Review Poster

First Derivative Test [Local min/max]

If x = c is a critical value on f and f’ changes sign at x = c…

(i) f has a local max at x = c if f’ changes from >0 to <0

(ii) f has a local min at x = c if f’ changes from < 0 to > 0

Number Line Analysis [instead of the big chart we used to

make]

(iii) No sign change at x = c, no local min/max

Second Derivative Test [Also a test for local min/max, not a test for concavity or points of inflection]

If x = c is a critical value on f [meaning f’(c) = 0 or is undefined] and f”(c) exists…

(i) If f”(c) > 0, x = c is a local min on f(ii) If f”(c) < 0, x = c is a local max on f(iii) If f”(c) = 0, then the test fails and we don’t

know anything.

Test for Concavity on f [Points of Inflection]

Evaluate f” at points where f’=0•Point of inflection on f at x = c, f changes from

concave down to concave up

• Point of inflection on f at x = c, f changes from concave up to concave down

Linear Approximation

Use equation of a line tangent to f at a point (x, f(x)) to estimate values of f(x) close to the

point of tangency1) If f is concave up (f” > 0), then the linear approximation will

be less than the true value

2) If f is concave down (f”< 0), then the linear approximation will be greater than the true value.

Properties of f(x) = ex

• Inverse is y = lnx • • Domain (-∞,∞) • ea eb = ea+b

• Range (0, ∞) • • • • elnx = x

• ln ex = x

Properties of f(x) = lnx• Inverse is y = ex • ln(ab) = lna + lnb• Domain (0, ∞) • ln (a/b) = lna - lnb• Range (- ∞, ∞) • ln(ak) = klna

• Always concave • ln x <0 if 0<x<1 down• Reflection over x-axis: -lnx• reflect over y-axis: ln(-x) • horizon. shift a units • vert. shift a units left: ln(x + a) up: ln(x) + a• horizon. shift a units • vert. shift a units right: ln(x – a) down: ln (x) - a

Chain Rule (Composite Functions)

Rules for Differentiation

Product rule:

Quotient rule:

Implicit Differentiation

When differentiating with respect to x (or t or θ)1) Differentiate both sides with respect to x, t,

or θ.2) Collect all terms with on one side of the

equation.3) Factor out . 4) Solve for .

An Example of Implicit Differentiation

Find if 2xy + y3+ x2 = 7

• 2y + 2x + 3y2 + 2x = 0

• (2x + 3y2)=-2x - 2y

• =

Another (slightly different) example of implicit differentiation

If x2 + y2 = 10, find .

I will pause here to let you catch up on copying and try to solve this problem on your own.

2x + 2y = 0

= -x/y

=

= = = (-y2 – x3 )/y3

Line Tangent to Curve at a Point

- Need slope (derivative) at a point (original function)

- A line normal to a curve at a point is ______________ to the tangent line at that point. (The slopes of these lines will be ___________ _____________)

Related Rates (The rates of change of two items are dependent)

1. Sketch2. Identify what you know and what you want

to find.3. Write an equation.4. Take the derivative of both sides of the

equation.5. Solve.

Big Section: Integrals

- Approximate area under a curve- Riemann Sum =- Left endpoint- Right endpoint- Midpoint

Inscribed and Circumscribed Rectangles

Inscribed Rectangles - underestimate - happens when f is decreasing and you use

right end point OR when f is increasing and you use left endpoint

• Circumscribed Rectangles-overestimate

- happens when f is decreasing and you use left end point OR when f is increasing and you use right endpoint

Trapezoidal Rule – most accurate approximation

If f is continuous on [a,b]

•As n ∞, this estimate is extremely accurate• Trapezoidal rule is always the average of left

and right Riemann sums

Fundamental Theorem of CalculusPart I

If F’ = f,

Fundamental Theorem of CalculusPart II

where a is a constant and x is a function

Properties of Definite Integrals

• • • •

Oh man. This is taking me so long to type. So. Long.

• If f is an odd function (symmetric about the origin, (a,b) (-a,-b)) , then

• If f is an even function (symmetric about the y-

axis, (a,b) (-a,b))), then • If f(x) ≥ g(x) on [a,b] then

Definition of Definite Integral

Average Value of a Function on [a,b]

• M(x) = Average Value =• • So, (b-a)(Avg. Val.)=

Total Distance Vs. Net Distance

Net Distance over time [a,b] =

Total Distance over time [a,b] =

Area Between Two Curves

If f(x) and g(x) are continuous on [a,b] such that f(x) g(x), then the area between f(x) and g(x) is given by

Area =

VolumesThere are basically three types of volume problems…

1. Volume by Rotation – Disc - (x-section is a circle)

(where R is the radius from the axis of rev.)

2. Volume by Rotation – Washer -(x-section is a circle)

R(x) = radius from axis of revolution to outer

figure r(x) = radius from axis of revolution to

inner figure

3. Volume of a Known Cross-Section (foam projects)

V =

A(x) is the area of a known cross-section

You’re almost there! Only two more slides! (After this one)

Next Big Section: Differential Equations and Slope Fields

Differential Equations

1. Separate dy and dx algebraically. [Separation of variables.]

2. both sides. This will create a c value. The general solution has a c in it.

3. Solve for c using the initial conditions. Use this c value to write the particular solution.

Slope Fields-Show a graphical solution to differential equations- Big picture made of tangent segments is the

solution- The slope of each individual tangent segment is

the value of at that point

The End