Math 19 Practice Exam 2B, Winter 2011

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1 Math 19 Practice Exam 2B, Winter 2011 Name: SUID#: Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your answers unless specifically instructed to do so. You may use any result from class that you like, but if you cite a theorem be sure to verify the hypotheses are satisfied. You have 2 hours. This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted. If you finish early, you must hand your exam paper to a member of teaching staff. Please check that your copy of this exam contains 10 numbered pages and is correctly stapled. If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam. Please sign the following: On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination. Signature: The following boxes are strictly for grading purposes. Please do not mark. Question: Total Points: Score:

2 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 1 of (10 points) Find each of the following limits, with justification (show steps). If there is an infinite limit, then explain whether it is or. ( (a) (6 points) lim x 3 x 3 x) x ( x 2 ) (b) (4 points) lim ln 3 x 2 x 1

3 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 2 of (8 points) Use the definition of the derivative as a limit to find the derivative of f(x) = x 2 + 3x + 2. No marks will be awarded if you get the right answer using other methods.

4 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 3 of (12 points) The following is the graph of g(x) (a) (3 points) Where is g (x) positive? And negative? Use the interval notation for your answer. (b) (3 points) Where is g (x) increasing? And decreasing? Use the interval notation for your answer.

5 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 4 of 10 (c) (6 points) Sketch the graph of g (x) on the axes provided below.

6 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 5 of (12 points) Find the derivatives of the following functions using any method you like. You do not need to simplify your answers. (a) (4 points) f(x) = ln(x 2 )e x (b) (4 points) g(x) = 1 tan(x) 1 x 4 (c) (4 points) Z(t) = 5t 6 3t 7 + 8t 2/3 + cos(t)

7 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 6 of (12 points) Careful: These two problems are unrelated. (a) (6 points) Find the tangent line to f(x) = ln(e x e + 1) at x = 1. (b) (6 points) Given that f (5) = 2, g (5) = 1, f(5) = 3, g(5) = 4, f(4) = 2 and f (4) = 8, find the derivative of h(x) = f(g(x)) at x = 5. f(x) + g(x)

8 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 7 of (8 points) Show that the function f(t) = t 2 1 is not differentiable at t = 1.

9 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 8 of (12 points) Find the derivatives of the following functions using any method you like. You do not need to simplify your answers. (a) (4 points) g(x) = (5x 3 2x)(tan(x)) (b) (4 points) Z(t) = sin(te t ) (c) (4 points) h(x) = x 2 + 5x 1 x

10 Math 19, Winter 2011 Practice Exam 2B, Winter 2011 Page 9 of (13 points) Miscelaneous. Careful: These two problems are unrelated. (a) (5 points) Give an example of a function that satisfies f(2) = 3, f(4) = 2 and that there is no number c in (2, 4) such that f(c) = 0. (Graph = 3 points, Formula = 5 points) (b) (8 points) Find the derivative of g(x) = tan(x) cos(x)

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Math 41 First Exam October 12, 2010

Math 41 First Exam October 12, 2010 Name: SUID#: Circle your section: Olena Bormashenko Ulrik Buchholtz John Jiang Michael Lipnowski Jonathan Lee 03 (11-11:50am) 07 (10-10:50am) 02 (1:15-2:05pm) 04 (1:15-2:05pm)