladder problem.Example: A 5 m ladder is propped against the wall. You climb to top, butthen it starts to slip; the tip of the ladder is moving downwards at 1 m/s. Findthe rate at which the bottom end of the ladder is sliding along the ground,

• when the bottom of the ladder is 3 m away from the wall, and

• when the top of the ladder hits the ground.


(Continued)

Related Rates - Ideal Gas Law - 1

Example: The ideal gas law states that

PV = nRT

0.1 moles of gas are held in a piston, and the temperature is held constantat 273o K. The piston moves to decrease the volume from 3 l to 1 l over 60seconds. What is the pressure half-way through this process, and at what rateis the pressure changing at this time?

Related Rates - Ideal Gas Law - 2

(Continued)

Related Rates - Radar Tracking - 1

Example: A radar tracking site is following a plane flying in a straight line.At its closest approach, the plane will be 150 km away from the radar site. Ifthe plane is traveling at 600 km/h, how quickly is the radar rotating to trackthe plane when the plane is closest?


Does the angular rotation speed up or slow down as the plane moves awayfrom the radar station?


Sketch a graph of the angular rotation rate against the rotation angle.

Unit #6 : Families of Functions, Taylor Polynomials, l’Hopital’sRule

Goals:

• To use first and second derivative information to describe functions.

• To be able to find general properties of families of functions.

• To extend our tangent line formula to higher-degree polynomial approximations(Taylor polynomials)

• To explore more advanced ways to evaluate limits.

Textbook reading for Unit #6 : Study sections 4.1, 4.3, 10.1, 4.7

Interpreting First and Second Derivatives - 1

Interpreting First and Second Derivatives

Reading: Section 4.1The information about the graph of a function f provided by the sign of f ′(x) andf ′′(x) on an interval (a, b) is expressed in the following table. (a and b are assumedto be finite.)


f ′(x) > 0 on (a, b) f increasing on [a, b]

f ′(x) < 0 on (a, b) f decreasing on [a, b]

f ′′(x) > 0 on (a, b) f concave up on [a, b]

f ′′(x) < 0 on (a, b) f concave down on [a, b]

[H-H, p. 176]


All the indicators above deal with non-zero values of f ′(x) and f ′′(x). Whatis distinctive about the zero values of these derivatives?


The intervals of increasing and decreasing described in the table earlier can lead toa surprising or counter-intuitive technical point.Example: For the function f (x) = x2, find the derivative.

Find the intervals f ′(x) > 0 and f ′(x) < 0.


Based on this analysis, describe the intervals on which f (x) is increasing, andintervals where it is decreasing.

Critical Points - 1

Critical Points

If f (x) is defined on the interval (a, b), then we call a point c in the interval acritical point if:

• f ′(c) = 0, or

• f ′(c) does not exist.

We will also refer to the point (c, f (c)) on the graph of f (x) as a critical point.We call the function value f (c) at a critical point c a critical value.

Critical Points - 2

Technical Notes:

1. By this definition, f (c) must be defined for c to be a critical point.

Sketch f (x) = 1/x, and decide whether x = 0 is a critical point.

Sketch g(x) = |x|, and decide whether x = 0 is a critical point.

Critical Points - 3

2. By the definition, if a function is defined on a closed interval, the endpoints ofinterval cannot be critical points.

Sketch the graph of f (x) =√x and decide whether x = 0 is a critical

point.

Critical Points - 4

Sketch the graph of g(x) = 3√x and decide whether x = 0 is a critical

point.

Critical Points - 5

Example: Identify all the critical points on the graph below, and character-ize any other interesting points by continuity, limits, or other properties.

First and Second Derivative Sign Chart Example - 1

Example: Consider the function

f (x) =x

x2 + 1

Construct a sign chart for both f ′ and f ′′, and use this information to sketchf (x).

First and Second Derivative Sign Chart Example - 2 Families of Functions - Example 1 - 1

Families of Functions

Reading: Section 4.3The approach taken in the previous example can be generalized to allow us tosketch families of functions, rather than a single function at a time. (A family offunctions is a set of functions that share a common mathematical form, but differin the particular value they might have for one or more parameters.)

Families of Functions - Example 1 - 2

Example: Consider the family of functions

f (x) =ax

x2 + b

Let b = 1, then sketch several members of the family with different positivevalues of a.


f (x) =ax

x2 + bSuppose a = 1 now. Create a sign chart for f ′(x), given that b can change.

Find the (x, y) coordinates of the critical points of f (x), in terms of b.


Sketch several members of the family using different positive values of b.


f (x) =ax

x2 + b

Would the family change substantially if a or b could be negative? If so, whatwould the change look like?


f (x) =ax

x2 + bFind the member of this family which has its maximum at (10, 1).


Example: Show that, for positive constants a and b, g(x) = a(1 − e−bx) isboth increasing and concave down for all x.


g(x) = a(1− e−bx)

Evaluate g(0) and the limit limx→∞

g(x).


Sketch what members of the family g(x) = a(1 − e−bx) might look like forx ≥ 0.

Taylor Polynomial Intro - 1

Taylor Polynomials

Reading: Section 10.1A more technical, but incredibly powerful, application of derivative information isthe construction of polynomial approximations to more complicated functions.Previously we found a formula for linear approximations to functions f (x) arounda point x = a:

This linear approximation, or tangent line formula, can also be called the Taylorpolynomial of degree 1 approximating f (x) near x = a.


Sketch the graph of cos(x) around x = 0, and add its tangent line based atx = 0.

The linearization or tangent line is clearly a very limited approximation tothis function. What might be a slightly more complex form of function thatwould work better in this case?


Taylor Polynomial of Degree 2

f (x) ≈ f (a) + f ′(a)(x− a) +f ′′(a)

2(x− a)2

is a quadratic approximation to f (x) near x = a.

For values of x close to a do you think this quadratic approximation will be abetter or worse approximation than the tangent line? Why?

Taylor Polynomials - Examples - 1

Example: Find the quadratic Taylor approximation to f (x) = cos(x) nearx = 0.


Sketch the graph of cos(x) around x = 0, and add both its 1st and 2nd degreeTaylor polynomial approximations for x near 0.


There is a very good reason for the particular form of the Taylor polynomial.What mathematical features will f (x) and its 2nd degree Taylor approximationshare at x = a?


If we wanted a still-better approximation for f (x) near a specific point x = a,how could we generalize our earlier 1st and 2nd degree Taylor polynomials?

Taylor Polynomials - Inverting the Process - 1

Example: You are told that f (x) ≈ 7− (x−5)+(x−5)3 for x near 5. Whatcan you say about the value and derivatives of f (x) at x = 5?

Applications of Taylor Polynomials - 1

Applications of Taylor Polynomials

Reading: Section 10.1It is not immediately obvious to most students why we would ever want to replacea perfectly good function like y = ex with its approximation y ≈ 1 + x. However,it can be argued that these Taylor approximations (and related ones like Fourierseries) are comparable in importance to the fundamental calculus ideas of thederivative and integral.Let us see how Taylor polynomials can help us answer previously unanswerablequestions.


Example: Evaluate the limit limx→0

e−2x − 1

x.



sin(5x)

x.


Have any students seen an alternative method for evaluating limits like this?If so, what was it called?

L’Hopital’s Rule - 1

L’Hopital’s Rule

Use Taylor polynomials to show that if the ratio limx→a

f (x)

g(x)gives the form

0

0,

then

limx→a

f (x)

g(x)= lim

x→a

f ′(x)

g′(x)


L’Hopital’s rule can be applied in slightly more general circumstances as well.

L’Hopital’s Rule

When a limit limx→a

or limx→±∞

off (x)

g(x)yields an indeterminate ratio form of

0

0or±∞±∞, then

limx→a

or limx→±∞

f (x)

g(x)= lim

x→aor lim

x→±∞f ′(x)

g′(x)



1− e−2x

x.


Example: Evaluate the limit limx→∞

1− e−2x

x.



1− cos(4x)

x2.


Example: Evaluate the limit limx→0+

x ln(x).

Unit #7 : Optimization, Optimal Marginal Rates

Goals:

• Formalize the first derivative test and the second derivative test for identifyinglocal maxima and minima.

• Distinguish global vs. local extrema.

• Practice optimization word problems.

• Study generalized optimization problems related to marginal rates.

Textbook reading for Unit #7 : Study Sections 4.1, 4.2, 4.4

Classifying Critical Points - Review - 1

Identifying Types of Critical Points

Reading: Section 4.1Last week we found critical points by looking for points on the graph where

• f ′(x) = 0, or

• f ′(x) was undefined (but f (x) was defined).

We will now formalize two ways to determine if a critical point is a local min, max,or neither. This avoids the need for a sketch of the graph.


First Derivative TestOne way to decide whether at a critical point there is a local maximum or minimumis to examine the sign of the derivative on opposite sides of the critical point. Thismethod is called the first derivative test. Complete this table:

f ′ sign left of c f ′ sign right of c Sketch

local minimum at c

local maximum at c

neither local max nor min


Example: Find the critical points of the function f (x) = 2x3−9x2+12x+3.Use the first derivative test to show whether each critical point is a localmaximum or a local minimum.


Using your answer to the preceding question, determine the number of realsolutions of the equation 2x3 − 9x2 + 12x + 3 = 0.


Second Derivative TestYou may also use the Second Derivative Test to determine if a critical point is alocal minimum or maximum.

• The first derivative test uses the first derivative around the critical point.

• The second derivative test uses the second derivative at the critical point.

• If f ′(c) = 0 and f ′′(c) > 0 then f has a local minimum at c.

• If f ′(c) = 0 and f ′′(c) < 0 then f has a local maximum at c.

• If f ′(c) = 0 and f ′′(c) = 0 then the test is inconclusive.


Example: A function f has derivative f ′(x) = cos(x2)+2x−1. Does it havea local maximum, a local minimum, or neither at its critical point x = 0?

Global vs. Local Optimization - 1

Global vs. Local Optimization

Reading: Section 4.2The first and second derivative tests only give us local information in most cases.However, if there are multiple local maxima or minima, we usually want the globalmax or min. The ease of determining when we have found the global max or minof a function depends strongly on the properties of the question.

Local vs Global Extrema

A local max occurs at x = c when f (c) > f (x) for x values near c.A global max occurs at x = c if f (c) ≥ f (x) for all values of x in the domain.It is possible to have several global maxima if the function reaches its peak valueat more than one point.Corresponding definitions apply for local and global minima.


Example: Give an example of a simple function with multiple global max-ima.

Example: Give an example of a simple function with a single global maxi-mum, but no global minimum.

Example: Give an example of a simple function with neither a global max-imum nor a global minimum.


Example: Earlier we worked with the function f (x) = 2x3 − 9x2 + 12x + 3.If we limit the function to the interval x ∈ [0, 2.5], what are the global maxand global minimum values on that interval?

Global Extrema on Closed and Open Intervals - 1

Global Extrema on Closed Intervals

A continuous function on a closed interval will always have a global max and aglobal min value. These values will occur at either

• a critical point or

• an end point of the interval.

To find which value is the global extrema, you can compute the original function’svalues at all the critical points and end points, and select the point with thehighest/lowest value of the function.


Global Extrema on Open Intervals

A function defined on an open interval may or may not have global maxima orminima.

If you are trying to demonstrate that a point is a global max or min, and you areworking with an open interval, including the possible interval (−∞,∞), provingthat a particular point is a global max or min requires a careful argument. Arecommendation is to look at either:

• values of f when x approaches the endpoints of the interval, or ±∞, as appro-priate; or

• if there is only one critical point, look at the sign of f ′ on either side of thecritical point.

With that information, you can often construct an argument about a particularpoint being a global max or min.


Example: Determine whether the function f (x) = 2x3 − 9x2 + 12x + 3 hasa global max and/or min.


Example: Determine whether the function f (x) = (x − 2)4 has a globalmax/and or min.

Optimization - Fencing Example - 1

Optimization

Reading: Section 4.4

An optimization problem is one in which we have to find the maximum or minimumvalue of some quantity. In principle, we already know how to find the maximumand minimum values of a function if we are given a formula for the function andthe interval on which the maximum or minimum is sought. Usually the hard partin an optimization problem is interpreting the word problem in order to find theformula of the function to be optimized.


Example: A farmer wants to build a rectangular enclosure to containlivestock. The farmer has 120 meters of wire fencing with which to build afence, and one side of the enclosure will be part of the side an already existingbuilding (so there is no need to put up fence on that side). What should thedimensions of each side be to maximize the area of the enclosure?

What is the quantity to be maximized in this example?


What are the variables in this question, and how are they related? You maywant to draw a picture.

Express the quantity to be optimized in terms of the variables. Try to eliminateall but one of the variables.

What is the domain on which the one remaining variable makes sense?


Use the techniques learned earlier in the course to maximized the function onthis domain. Give reasons explaining why the answer you found is the globalmaximum.


(continued)

Optimization - Storage Example - 1

Example: (Storage Container)A rectangular storage container with an open top is to have a volume of 10m3. The length of its base is to be twice its width. Material for the base costs$10.00 per m2, and material for the sides costs $6.00 per m2. Determine the

cost of the material for the cheapest such container.

Optimization - Storage Example - 2 Optimization - Storage Example - 3

Optimization - Fisher Example - 1

Example (Taken from 2004 Dec Exam)A fisher is in a boat at point A, which is 2 km from the nearest point on theshoreline. He is to go to a lighthouse at point B, which is 3 km down the coast (seefigure below).

A

B

Landing Point

2 km

3 km


If the fisher can row at 4 km per hour, and walk at 5 km per hour, find anexpression for T (x), the travel time if the fisher lands the boat x km down theshore from the nearest approach.

A

B

Landing Point

2 km

3 km


If the fisher can row at 4 km per hour, and walk at 5 km per hour, how farfrom the point B should he land the boat to minimize the time it takes to getto the lighthouse? Make sure to indicate how you know your answer is theglobal minimum.

A

B

Landing Point

2 km

3 km


A

B

Landing Point

2 km

3 km

Optimization - Fisher General Case - 1

Often, the numerical values in an optimization problem are somewhat arbitrary, orestimated using best guesses. It is often more important to discover the responsein the solution to a range of possible problem values. In that vein, we now supposethe fisher has a motor that will drive his boat at a speed of v km per hour.If the fisher’s walking speed is still 5 km per hour, for what values of v willit be fastest to simply drive the boat directly to the lighthouse (i.e. do nowalking)?

A

B

Landing Point

2 km

3 km

Optimization - Fisher General Case - 2

A

B

Landing Point

2 km

3 km

Optimization Without Formulas - 1

Optimization Without Formulas

In the preceding examples, we used derivative calculations on explicit formulas tofind critical points. Unfortunately, such formulas aren’t always available. Some-times, the function we are given isn’t the one we need to optimize directly. Thefollowing example showcases an important class of problems from both biology andeconomics.


Example: A chipmunk stocks its nest with acorns for the winter. Thechipmunk wants to be as efficient as possible while foraging. There is a rela-tionship between the number of acorns in the chipmunk’s mouth and the timeit takes to pack another into its cheeks. Clearly, when the cheeks are alreadyvery full, it becomes harder to add another acorn without dropping one alreadycollected. So, we can imagine that the graph of “load size” versus time lookslike:

τ

0


This graph shows F (t), the amount of food collected in collection time t. Alsoshown is the travel time, τ , before the chipmunk reaches the collection site.(Total trip time is t + τ .)


How would we compute the average collection rate based on collecting nutsfor time t? (call this average rate R(t)).


An important consequence of this formula is that the value of the functionR(t) can be seen as a slope in the F (t) graph. Sketch it on the graph.

τ

0


While there is some merit in studying this example using particular numbers, animportant insight can be arrived at by studying the problem using an arbitraryfunction F (t).Without knowing the feeding function, F (t), find the derivative of R(t), andfind its maximum value.


Question: Based on those calculations, which of the following points maximizesthe value of R(t)?

τ

0

+

+

+

++

A

B

C

DE


This result, that the optimal time to stop harvesting occurs when the

current instantaneous rate of harvest equals thecurrent average rate of harvest,

is known as theMarginal Value Theorem.

It applies just as well to mining, oil wells, and crop harvesting as it does to biology.

Unit #8 : The Integral

Goals:

• Determine how to calculate the area described by a function.

• Define the definite integral.

• Explore the relationship between the definite integral and area.

• Explore ways to estimate the definite integral.

Textbook reading for Unit #8 : Study Sections 5.1, 5.2, 7.5, and (for enrich-ment) 7.6.

Distance and Velocity - 1

Distance and Velocity

Recall that if we measure distance x as a function of time t, the velocity is deter-mined by differentiating x(t), i.e. finding the slope of the graph.Alternatively, suppose we begin with a graph of the velocity with respect to time.How can we determine what distance will be traveled? Does it “appear” in thegraph somehow?Let’s begin with the simple case of constant velocity...

distance = velocity× time

For a constant velocity, the distance traveled in a certain length of time was simply

Distance and Velocity - 2

the area of the rectangle underneath the velocity vs. time graph.

What if the velocity is changing?

We can’t determine the exact distance traveled, but maybe we can estimate it.Let’s assume that the velocity is not changing too quickly, so over a short amountof time it’s roughly constant. We know how to find the distance traveled in thatshort time...

Making many of these approximations, we could come up with a rough estimateof the total distance. How does this estimate relate to the graph?

Calculating Areas - 1

Calculating Areas

It appears that the distance traveled is the area under the graph of velocity, evenwhen the velocity is changing. We’ll see exactly why this is true very soon.If we are simply interested in the area under a graph, without any physical inter-pretation, we can already do so if the graph creates a shape that we recognize.Example: Calculate the area between the x-axis and the graph of y = x + 1from x = −1 and x = 1.

Calculating Areas - 2

Example: Calculate the area between the x-axis and the graph of y =√1− x2 from x = −1 to x = 1.

−1 1

−1

1

What shapes do you know, right now, for which you can calculate the exactarea?

Estimating Areas - 1

Estimating Areas

Unfortunately, many or most arbitrary areas are essentially impossible to find thearea of when the shape isn’t a simple composition of triangles, rectangles, or circles.In these cases, we must use less direct methods. We start by making an estimateof the area under the graph using shapes whose area is easier to calculate.

Suppose we are trying to find the area underneath the graph of the function f (x)given below between x = 1 and x = 4. Call this area A.

1 2 3 4 5

−1

1

2

3

4

5


We can make a rough estimation of the area by drawing a rectangle that completelycontains the area, or a rectangle that is completely contained by the area.Calculate this overestimate and underestimate for the area A.

1 2 3 4 5

−1

1

2

3

4

5


The next step is to use smaller rectangles to improve our estimate the area. Wecan divide the interval from x = 1 to x = 4 into 3 intervals of width 1, and usedifferent size rectangles on each interval.

Estimate the area A by using 3 rectangles of width 1. Use the function valueat the left edge of the interval as the height of each rectangle.

1 2 3 4 5

−1

1

2

3

4

5


We can repeat this process for any number of rectangles, and we expect that ourestimation of the area will get better the more rectangles we use. The method weused above, choosing for the height of the rectangles the function at the left edge,is called the left hand sum, and is denoted LEFT(n) if we use n rectangles.Suppose we are trying to estimate the area under the function f (x) from x = a tox = b via the left hand sum with n rectangles. Then the width of each rectangle

will be ∆x =b− a

n.


If we label the endpoints of the intervals to be a = x0 < x1 < · · · < xn−1 < xn = b,then the formula for the left hand sum will be

LEFT (n) =f (x0)∆x + f (x1)∆x + . . . + f (xn−1)∆x =

n∑

i=1

f (xi−1)∆x.


We have a similar definition for the right hand sum, or RIGHT(n), calculatedby taking the height of each rectangle to be the height of the function at the righthand endpoint of the interval.

RIGHT (n) =f (x1)∆x + f (x2)∆x + . . . + f (xn)∆x =

n∑

i=1

f (xi)∆x.


Calculate LEFT(6) and RIGHT(6) for the function shown, between x = 1and x = 4. You will need to estimate some rectangle heights from the graph.

1 2 3 4 5

−1

1

2

3

4

5

1 2 3 4 5

−1

1

2

3

4

5


In general, when will LEFT(n) be greater than RIGHT(n)?

When will LEFT(n) be an overestimate for the area?

When will LEFT(n) be an underestimate?

Riemann Sums - 1

Riemann Sums

Area estimations like LEFT (n) and RIGHT (n) are often called Riemannsums, after the mathematician Bernahrd Riemann (1826-1866) who formalizedmany of the techniques of calculus. The general form for a Riemann Sum is

f (x∗1)∆x + f (x∗2)∆x + . . . + f (x∗n)∆x =

n∑

i=1

f (x∗i )∆x

where each x∗i is some point in the interval [xi−1, xi]. For LEFT (n), we choosethe left hand endpoint of the interval, so x∗i = xi−1; for RIGHT (n), we choosethe right hand endpoint, so x∗i = xi.

Riemann Sums - 2

The common property of all these approximations is that they involve

• a sum of rectangular areas, with

• widths (∆x), and

• heights (f (x∗i ))

There are other Riemann Sums that give slightly better estimates of the areaunderneath a graph, but the often require extra computation. We will examinesome of these other calculations a little later.

The Definite Integral - 1

The Definite Integral

We observed that as we increase the number of rectangles used to approximatethe area under a curve, our estimate of the area under the graph becomes moreaccurate. This implies that we want to use a limit if we want to calculate theexact area.

The area underneath the graph of f (x) between x = a and x = b is equal to

limn→∞

LEFT (n) = limn→∞

n∑

i=1

f (xi−1)∆x, where ∆x =b− a

n.

This limit is called the definite integral of f (x) from a to b, and is equal to thearea under curve whenever f (x) is a non-negative continuous function. The definiteintegral is written with some special notation.


Notation for the Definite IntegralThe definite integral of f (x) between x = a and x = b is denoted by the symbol

∫ b

a

f (x) dx

We call a and b the limits of integration and f (x) the integrand. The dxdenotes which variable we are using; this will become important for using sometechniques for calculating definite integrals. Note that this notation shares thesame common structure with Riemann sums:

• a sum (

∫sign)

• widths (dx), and

• heights (f (x))


Example: Write the definite integral representing the area underneath thegraph of f (x) = x + cosx between x = 2 and x = 4.


We might be concerned that we defined the area and the definite integral usingthe left hand sum. Would we get the same answer for the definite integral if weused the right hand sum, or any other Riemann sum? In fact, the limit using anyRiemann sum would give us the same answer.

Let us look at the left and right hand sums for the function 2x on the interval fromx = 1 to x = 3.

Calculate LEFT (2)−RIGHT (2)

Calculate LEFT (4)−RIGHT (4)


Calculate LEFT (n)−RIGHT (n).


What will the limit of this difference be as n → ∞? What does this tell usabout what would happen if we defined the definite integral in terms of theright hand sum?

Estimating Area Between Curves - 1

More on the Definite Integral and Area

We can use the definite integral to calculate other areas, as well. Suppose we wantto find the area between the curves y = x2 and the line y = 2 − x. It is easy tosee that the two intersect as shown in the following graph.

−3 −2 −1 0 1 2

−1

1

2

3

4

5

6

We can again calculate this area by estimating via rectangles and the taking thelimit to get the definite integral.


−3 −2 −1 0 1 2

−1

1

2

3

4

5

6

If we estimate this area using the left hand sum, what will be the height of therectangle on the interval [xi−1, xi]?


Write the formula for LEFT (n), using this height as the function value. Whatwill ∆x be?

Write the definite integral representing the area of this region.

Negative Integral Values - 1

Negative Integral Values

So far we have only dealt with positive functions. Will the definite integral still beequal to the area underneath the graph if f (x) is always negative? What happensif f (x) crosses the x-axis several times?

Example: Suppose that f (t) has the graph shown below, and that A, B, C,D, and E are the areas of the regions shown.

If we were to partition [a, b] into small subintervals and construct a correspondingRiemann sum, then the first few terms in the Riemann sum would correspond tothe region with area A, the next few to B, etc.


Which of these sets of terms have positive values?

Which of these sets have negative values?


Express the integral

∫ b

a

f (t)dt in terms of the (positive) areas A, B, C, D,

and E.

If f (t) represents a velocity, what do the “negative areas” represent?

Estimating Integrals - Midpoint Rule - 1

Better Approximations to Definite Integrals

To prepare for this section, you should read Section 7.5 in the textbook.

•We saw how to approximate definite integrals using left and right hand Riemannsums. Unfortunately, these estimates are very crude and inefficient

• Even when we have sophisticated techniques for evaluating integrals, thesemethods will not apply to all functions, for example:

·∫

e−x2dx - used in probability

·∫

sin(x)

xdx - used in optics


• To evaluate definite integrals of such functions, we could use left or right handRiemann sums. However, it would be preferable to develop more accurateestimates.

• But more accurate estimates can always be made by using more rectangles

• More precisely, we want to develop more efficient estimates: estimates that aremore accurate for similar amounts of work.

[In Section 7.6, there is a discussion on the relative accuracy of the methods whichfollow.]


Midpoint Rule

The accuracy of a Riemann sum calculation will usually improve if we choose themidpoint of each subdivision rather than the right or left endpoints.Sketch 3 copies of a diagram for a short interval of an arbitrary functionf (x).

Compare the accuracy of the left-hand, right-hand and midpoint rules forestimating the area on the interval.


For what kinds of functions f will the midpoint rule always give a value thatis exactly equal to the integral?


Try to sketch a function, and a subdivision of an interval, for which the mid-point rule gives a less accurate answer than either the right-hand sum or theleft-hand sum.

Would the midpoint still be less accurate if you doubled the number of intervalsin your example?

Estimating Integrals - Trapezoidal Rule - 1

The Trapezoidal Rule

The Midpoint Rule is only one possible variation on Riemann sums. Anotherapproach is to use a shape other than a rectangle to estimate the area on aninterval. For the appropriately named Trapezoidal Rule, we use a trapezoid oneach interval.Sketch one interval of a function, and a trapezoidal approximation to the areaunder the graph.

Write a formula for the area of the single trapezoid.


Write a formula for the full Trapezoid Rule (written TRAP (n)), estimatingthe entire area under a graph with multiple intervals.


Express the trapezoidal rule in terms of the left and right hand Riemann sums(LEFT (n) and RIGHT (n)).


‘The trapezoidal rule is especially accurate for functions f for which |f ′′(x)| issmall for all x.’ Explain from an intuitive point of view why you would expectthis statement to be correct.

Estimating Integrals - Trapezoidal Rule Examples - 1

Use the trapezoidal rule to estimate

∫ 10

0

f (x) dx, if we have measured the

values in the following table for f (x).

x 0 2 4 6 8 10f(x) 1 3 4 5 4 2

Estimating Integrals - Trapezoidal Rule Examples - 2

Sketch the graph of a function, f (x), on the interval 0 ≤ x ≤ 1 for which

1. f (x) is decreasing on [0, 1] andfor any number of intervals we choose,

2. MID(n) >

∫ 1

0

f (x) dx

3. TRAP (n) <

∫ 1

0

f (x) dx

Unit #9 : Definite Integral Properties, Fundamental Theorem ofCalculus

Goals:

• Identify properties of definite integrals

• Define odd and even functions, and relationship to integral values

• Introduce the Fundamental Theorem of Calculus

• Compute simple anti-derivatives and definite integrals

Reading: Textbook reading for Unit #9 : Study Sections 5.4, 5.3, 6.2

Definite Integrals in Modeling - 1

Definite Integrals in Modeling

One of the primary applications of integration is to use a known rate of change,and compute the net change over some time interval.Example: Suppose water is flowing into/out of a tank at a rate given byr(t) = 200 − 10t L/min, where positive values indicate the flow is into thetank.

Write an integral that expresses the change in the volume of water in thetank during the first 30 minutes of filling.


Estimate the integral using a left-hand rule with three intervals.

Does this information tell you the actual volume in the tank after 30 minutes?Why or why not?


Question: If h(t) represents the height of a child (in cm) at time t (in years),and the child is 120 cm tall at age 10, how would you represent the amount thechild grew between t = 10 and t = 18 years?

A.

∫ 18

10

h(t) dt

B.

∫ 18

10

h(t) dt + 120

C.

∫ 18

10

h′(t) dt

D.

∫ 18

10

h′(t) dt + 120

Limit Properties for Integrals - 1

Properties of Definite Integrals

Example: Sketch the area implicit in the integral

∫ π/3

−π/3cos(x) dx

If you were told that

∫ π/3

0

cos(x) dx =

√3

2, find the size of the area you

sketched.


This example highlights an important and intuitive general property of definiteintegrals.

Additive Interval Property of Definite Integrals

∫ b

a

f (x) dx =

∫ c

a

f (x) dx +

∫ b

c

f (x) dx

Explain this general property in words and with a diagram.


A more rarely helpful, but equally true, corollary of this property is a secondproperty:

Reversed Interval Property of Definite Integrals

∫ b

a

f (x) dx = −∫ a

b

f (x) dx

Use the integral

∫ π/3

0

cos(x) dx+

∫ 0

π/3

cos(x)dx, and the earlier interval prop-

erty, to illustrate the reversed interval property.


Give a rationale related to Riemann sums for the Reversed Interval property.

Integrals of Even and Odd Functions - 1

Even and Odd Functions

These properties can be helpful especially when dealing with even and odd func-tions.Define an even function. Give some examples and sketch them.


Define an odd function. Give some examples and sketch them.


Integral Properties of Even and Odd FunctionsFind a property of odd functions when you integrate on both sides of x = 0.


Find a property of even functions when you integrate on both sides of x = 0.

Scaling, Adding Definite Integrals - 1

Linearity of Definite Integrals

Example: If

∫ b

a

f (x) dx = 10, then what is the value of

∫ b

a

5f (x) dx?

Sketch an area rationale for this relation.


Example: If

∫ b

a

f (x) dx = 2, and

∫ b

a

g(x) dx = 4 then what is the value of∫ b

a

f (x) + g(x) dx? Again, sketch an area rationale for this relation.


Linearity of Definite Integrals

∫ b

a

kf (x) dx = k

∫ b

a

f (x) dx

∫ b

a

f (x)± g(x) dx =

∫ b

a

f (x) dx±∫ b

a

g(x) dx

Bounds on Integrals - 1

Simple Bounds on Definite IntegralsExample: Sketch a graph of f (x) = 5 sin(2πx), then use it to make an areaargument proving the statement that

0 ≤∫ 1

2

0

5 sin (2πx) dx ≤ 5

2


Simple Maximum and Minimum Values for Definite IntegralsIf a function f (x) is continuous and bounded between y = m and y = M on theinterval [a, b], i.e. m ≤ f (x) ≤M on the interval, then

m(b− a) ≤∫ b

a

f (x) dx ≤M(b− a)

Note that the maximum and minimum values we get with the method above arequite crude. Sometimes you will be asked for much more precise values which canoften be just as easy to find.

Example: Use the graph to find the exact value of

∫ 1

0

5 sin(2πx) dx i.e. not

just a range, but the single correct area value.


Relative Sizes of Definite IntegralsExample: Two cars start at the same time from the same starting point.For the first second,

• the first car moves at velocity v1 = t, and

• the second car moves at velocity v2 = t2.

Sketch both velocities over the relevant interval.


Which car travels further in the first second? Relate this to a definite integral.

Comparison of Definite IntegralsIf f (x) ≤ g(x) on an interval [a, b], then

∫ b

a

f (x) dx ≤∫ b

a

g(x) dx

The Fundamental Theorem of Calculus - Theory - 1

The Fundamental Theorem of Calculus

Reading: Section 5.3 and 6.2We have now drawn a firm relationship between area calculations (and physicalproperties that can be tied to an area calculation on a graph). The time has nowcome to build a method to compute these areas in a systematic way.

The Fundamental Theorem of Calculus

If f is continuous on the interval [a, b], and we define a related function F (x) suchthat F ′(x) = f (x), then

∫ b

a

f (x) dx = F (b)− F (a)


The fundamental theorem ties the area calculation of a definite integral back toour earlier slope calculations from derivatives. However, it changes the directionin which we take the derivative:

• Given f (x), we find the slope by finding the derivative of f (x), or f ′(x).

• Given f (x), we find the area

∫ b

a

f (x) dx by finding F (x) which is the anti-

derivative of f (x); i.e. a function F (x) for which F ′(x) = f (x).


In other words, if we can find an anti-derivative F (x), then calculating the valueof the definite integral requires a simple evaluation of F (x) at two points (F (b)−F (a)). This last step is much easier than computing an area using finite Riemannsums, and also provides an exact value of the integral instead of an estimate.

The Fundamental Theorem of Calculus - Example - 1

Example: Use the Fundamental Theorem of Calculus to find the areabounded by the x-axis, the line x = 2, and the graph y = x2. Use the fact thatd

dx

(1

3x3)

= x2.


We used the fact that F (x) =1

3x3 is an anti-derivative of x2, so we were able

use the Fundamental Theorem.

Give another function F (x) which would also satisfyd

dxF (x) = x2.

Use the Fundamental Theorem again with this new function to find the area

implied by

∫ 2

0

x2 dx.


Did the area/definite integral value change? Why or why not?

Based on that result, give the most general version of F (x) you can think of.

Confirm thatd

dxF (x) = x2.

Basic Anti-Derivatives - Reference - 1

With our extensive practice with derivatives earlier, we should find it straightfor-ward to determine some simple anti-derivatives.Complete the following table of anti-derivatives.

function f (x) anti-derivative F(x)

x2x3

3+ C

xn

x2 + 3x− 2



cosx

sinx

x + sinx



ex

2x

1√1− x2

1

1 + x2

1

x


The chief importance of the Fundamental Theorem of Calculus (F.T.C.)is that it enables us (potentially at least) to find values of definite integrals moreaccurately and more simply than by the method of calculating Riemann sums.In principle, the F.T.C. gives a precise answer to the integral, while calculating a(finite) Riemann sum gives you no better than an approximation.

Basic Anti-Derivatives - Examples - 1

Example: Consider the area of the triangle bounded by y = 4x, x = 0 andx = 4. Compute the area based on a sketch, and then by constructing anintegral and using anti-derivatives to compute its value.

Basic Anti-Derivatives - Examples - 2

Example: Use a definite integral and anti-derivatives to compute the areaunder the parabola y = 6x2 between x = 0 and x = 5.

Basic Anti-Derivatives - 1/x - 1

The last entry in our anti-derivative table was f (x) =1

x. It is a bit of a special

case, as we can see in the following example.

Example: Sketch the area implied by the integral

∫ −1

−3

1

xdx.


Example: Now use the anti-derivative and the Fundamental Theorem of

Calculus to obtain the exact area under f (x) =1

xbetween x = −3 and x = −1.

Make any necessary adaptations to our earlier anti-derivative table.


Definite vs. Indefinite Integrals - 1

Anti-derivatives and the Fundamental Theorem ofCalculus

The F.T.C. tells us that if we want to evaluate∫ b

a

f (x) dx

all we need to do is find an anti-derivative F (x) of f (x) and then evaluate F (b)−F (a).

THERE IS A CATCH. While in many cases this really is very clever and straight-forward, in other cases finding the anti-derivative can be surprisingly difficult. Thisweek, we will stick with simple anti-derivatives; in later weeks we will develop tech-niques to find more complicated anti-derivatives.

Some general remarks at this point will be helpful.


Remark 1

Because of the importance of finding an anti-derivative of f (x) when you want to

calculate

∫ b

a

f (x) dx, it has become customary to denote the anti-derivative itself

by the symbol ∫f (x) dx

The symbol

∫f (x) dx (with no limits on the integral) refers to the

anti-derivative(s) of f (x), and is called the indefinite integral of f (x)

Note that the definite integral is a number, but the indefinite integral is a function(really a family of functions).


Remark 2

Since there are always infinitely many anti-derivatives, all differing from each otherby a constant, we customarily write the anti-derivative as a family of functions, inthe form F (x) + C. For example,

∫x2 dx =

x3

3+ C

Note that an anti-derivative is a single function, while the indefinite integral is

a family of functions.


Remark 3

Since the last step in the evaluation of the integral

∫ b

a

f (x) dx, once the anti-

derivative F (x) is found, is the evaluation F (b) − F (a), it is customary to write

F (x)∣∣∣b

ain place of F (b)− F (a), as in

∫ 4

0

x2 dx =x3

3

∣∣∣∣4

0

=43

3− 03

3

Remark 4

The variable x in the definite integral

∫ b

a

f (x) dx is called the variable of inte-

gration. It can be replaced by another variable name without altering the value ofthe integral. ∫ b

a

f (x) dx =

∫ b

a

f (u) du =

∫ b

a

f (θ) dθ

Unit #10 : Graphs of Antiderivatives, Substitution Integrals

Goals:

• Relationship between the graph of f (x) and its anti-derivative F (x)

• The guess-and-check method for anti-differentiation.

• The substitution method for anti-differentiation.

Reading: Textbook reading for Unit #10 : Study Sections 6.1, 7.1

Graphs of Antiderivatives - 1

The Relation between the Integral and the Derivative Graphs

The Fundamental Theorem of Calculus states that∫ b

a

f (x) dx = F (b)− F (a)

if F (x) is an anti-derivative of f (x).Recognizing that finding anti-derivatives would be a central part of evaluatingintegrals, we introduced the notation

∫f (x) dx = F (x) + C ⇔ F ′(x) = f (x)

In cases where we might not be able to find an anti-derivative by hand, we can atleast sketch what the anti-derivative would look like; there are very clear relation-ships between the graph of f (x) and its anti-derivative F (x).


Example: Consider the graph of f (x) shown below. Sketch two possibleanti-derivatives of f (x).

0 1 2 3 4

−4

−2

2

4

6

0 1 2 3 4


Example: Consider the graph of g(x) = sin(x) shown below. Sketch twopossible anti-derivatives of g(x).


The Fundamental Theorem of Calculus lets us add additional detail to the anti-derivative graph:

∫ b

a

f (x) dx = F (b)− F (a) = ∆F

What does this statement tell up about the graph of F (x) and f (x)?


Re-sketch the earlier anti-derivative graph of sin(x), find the area underneathone “arch” of the sine graph.

Antiderivative Graphs - Example 1 - 1

Example: Use the area interpretation of ∆F or F (b)− F (a) to constructa detailed sketch of the anti-derivative of f (x) for which F (0) = 2.

−4 −3 −2 −1 0 1 2 3 4

−4

−3

−2

−1

1

2

3

4

−4 −3 −2 −1 0 1 2 3 4


Example: f (x) is a continuous function, and f (0) = 1. The graph of f ′(x)is shown below.

0 1 2 3 4 5 6

−2

2

Based only on the information in the f ′ graph, identify the location and typesof all the critical points of f (x) on the interval x ∈ [0, 6].


Sketch the graph of f (x) on the interval [0, 6].

0 1 2 3 4 5 6

−2

2


0 1 2 3 4 5

−2

−1

1

2

Question: Given the graph of f (x) above, which of the graphs below is ananti-derivative of f (x)?

0 1 2 3 4 5

−2

−1

1

2

3

4

5

6

0 1 2 3 4 5

−2

−1

1

2

3

4

5

6

0 1 2 3 4 5

−2

−1

1

2

3

4

5

6

0 1 2 3 4 5

−2

−1

1

2

3

4

5

6

A B C D

Integration Method - Guess and Check - 1

We now return to the challenge of finding a formula for an anti-derivative function.We saw simple cases last week, and now we will extend our methods to handle morecomplex integrals.

Anti-differentiation by Inspection:The Guess-and-Check Method

Reading: Section 7.1

Often, even if we do not see an anti-derivative immediately, we can make an edu-cated guess and eventually arrive at the correct answer.

[See also H-H, p. 332-333]


Example: Based on your knowledge of derivatives, what should the anti-

derivative of cos(3x),

∫cos(3x) dx, look like?


Example: Find

∫e3x−2 dx.


Example: Both of our previous examples had linear ‘inside’ functions. Hereis an integral with a quadratic ‘inside’ function:

∫xe−x

2dx

Evaluate the integral.

Why was it important that there be a factor x in front of e−x2

in this integral?

Integration Method - Substitution - 1

Integration by Substitution

We can formalize the guess-and-check method by defining an intermediate variablethe represents the “inside” function.Reading: Section 7.1

Example: Show that

∫x3√x4 + 5 dx =

1

6(x4 + 5)3/2 + C.


Relate this result to the chain rule.


Now use the method of substitution to evaluate

∫x3√x4 + 5 dx

Substitution Integrals - Example 1 - 1

Steps in the Method Of Substitution

1. Select a simple function w(x) that appears in the integral.

• Typically, you will also see w′ as a factor in the integrand as well.

2. Finddw

dxby differentiating. Write it in the form . . . dw = dx

3. Rewrite the integral using only w and dw (no x nor dx).

• If you can now evaluate the integral, the substitution was effective.

• If you cannot remove all the x’s, or the integral became harder instead of eas-ier, then either try a different substitution, or a different integration method.


Example: Find

∫tan(x) dx.


Though it is not required unless specifically requested, it can be reassuring to checkthe answer.Verify that the anti-derivative you found is correct.


Example: Find

∫x3ex

4−3dx.


Example: For the integral,∫

ex − e−x(ex + e−x)2

dx

both w = ex− e−x and w = ex + e−x are seemingly reasonable substitutions.Question: Which substitution will change the integral into the simpler form?

1. w = ex − e−x

2. w = ex + e−x


Compare both substitutions in practice.∫

ex − e−x(ex + e−x)2

dx

with w = ex − e−x with w = ex + e−x


Example: Find

∫sin(x)

1 + cos2(x)dx.

Substitutions and Definite Integrals - 1

Using the Method of Substitution for Definite Integrals

If we are asked to evaluate a definite integral such as∫ π/2

0

sinx

1 + cos xdx ,

where a substitution will ease the integration, we have two methods for handlingthe limits of integration (x = 0 and x = π/2).

a) When we make our substitution, convert both the variables x and the limits(in x) to the new variable; or

b) do the integration while keeping the limits explicitly in terms of x, writing thefinal integral back in terms of the original x variable as well, and then evaluating.


Example: Use method a) to evaluate the integral∫ π/2

0

sinx

1 + cosxdx


Example: Use method b) method to evaluate∫ 64

9

√1 +√x√

xdx .

Non-Obvious Substitutions Integrals - 1

Non-Obvious Substitution Integrals

Sometimes a substitution will still simplify the integral, even if you don’t see anobvious cue of “function and its derivative” in the integrand.Example: Find ∫

1√x + 1

dx .

Non-Obvious Substitutions Integrals - 2

∫1√x + 1

dx

Unit #11 : Integration by Parts, Average of a Function

Goals:

• Learning integration by parts.

• Computing the average value of a function.

Reading: Sections 7.2, 5.3

Integration Method - By Parts - 1

Integration by Parts

Reading: Section 7.2So far in studying integrals we have used

• direct anti-differentiation, for relatively simple functions, and

• integration by substitution, for some more complex integrals.

However, there are many integrals that can’t be evaluated with these techniques.

Try to find

∫xe4x dx.


This particular integral can be evaluated with a different integration technique,integration by parts. This rule is related to the product rule for derivatives.Expand

d

dx(uv) =

Integrate both sides with respect to x and simplify.

Express

∫udv

dxdx relative to the other terms.


Integration by PartsFor short, we can remember this formula as

∫udv = uv −

∫vdu

Integration by parts:

• Choose a part of the integral to be u, and the remaining part to be dv.

• Differentiate u to get du.

• Integrate dv to get v.

• Replace

∫u dv with uv −

∫vdu.

• Hope/check that the new integral is easier to evaluate.


Use integration by parts to evaluate

∫xe4x dx.


Verify that your anti-derivative is correct.

Integration By Parts - Examples - 1

Guidelines for selecting u and dv

• Try to select u and dv so that either

– u′ is simpler than u or

–∫dv is simpler than dv

• Ensure you can actually integrate the dv part by itself


Example: Find

∫x cosx dx.


Example: Now evaluate the slightly more challenging integral∫x2 cosx dx


∫x2 cosx dx

Integration By Parts - Definite Integrals - 1

When using integration by parts to evaluate definite integrals, you need to applythe limits of integration to the entire anti-derivative that you find.

Example: Evaluate

∫ π

0

x sin 4x dx

Integration By Parts - Definite Integrals - 2

Don’t forget that dv does not require any other factors besides dx. That can helpwhen there is only a single factor in the integrand.

Example: Find

∫ 2

1

lnx dx

Integration By Parts - Circular Case - 1

There are some classical problems that can be solved by integration by parts, butnot in the direct way we have seen so far.

Example: Integrate by parts twice to find

∫cos(x)ex dx.

Integration By Parts - Circular Case - 2

∫cos(x)ex dx

Applications of Integrals - Average Value - 1

Applications of Integrals - Average Value

Example: If the following graph describes the level of CO2 in the air in agreenhouse over a week, estimate the average level of CO2 over that period.

0 1 2 3 4 5 6 7

35

04

00

45

0

Time (Days)

CO

2 (

ppm

)

Give the units of the average CO2 level.


Example: Sketch the graph of f (x) = x2 from x = −2 to 2, and estimatethe average value of f on that interval.


What makes the single “average” f value different or distinct from otherpossible f values?

Use this property to find a general expression for the average value of f (x)on the interval x ∈ [a, b].


Find the exact average of f (x) = x2 on the interval x ∈ [−2, 2] using thisformula.

Average Value - Examples - 1

Average Value of a Function on [a, b]The average value of a function f (x) on the interval [a, b] is given by

A =1

b− a

∫ b

a

f (x) dx


Example: The temperature in a house is given by H(t) = 18 + 4 sin(πt/12),where t is in hours and H is degrees C. Sketch the graph of H(t) from t = 0to t = 12, then find the average temperature between t = 0 and t = 12.


Example: You are told that the average value of f (x) over the interval

x ∈ [0, 3] is 5. What is the value of

∫ 3

0

f (x) dx?

Average Value - Concepts - 1

Example: If f (x) is even, what is

∫ 3

−3f (x) dx? Sketch such a possible

function.

Example: If f (x) is odd, what is

∫ 3

−3f (x) dx? Sketch such a possible

function.


Example: Without computing any integrals, explain why the average valueof cos(x) on x ∈ [0, π/2] must be greater than 0.5.


Can you make a general statement about the average value of decreasing func-tions which are concave down?

MATH 121 -Calculus Interactive Notes Fall 2014math121/Notes/notes_fall_math121.pdf · MATH 121...

Documents

Transcript of MATH 121 -Calculus Interactive Notes Fall 2014math121/Notes/notes_fall_math121.pdf · MATH 121...