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Final Exam Review great review for final!

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University of Waterloo
MATH 137
Nico Spronk

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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