math 137 midterm

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: