Trinity College December 03, 2015 MATH 131-01 Calculus I By signing below, you attest that you have neither given nor received help of any kind on this exam. Signature: Printed Name: Instructions: Show work to get full credit (the correct answer may NOT be enough). Do all your work on the paper provided. Write clearly! Double check your answers! You will not receive full credit for using methods other than those discussed in class. Exam 3 MATH 131-01 Calculus I Problem Available Your Number Points Points 1 9 2 4 3 7 4 6 5 8 6 6 7 6 8 12 9 5 10 8 11 6 Total 77
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1. Find the derivative dy using any method. Do not simplify. dx (a) ỵ = sin 3 (ln x) [3] (b) ỵ = 3 sec(x) tan(x) [3] (c) ỵ = sin 1 (x 2 ) [3] Page 3 of 13
2. Choose the differential equation from the list below which corresponds to each of the following slope [4] fields. Write your answer on the line below the slope field. dy dx = xy dy dx = 1 dy dx = 1 dy dx = y dy dx = 1 x dy dx = 1 y (a) (b) (a) (b) (c) (d) (c) (d) Page 4 of 13
3. Consider the function f(x) = x 2 3. [7] (a) Use two iterations of Newton s method to approximate the root of this function, starting with the initial guess x 0 = 1. (b) On the graph of f(x) below, label the points x 0 and x 1 as well as the relevant tangent line used for the first iteration. Page 5 of 13
4. Let y(x) be a solution to the differential equation whose slope field is given below. (a) What are the x-values of the critical points of the function y(x). (b) Classify the critical points as local maxima, local minima or neither. (c) Sketch the solution which satisfies y(0) = 1 on the slope field above. Page 6 of 13
5. For the following questions indicate whether the statement is TRUE or FALSE. If true, provide a one sentence explanation OR state the corresponding theorem. If false, provide a counterexample or explanation. (a) Assume f is a polynomial. If f (c) = 0 and f (c) > 0, then x = c is a local minimum. (b) Assume f is a polynomial. If f (c) = 0, then x = c is an inflection point of f. (c) Assume f is continuous. If x = c is a local extremum of f, then f (c) = 0. (d) Assume f is differentiable on [a, b] and f(a) = f(b), then there exists a c (a, b) such that f (c) = f(b) f(a). b a Page 7 of 13
6. Consider the two functions, f(x) = x 2 x and g(x) = x. (a) Does f satisfy the hypotheses of the Mean Value Theorem on the interval [ 1, 1]? If yes, explain briefly why. If no, explicitly state which hypothesis is not met. (b) Does f satisfy the hypotheses of Rolle s Theorem on the interval [ 1, 1]? If yes, explain briefly why. If no, explicitly state which hypothesis is not met. (c) Does g satisfy the hypotheses of Rolle s Theorem on the interval [ 1, 1]? If yes, explain briefly why. If no, explicitly state which hypothesis is not met. Page 8 of 13
7. Sketch the graph of a single function that satisfies the following conditions: [6] (a) f is continuous on (, ) (b) f(1) = 2 (c) f(3) = 4 (d) f(4) = 1 (e) f (1) = f (3) = 0 (f) f (4) DNE (g) f (x) > 0 on (1, 3) and (4, ) (h) f (x) < 0 on (, 1) and (3, 4) (i) f (x) > 0 on (, 2) (j) f (x) < 0 on (2, 4) and (4, ) Page 9 of 13
8. Suppose that f is a function whose first and second derivative are given by f (x) = 4 x2 (2x 1) 2 f 2x 16 (x) = (2x 1) 3 (a) Find the intervals on which f is increasing. (b) Find the intervals on which f is decreasing. (c) For what values of x does f have local maxima? Justify your answer. (d) For what values of x does f have local minima? Justify your answer. (e) Find the intervals on which f is concave up. (f) Find the intervals on which f is concave down. Page 10 of 13
9. Find the critical points for the piecewise function [5] { 3x 2/3 if x 1 f(x) = x 2 3x + 5 if x > 1 Page 11 of 13
10. A cup of coffee is poured from a pot whose contents are 95 C into a cup in a room whose temperature is 20 C. Let T (t) represent the temperature of the coffee in the cup at time t where t is the time since the coffee was poured into the cup. (a) Write a differential equation that represents the rate of change of the temperature as a function of time. (Use the constant k to represent the proportionality constant.) dt dt = (b) Write an equation for the solution, T (t), that satisfies the given initial temperature of the coffee. (Again, your answer will be in terms of k.) (c) After 1 minute, the coffee has cooled to 90 C. Use this information to find the value of k. (d) Evaluate the lim T (t). t Page 12 of 13
11. The graphs of f, f, and f are given below, in no particular order. Determine which graph corresponds [6] to f, f, and f and write the function next to the graph. 0.5 a.) 0 0.5 6 4 2 0 2 4 6 1 0.5 b.) 0 0.5 6 4 2 0 2 4 6 0.5 0 c.) 0.5 6 4 2 0 2 4 6 Page 13 of 13