Review for Final. The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study:

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1 Review for Final The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study: Chapter 2 Find the exact answer to a limit question by using the limit rules Determine if a function is continuous (or discontinuous) at certain points, and explain why Solve a limit problem involved with infinity Use the Squeeze Theorem to prove the answer to a limit question Use the Intermediate Value Theorem to prove a solution to an equation exists Estimate the slope of a tangent line (instantaneous rate of change) by drawing a tangent line and estimating the slope Estimate the slope of a tangent line using the "forward-backward" technique Find the derivative by using the formula Sketch the graph of the derivative of a function Know at what points the derivative does not exist Chapter 3 Find the equation of a tangent line and/or normal line (of an explicit or implicit equation) Find and simplify first and second derivatives (using the various rules) Use product and quotient rules (example: if f(3) = 4, f (3) = 5, g(3) = -1 and g (3) = 2, what is the Derivative of f(3)g(3)?) Implicit differentiation Derivatives of logarithmic and inverse trig functions Logarithmic differentiation Linear approximation Using a differential to estimate errors Prove the following using the definition of derivatives: derivative of trig functions. Prove the following: derivative of the natural logarithm, derivative of inverse trig functions, power rule Taylor series Chapter 4 Related rates (similar to questions from section 4.1) Use derivatives to determine critical points, inflection points, maxima, minima, direction and concavity (sections ) Use L'Hopital's rule to determine limits Solve optimization problems Newton s method Find the anti-derivative of a function

2 Sample Questions 1) Solve the following limit problems, using the method indicated: a) 2 t 9 lim x 3 2t 2 + 7t + 3 (use algebraic techniques, as in chapter 2) b) 1 x + lnx lim (use L Hospital s Rule) x cos πx c) 2 t 9 lim x 2t 2 + 7t + 3 (use any method) 2) For each function below, find the derivative dy/dx a) y = 4e 5x x b) y = 8x 3 2x c) 3xy 2 + 9x = 4xy d) y = tan( x) x 3 5 e) y = x x (hint: use logarithmic differentiation) 3) Consider the function f(x) = tan -1 (x) a) Find the linear approximation at the point a = 1. b) Use the linear approximation from part (a) to estimate tan -1 (1.1) c) Find the 3 rd order Taylor polynomial (let n = 3, a = 1) d) Use the Taylor polynomial from part (c) to estimate tan -1 (1.1) e) the exact value of tan -1 (1.1) is Which answer, from part (b) or (d), is closer? 4) Two straight roads intersect at right angles in Newtonville. Car A is on one road moving toward the intersection at a speed of 60 miles per hour. Car B is on the other road moving away from the intersection at a speed of 40 miles per hour. (See the diagram below). When Car A is 2 miles from the intersection and Car B is 4 miles from the

3 intersection, how fast is the distance between the cars changing? Is the distance increasing or decreasing? 5) Find the absolute minimum and maximum values of y = x 3 9x + 8 on the interval [-3,1]. 6) On what intervals is f(x) = 2 x e concave up? 7) A function has a second derivative f (x) = 5x 2 + 4x. Furthermore, f (0) = 2 and f(1) = 6. Find the function. 8) Find the dimensions of a rectangle of largest area that can be inscribed in an equilateral triangle of side L = 12 cm, if one side of the rectangle lies on the base of the triangle. 9) Use the definition of the derivative to prove that if y = sin x, then y = cos x 10) Use logarithmic differentiation to prove the power rule. 1 11) Prove that if y = sin -1 (x), then y = 2 1 x. (hint: re-write as x = sin y) 12) Find the antiderivative F(x) of the following: a) f(x) = 4x 2 2x + 6 b) f(x) = cos(x) c) f(x) = 1 x x + 3 / 4

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