Mathematics 111, Spring Term 2010 Topics List and Sample Problems for the Final Exam This review sheet consists of main two parts: a list of the topics from which the exam will be built, and some sample problems. The final exam will be given on Monday, 10 May 2010 at 6:00 p.m. in Harder 202. It will be a cumulative exam, comprising the entire semester. We have studied parts of the following sections of the text: Chapter 1, Functions and Models; Chapter 2, Limits and Derivatives; Chapter 3, Differentiation Rules; Chapter 4, Applications of Differentiation; Chapter 5, Integrals. Midterm 1: 1) Functions (Sections 1.1-1.3, 1.5-1.6). a) The concept of a function: input, process, output. b) Linear functions, slope, intercepts, equations of a line. c) Functional notation: the function f given by f (x) = (expression in x). d) Ways to represent functions, such as tables, graphs, expressions, and verbal descriptions. e) Graph transformations (horizontal and vertical translations, reflections, and stretchings). f) Composition of functions, inverse functions. g) Important families of functions: transcendental functions (exponential and logarithmic functions, trigonometric functions, etc.), and algebraic functions (power functions, polynomial functions, and rational functions). h) Idea of mathematical modeling with functions. 2) Limits and Derivatives (Sections 2.1-2.3, 2.5-2.8). a) Average rates of change and instantaneous rates of change. Units of a rate of change. b) Concept of limit of a function at a point. Meaning of lim f ( x) , lim f ( x) , etc. x a x a c) Limit “rules” and exact computation of limits using algebraic simplification. d) Infinite limits and limits at (or vertical and horizontal asymptotes). e) The derivative at a point: what it is (i.e., formal definition), how to estimate it, how to compute it exactly (in certain cases) using algebra, interpretation as slope and instantaneous rate of change, difference quotients. f) The derivative function f : what it is, domain of f , ways to approximate or compute f exactly, tangent slopes and tangent lines. Midterm 2: 1. 2. 3. 4. 5. Basic rules of differentiation: the power rule, the sum rule, and the constant multiple rule. Derivatives of exponential functions. The product and quotient rules. The chain rule for differentiating composite functions. Derivatives of the trigonometric functions and the inverse trigonometric functions (arcsin, arctan). 6. Derivatives of logarithmic functions and logarithmic differentiation. 7. Related rate problems. 8. Implicit functions and implicit differentiation. 9. Higher derivatives ( f , f ,... ). 10. Exponential growth and decay. (Not in Spring 2010.) 11. Linear approximation of a function f at a point (where f is differentiable). Since the Second Midterm: 1. Maximum and minimum values of a function (both global and local); critical points; extreme value theorem. 2. Mean value theorem. 3. Use of the first and second derivatives in curve sketching; concavity & points of inflection. 4. L’Hospital’s rule for limits of the form “0/0” and “ / ”; other indeterminate forms. 5. Optimization problems. 6. Antiderivatives; symbolic computation and graphical approximation. 7. The area (under a graph) problem, rectangular approximations to area. 8. The concept of the definite integral of a continuous function (the limit of the Riemann sums as the number of rectangles n and the width x of the subintervals 0 ). 9. The geometric interpretation of the definite integral (“signed area”). 10. Properties of definite integrals (linearity, comparison, etc.). x 11. The Fundamental Theorem of Calculus, parts I and II: Part I: If g ( x) f with f is a b continuous, then g ' ( x) f ( x). Part II: f ( x)dx F (b) F (a) , where f is continuous and a f. F Sample Problems: 1. Evaluate each of the following: lim x 3 lim h x2 0 2x 3 3 x lim __________ sin( / 2 h) sin( / 2) h t t 3 5t 2 716 . 4 6t 18.3t 21 __________ __________ 2. Find the derivatives of the following functions: f ( x) m(t ) 3. x2 (t 3 5x 3 g( x) cos(t ))12 n( x) x 2 / 3 tan( x) h( x) (t 1) / (t 2 1) arcsin(tan(e x )) Find y dy / dx by implicit differentiation, where y is a function of x defined by the relation xy 3 sin( y) 0 . 4. Evaluate the following using the FTC: 3 1 1 x 7 dx x cos( x )dx 0 1 dx 1 x2 0 x 5. Let f be the function whose graph is shown, and let F be defined by F ( x ) f (t )dt 3 a) Estimate F(5) and F(1) . b) Where is F increasing? Explain your answer. c) At which value(s) of x does F have local minima? Explain. d) Where is F concave down? Explain. 6. Referring to the function f whose graph is given in the previous problem, sketch the graph of g , where g ( x) f ( x 2) 1 . Also sketch the graph of F that was discussed in the previous problem. 7. Use the definition of the derivative (limit process) to compute h (x) , where h( x) 1 / x . Then find the equation of the line tangent to the graph at the point (2, 1/2). 8. Sketch a graph of a single function f with the following properties: f(0) = 1, f(1) = 4, f(-2) = 0; f(x) < 0 for x < -2 and h(x) > 0 for x > -2; f '(-2) = f '(3) = 0, and f '(1) is not defined; f '(x) > 0 for x < -2, -2 < x < 1, and x > 3; f '(x) < 0 for 1 < x < 3. 9. A cannonball is fired directly upward, starting from ground level (height = 0), at time t 0 seconds. Let h(t ) represent the cannonball's height above the ground level (in feet) at time t seconds, v(t ) 192 32t the cannonball's vertical velocity (in feet per second) at time t, and a(t ) the vertical acceleration at time t seconds. 4 v (t )dt . What does this answer tell you about the cannonball? a) Calculate 1 b) Showing all work, find a formula for h(t ) . c) Showing all work, find a formula for a(t ) . d) At what time is the cannonball at the highest point? How high is it at that time? 10. Use the average of the left and right sums with four subintervals to approximate the value of 3 6 dx . x 1 2 2 | x| dx 11. Evaluate the following using geometry: 1 (2 3x )dx 1 12. Find the local maxima and minima of the function f , given by f ( x) maximum and minimum of f on the interval [-2,3]. 3x 3 9 x , and the global 13. You wish to mail a cylindrical package whose combined length and girth (the circumference of a cross section perpendicular to the length) is 84 inches. What are the length and girth of the cylindrical tube with the largest volume that you can mail? 1 14. Compute the definite integral of the function h , given by h( x) , on the interval [1/2, 5/2]. x2 15. Use linear approximation to estimate 9.01 . [Hint: find the best linear approximation to f ( x) x at x 9 .] 16. Find the point that the Mean Value Theorem guarantees will exist for the function f ( x) x on the interval [1, 9]. x 3 12 x 5 17. Sketch the graph of y f ( x) on the axes provided, given graphs of f & f . x2 1 Note that can be used as a convenient “starting point.” Graph of f Graph of f Graph of f