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Final

# Final Exam Review great review for final!

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University of Waterloo

Mathematics

MATH 137

Nico Spronk

Winter

Description

MATH 137 – Final Exam Review
Saturday December 18, 9-11:30 am
The final exam will cover materials from Section 3.4 to Section 6.1 of the textbook. For the detailed list
of sections covered after Section 3.4, please see the last page of the syllabus. You still must know how
to use all definitions, theorems and formulas that were covered in the midterm. For example, you must
know the derivatives of trigonometric functions on page 193 and the double-angle formulas (15a, 15b,
16a, 16b) on page A29.
You will be asked to reproduce the proof of one of the following:
• Fermat's theorem, local maximum case only (page 273-274)
• Constant function theorem (Theorem 5, page 284)
• Increasing Test, part (a) only (page 287)
• Fundamental Theorem of Calculus, Part 2 only (page 384)
Alternative proofs will be accepted as long as they are mathematically correct.
You will not be asked to state any definition or any theorem. But you are required to know how to use
all the definitions and theorems that were covered in class to solve concrete problems.
Chapter 1
Interval Notation: (a,b) – open interval; endpoints not included
[a,b] – closed interval; endpoints included
[a,ⱷ) – infinite intervals; other variations as well
Function: is a machine which takes each element and assigns that to value a UNIQUE element
Vertical Line Test: given a set, the graph of the function is the set for each vertical line x=a, the line
intersects the set, at most once
Horizontal Line Test: f is one-to-one if no horizontal line y=a meets the graph at most once
Properties of Functions:
odd if f(-x) = -f(x); symmetric about the y-axis
even if f(-x) = f(x); symmetric about the origin
Composition of Functions:
1. Consider the inside function first
2. Amongst the points, figure out which of those points lies in the first function
3. The domain of the composition will equal to those elements in the second function such that the
second function is contained inside the first functions domain
Inverse Functions: f is one-to-one if distinct inputs give distinct outputs ; undoes what the function has
done; mirror image of the original graph about the line y=x
1. Y=f(x) 2. X=f(y) 3. Solve for y Exponential Functions: is the unique continuous function which satisfies the following that f(p/q) =
a^(p/q) whenever p is an integer and q is a natural number. Special case: e^x; loge=log=ln; it is its own
inverse
Trigonometric Functions: 2π periodic; cosine is an even function; sine is an odd function; cos^2(x) +
sin^2(x) = 1
Half-Angle Formula: sin (x+y) = sinxcosx + sinycosx cos(x+y) = cosxcosy-sinxsiny
Chapter 2
Error Game: the lim of f(x) = L if for any declaration of error, ε > 0, we can find δ > 0 such that |f(x) –L} <
ε wherever 0a of x = a
2. Lim x->a of a constant = constant
Squeeze Theorem: suppose you have 3 functions, f g and h. if fa of f and h = L
then the lim x->a of g =L as well
Intermediate Value Theorem: if a function in the interval is continuous over the entirety of the interval
and is negative at f(a) and positive at f(b), then there is a value N that exists between a and b where f(N)
= 0; it crosses the x-axis
Chapter 3
Tangent Line: the line that best approximates f(x) at x=a
Derivative: a function on an open interval is differentiable on the open interval if f for each value of x on
the open interval, f’(x) exists. The function f’(x) is called the derivative function
Derivative of Sums: (cf +g) (x) = cf(x) + g(x) and is differentiable at a with, (cf +g)’ (x) = cf’(x) + g’(x)
Differentiability => Continuity: if f is differentiable at a, then f is continuous at a.
Product Rule of Derivatives: (fg)’ (a) = f’(a) g(a) + g’(a) f(a)
Chain Rule: (fg)’(a) = f’(g(a)) g’(a)
Half-Life Equation: M(t) = Moe^(-kt)
Implicit Differentiation: allows us to compute dy/dx (derivative of y in terms of x)
Chapter 4
Minimum/Maximum Values:
- an absolute maximum of f on D is the largest value that f(x), a point Cmax such that f(x) < f(Cmax) for
the whole domain - an absolute minimum of f on D is the smallest value that f(x), a point Cmin such that f(x) > f(Cmin) for
the whole domain
Extreme Value Theorem: if a function which is continuous on a closed bounded interval, then f has both
an absolute maximum and absolute minimum on the interval
Local Maximum/Minimums:
- if f(c) > f(x) within some wiggle room of the domain then c is the local maximum
-if f(c) < f(x) within some wiggle room of the domain then c is the local minimum
Critical Numbers: if a function contains a number, c, such that c has some wiggle room in the domain,
and f’(c) = 0 or f’(c) DNE then c is a critical number (point)
1. Find all critical numbers for f on the domain, eval

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