math 137 midterm
DESCRIPTION
MATH 137 MIDTERM. 2010 Outreach Trip. Summary Date Aug 20 – Sept 4 Location Cusco, Peru # Students 22 Project Cost $16,000. Building Projects Kindergarten Classroom provides free education Sewing Workshop enables better job prospects ELT Classroom enables better job prospects. - PowerPoint PPT PresentationTRANSCRIPT
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MATH 137 MIDTERM
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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
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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!
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Agenda
• Functions and Absolute Value• One-to-One Functions and Inverses• Limits• Continuity• Differential Calculus• Proofs (time permitting)
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Functions and Absolute Value
REVIEW OF FUNCTIONS
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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
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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.
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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).
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Functions and Absolute Value
Even Function Odd Function
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Functions and Absolute Value
ABSOLUTE VALUE
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Functions and Absolute Value• Definition:
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Functions and Absolute ValueExample. Given that show that
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Functions and Absolute Value
SKETCHING – THE USE OF CASES
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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)
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Functions and Absolute Value
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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
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Functions and Absolute Value
Cases:
In case 1, we have .In case 2, we have .In case 3, we have
Example. Sketch
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Functions and Absolute Value
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Functions and Absolute Value
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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 .
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Functions and Absolute Value
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Functions and Absolute Value
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One-to-One Functions & Inverses
ONE-TO-ONE FUNCTIONS
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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.
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One-to-One Functions & Inverses
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One-to-One Functions & Inverses
y = ln(x) y = cos(x)
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One-to-One Functions & Inverses
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Functions and Absolute Value
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One-to-One Functions & Inverses
INVERSE FUNCTIONS
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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
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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.
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One-to-One Functions & Inverses
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One-to-One Functions & Inverses
11
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One-to-One Functions & Inverses
INVERSE TRIGONOMETRIC FUNCTIONS
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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.
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One-to-One Functions & Inverses
Trig Function
Domain Restriction
Inverse Trig Function
Domain/Range
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One-to-One Functions & Inverses
rgregr
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One-to-One Functions & Inverses
Let .
Then, . Constructing a diagram:
By Pythagorean Theorem, missing side has length Thus, egegge
• Example. Simplify .
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Limits
EVALUATING LIMITS
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LimitsLimit LawsGiven the limits exist, we have:
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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.
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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 )
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Limits
lim𝑥→∞
𝑥3
𝑥3+5 𝑥𝑥3
2𝑥3
𝑥3− 𝑥
2
𝑥3+ 4𝑥3
¿ lim𝑥→∞
1+ 5𝑥2
2− 1𝑥 + 4𝑥3
¿12 111
Example. Evaluate
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Limits
THE FORMAL DEFINITION OF A LIMIT
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Limits
if given any , we can find a such that:
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Limits
Set
}Select
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Limits
SQUEEZE THEOREM
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Limits
Squeeze Theorem:
and
then
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Limits
----
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Limits
Fundamental Trigonometric Limit:
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Limits
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Limits
11
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Continuity
THEOREMS OF CONTINUITY
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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.
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Continuity
Therefore, Now,
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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
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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]
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Continuity
Infinite
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Continuity
Jump
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Continuity
Removable
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Continuity
Infinite Oscillations
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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 .
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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.
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Differential Calculus
DEFINITION OF THE DERIVATIVE
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Differential Calculus
First principles:
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Differential Calculus
¿ limh→0
−2 h𝑎 −h2
(𝑎+h )2𝑎2
h
111
Example. Use the definition of the derivative to find for
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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...)
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Differential Calculus
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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.
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Differential Calculus
DERIVATIVE RULES
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Differential CalculusPower Rule.
Product Rule.
Quotient Rule.
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Differential Calculus
Example. Differentiate using Quotient Rule.
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ProofsLIMIT SUM LAWLet > 0 be given.If , then By Triangle Inequality:
if and Then, there exist such that:If , then If , then
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ProofsLIMIT SUM LAW (continued)Let Thus, if , then and Therefore, Hence,
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ProofsDIFFERENTIABILITY IMPLIES CONTINUITYFor x close to a point a, we have:
Taking limits, we have:
Therefore, is continuous at
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ProofsPRODUCT RULEUsing first principles:
Adding and subtracting in the numerator: