math 137 midterm

MATH 137 MIDTERM

2010 Outreach TripSummaryDate Aug 20 – Sept 4Location Cusco, Peru# Students 22Project Cost$16,000

Building ProjectsKindergarten Classroom provides free educationSewing Workshop enables better job prospectsELT Classroom enables better job prospectsMore info @

studentsofferingsupport.ca/blog

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


• 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


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.


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).


Even Function Odd Function


ABSOLUTE VALUE

Functions and Absolute Value• Definition:

Functions and Absolute ValueExample. Given that show that


SKETCHING – THE USE OF CASES


• 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)


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


Cases:

In case 1, we have .In case 2, we have .In case 3, we have

Example. Sketch


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 .

One-to-One Functions & Inverses

ONE-TO-ONE FUNCTIONS


• 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.


y = ln(x) y = cos(x)


INVERSE FUNCTIONS


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


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.


11


INVERSE TRIGONOMETRIC FUNCTIONS


In order to define an inverse trigonometric function, we must restrict the domain of the corresponding trigonometric function to make it one-to-one.


Trig Function

Domain Restriction

Inverse Trig Function

Domain/Range


rgregr


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


First principles:


¿ limh→0

−2 h𝑎 −h2

(𝑎+h )2𝑎2

h

111

Example. Use the definition of the derivative to find for


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...)


--


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.


DERIVATIVE RULES

Differential CalculusPower Rule.

Product Rule.

Quotient Rule.


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:

math 137 midterm

Documents

absolute value88 functions

absolute value77 functions

absolute value1919 functions

absolute value2020 functions

absolute value1212 functions

absolute value1515 functions

absolute value1717 functions

functions2323 functions