Download Calculus I Exam 2 - October 2006 and more Exams Mathematics in PDF only on Docsity! Calculus I Common exam Exam 2 17 October 2006 Answer all of the following questions. Additional sheets are available if necessary. No books or notes may be used. You may use a calculator. You may not use a calculator which has symbolic manipulation capabilities. When answering these questions, please be sure to 1) check answers when possible, 2) clearly indicate your answer and the reasoning used to arrive at that answer (unsupported answers may not receive credit). Each question is followed by space to write your answer. Please lay out your solutions neatly in the space below the question. You are not expected to write each solution next to the statement of the question. You are to answer two of the last three questions. Please indicate which problem is not to be graded by crossing through its number on the table below. Name Section Last four digits of student identification number Question Score Total p. 1/Q1–3 14 p. 2/Q4 14 p. 3/Q5,6 14 p. 4/Q7,8 14 p. 5/Q9 14 p. 6/Q10 14 p. 7/Q11 14 p. 8/Q12 14 Free 2 2 100 1. Find the exact value of the angle θ in radians that satisfies both π2 < θ < 3π 2 and sin(θ) = −12 . θ = 2. Using the addition or subtraction formula for the sine function and the equality π 12 = π 3 − π 4 , obtain the exact value for sin( π 12 ). sin( π12 ) = 3. Compute lim x→0 sin(3x) sin(x) . Your solution should show the limit laws you use. lim x→0 sin(3x) sin(x) = 7. Suppose f(1) = 2, f ′(1) = 3, f ′′(1) = 5 and define a function h by h(x) = f(x2). Find the derivatives h′(1) and h′′(1). h′(1) = , h′′(1) = 8. Let f(x) = x√ 3 + x2 . (a) Compute f ′(x) and simplify your answer by writing it as a single fraction. (b) Find the linear approximation L(x) to f(x) at x = 1. (a) f ′(x) = , (b) L(x) = 9. (a) Define what it means to say that a number b is a critical number of a function f . (b) If the graph of f is as pictured below, circle all the letters that label critical numbers of f on the x-axis. You do not need to provide an explanation. A B C D E (c) Circle all the letters which correspond to values on the x-axis where the function f has a local maximum. You do not need to provide an explanation. A B C D E y CA EB x D Answer two of the following three questions. Indicate the question that is not to be graded by marking through this question on the front of the exam. 10. Suppose that f is a function that has derivatives of all orders at all values. (a) Express d dx [xf(x)] in terms of f and its derivative. (b) Prove using mathematical induction that dn dxn [xf(x)] = nf (n−1)(x) + xf (n)(x) for all natural numbers n = 1, 2, 3, . . .. Recall that f (0) = f and for n ≥ 1, f (n) is the nth derivative of f .
