MATH 137 MIDTERM
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Introduction
• Arjun Sondhi• 2A Statistics/C&O• First co-op in
Gatineau, QC
Root beer float at Zak’s Diner in Ottawa!
Agenda
• Functions and Absolute Value• One-to-One Functions and Inverses• Limits• Continuity• Differential Calculus• Proofs (time permitting)
Functions and Absolute Value
REVIEW OF FUNCTIONS
Functions and Absolute Value
• A function f, assigns exactly one value to every element x• For our purposes, we can use y and f(x) interchangeably • In Calculus 1, we deal with functions taking elements of
the real numbers as inputs and outputting real numbers
Functions and Absolute Value
Domain: The set of elements x that can be inputs for a function f
Range: The set of elements y that are outputs of a function f Increasing Function: A function is increasing over an interval A if for all , the property holds.
Decreasing Function: A function is decreasing over an interval A if for all , the property holds.
Functions and Absolute Value
Even Function: A function with the property that for all values of x:
Odd Function: A function with the property that for all
values of x:
• A function is neither even nor odd if it does not satisfy either of these properties.
• When sketching, it is helpful to keep in mind that even functions are symmetric about the y-axis and that odd functions are symmetric about the origin (0, 0).
Functions and Absolute Value
Even Function Odd Function
Functions and Absolute Value
ABSOLUTE VALUE
Functions and Absolute Value• Definition:
Functions and Absolute ValueExample. Given that show that
Functions and Absolute Value
SKETCHING – THE USE OF CASES
Functions and Absolute Value
• How to sketch functions involving piecewise definitions? • Start by looking for the key x-values where the function
changes value• Use these x-values to create different “cases”
• Recall: (Heaviside function)
Functions and Absolute Value
Functions and Absolute Value
𝐻 (𝑥+1 )={1𝑖𝑓 𝑥+1≥0⇒ 𝑥≥−10 𝑖𝑓 𝑥+1<0⇒𝑥<−1
• Therefore, key points are x = -1 and x = 0
342
Example. Sketch
Functions and Absolute Value
Cases:
In case 1, we have .In case 2, we have .In case 3, we have
Example. Sketch
Functions and Absolute Value
Functions and Absolute Value
Functions and Absolute Value
Case 1: , which implies that o We have o Isolating for :
Case 2: , which implies that o We have o Isolating for :
Example. Sketch the inequality .
Functions and Absolute Value
Functions and Absolute Value
One-to-One Functions & Inverses
ONE-TO-ONE FUNCTIONS
Functions and Absolute Value
• A function is one-to-one if it never takes the same y-value twice, that is, it has the property:
Horizontal Line Test: We can see that a function is one-to-one if any horizontal line touches the function at most once.
If a function is increasing and decreasing on different intervals, it cannot be one-to-one unless it is discontinuous.
One-to-One Functions & Inverses
One-to-One Functions & Inverses
y = ln(x) y = cos(x)
One-to-One Functions & Inverses
Functions and Absolute Value
One-to-One Functions & Inverses
INVERSE FUNCTIONS
One-to-One Functions & Inverses
A function that is one-to-one with domain A and range B has an inverse function with domain B and range A.
• reverses the operations of in the opposite direction
• is a reflection of in the line y = x
One-to-One Functions & Inverses
Cancellation Identity: Let and be functions that are inverses of each other. Then:
The cancellation identity can be applied only if x is in the domain of the inside function.
One-to-One Functions & Inverses
One-to-One Functions & Inverses
11
One-to-One Functions & Inverses
INVERSE TRIGONOMETRIC FUNCTIONS
One-to-One Functions & Inverses
In order to define an inverse trigonometric function, we must restrict the domain of the corresponding trigonometric function to make it one-to-one.
One-to-One Functions & Inverses
Trig Function
Domain Restriction
Inverse Trig Function
Domain/Range
One-to-One Functions & Inverses
rgregr
One-to-One Functions & Inverses
Let .
Then, . Constructing a diagram:
By Pythagorean Theorem, missing side has length Thus, egegge
• Example. Simplify .
Limits
EVALUATING LIMITS
LimitsLimit LawsGiven the limits exist, we have:
LimitsAdvanced Limit LawsGiven the limits exist and n is a positive integer, we have:
Indeterminate Form (can’t use limit laws)You must algebraically work with the function (by factoring, rationalizing, and/or expanding) in order to get it into a form where the limit can be determined.
Limits
lim𝑥→7
√2+𝑥−3𝑥−7 ∙ √2+𝑥+3
√2+𝑥+3
¿ lim𝑥→ 7
𝑥−7(𝑥−7 ) (√2+𝑥+3 )
¿ lim𝑥→7
1√2+𝑥+3
=16
111
Example. Evaluate
¿ lim𝑥→ 7
(2+𝑥 )−9(𝑥−7 ) (√2+𝑥+3 )
Limits
lim𝑥→∞
𝑥3
𝑥3+5 𝑥𝑥3
2𝑥3
𝑥3− 𝑥
2
𝑥3+ 4𝑥3
¿ lim𝑥→∞
1+ 5𝑥2
2− 1𝑥 + 4𝑥3
¿12 111
Example. Evaluate
Limits
THE FORMAL DEFINITION OF A LIMIT
Limits
if given any , we can find a such that:
Limits
Set
}Select
Limits
SQUEEZE THEOREM
Limits
Squeeze Theorem:
and
then
Limits
----
Limits
Fundamental Trigonometric Limit:
Limits
Limits
11
Continuity
THEOREMS OF CONTINUITY
Continuity
Definition of Continuity
A function is continuous at a point if .
A function is continuous over an interval A if it is continuous on every x in A.
Continuity
Therefore, Now,
---
Continuity
Continuity TheoremsIf are continuous functions at , then:• is continuous at • is continuous at • is continuous at (given that )• If is continuous at and is continuous at then is
continuous at
Continuity
TYPES OF DISCONTINUITIES Infinite
o When a function has a vertical asymptote Jump
o When the one-sided limits do not equal one another Removable
o When the limit does not equal the function value at a point Infinite Oscillations
o When there are an infinite number of oscillations in a neighbourhood of a point
o EX]
Continuity
Infinite
Continuity
Jump
Continuity
Removable
Continuity
Infinite Oscillations
Continuity
INTERMEDIATE VALUE THEOREM
If a function is continuous for all in an interval and and (or vice versa), then there exists a
point such that .
Continuity
is a polynomial function, so it is continuous on all
Thus, by the IVT, the function crosses the x-axis between 0 and 1.
---
Example. Show that has a root between 0 and 1.
Differential Calculus
DEFINITION OF THE DERIVATIVE
Differential Calculus
First principles:
Differential Calculus
¿ limh→0
−2 h𝑎 −h2
(𝑎+h )2𝑎2
h
111
Example. Use the definition of the derivative to find for
Differential Calculus
DIFFERENTIABILITY
In single-variable calculus, the differentiability of a function at a point refers to the existence of the derivative at that point.
(This is NOT so in multivariable calculus...)
Differential Calculus
--
Differential Calculus
Theorem. If a function is differentiable at a point, it is also continuous at that point.
By the Contrapositive Law from MATH 135, we also have the statement: “If a function is NOT continuous at a point, then it is NOT differentiable at the point”. The converse of the theorem, “If a function is continuous at a point, it is also differentiable at that point.” is FALSE! A function that is continuous, but not differentiable at a point is , at x = 0.
Differential Calculus
DERIVATIVE RULES
Differential CalculusPower Rule.
Product Rule.
Quotient Rule.
Differential Calculus
Example. Differentiate using Quotient Rule.
ProofsLIMIT SUM LAWLet > 0 be given.If , then By Triangle Inequality:
if and Then, there exist such that:If , then If , then
ProofsLIMIT SUM LAW (continued)Let Thus, if , then and Therefore, Hence,
ProofsDIFFERENTIABILITY IMPLIES CONTINUITYFor x close to a point a, we have:
Taking limits, we have:
Therefore, is continuous at
ProofsPRODUCT RULEUsing first principles:
Adding and subtracting in the numerator:
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