Math 121. Exam II. November 28 th, 2018

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1 Math 121 Exam II November 28 th, 2018 Name: Section: The following rules apply: This is a closed-book exam. You may not use any books or notes on this exam. For free response questions, you must show all work. Answers without proper justification will not receive full credit. Partial credit will be awarded for significant progress towards the correct answer. Cross off any work that you do not want graded. You have 50 minutes to complete this exam. When time is called, stop writing immediately and turn in your exam to the nearest proctor. You may not use any electronic devices including (but not limited to) calculators, cell phone, or ipods. Using such a device will be considered a violation of the university s academic integrity policy and, at the very least, will result in a grade of 0 for this exam. For multiple choice questions, circle the letter of the best answer. Make sure your circles include just one letter. These problems will be marked as correct or incorrect; partial credit will not be awarded for problems in this section Page: Total Points: Score:

2 Math 121 Exam II - Page 2 of 8 11/28/ (15 points) A 12 foot plank is leaning against a vertical wall. The bottom of the plank is pushed towards the wall at a constant rate of 3 ft/min. How fast is the acute angle that the plank makes with the ground changing at the instant when the bottom of the plank is 6 ft from the wall? Your answer should include the appropriate units. cos θ = x 12 We want to know dθ dt dθ dt sin θ = 1 dx 12 dt when x = 6. When x = 6, we have cos θ = 6 12 = 1 2 Now, we can solve for dx dt. θ = π/3 dθ dt sin θ = 1 dx 12 dt dθ dt sin π 3 = 1 12 ( 3) dθ 3 dt 2 = 1 4 dθ dt = The acute angle that the plank makes with the ground is increasing at a rate of radians/minute.

3 Math 121 Exam II - Page 3 of 8 11/28/ (15 points) Compute an equation for the line tangent to the curve x 2 sin(2y) + y + x = 1 at the point (1, 0). d ( x 2 sin(2y) + y + x ) = d dt dt (1) 2x sin(2y) + 2x 2 dy dy cos(2y) + dx dx + 1 = 0 Plugging in (1,0) we get 2(1) sin(2(0)) + 2(1) 2 dy dy cos(2(0)) + dx dx + 1 = dy dy (1) + dx dx + 1 = 0 3 dy dx = 1 dy dx = 1/3 Tangent line: y 0 = 1 3 (x 1) or y = 1 3 x + 1 3

4 Math 121 Exam II - Page 4 of 8 11/28/ Compute each of the following limits. (a) (10 points) lim x 0 cos (x) 1 3x 2 cos (x) 1 sin (x) lim x 0 3x 2 = lim x 0 6x cos(x) = lim x 0 6 = 1 6 (b) (10 points) lim x)1/x x 0 +(1 ( Consider the limit lim ln (1 x) 1/x) x 0 + ( lim ln (1 x) 1/x) 1 = lim ln ((1 x)) x 0 + x 0 + x ln ((1 x)) = lim x 0 + x 1 1 x x (0) 1 = lim = = 1 So, lim x 0 +(1 x)1/x = e 1

5 Math 121 Exam II - Page 5 of 8 11/28/ (10 points) Use local linear approximation to estimate the value of Express your answer as a fraction. Let f(x) = 3 x. We ll find the local linear approximation of f at x = 27 and use that to approximate f (x) = 1 3 x 2/3 = x 2 f (27) = 1 27 f(27) = 3 L(x) = 1 (x 27) L(26) 1 (26 27) ( 1)

6 Math 121 Exam II - Page 6 of 8 11/28/ (5 points) Let y = sin (3x). Which of the following is d74 y, i.e. the 74th derivative of y dx74 (a) 0 (b) 3 74 cos (3x) 3 74 sin (3x) (d) 3 74 cos (3x) (e) 3 74 sin (3x) 6. (5 points) Let y = log 2 (f(x)). Which of the following is dy dx? (a) 1 f (x) (b) f (x) f(x) (d) ln (f(x)) ln(2) f (x) ln(2)f(x) (e) ln(2)f (x) f(x) 7. (5 points) Which of the following is d [arcsin (4x)]? dx (a) arccos(4x) (b) 4 arccos(4x) (d) (e) x x x 2

7 Math 121 Exam II - Page 7 of 8 11/28/ (5 points) Which of the following is d [arctan (4x)]? dx (a) arcsec 2 (4x) (b) 4arcsec 2 (4x) (d) (e) x x x 2 9. (5 points) Suppose f(x) = e x cot x. Which of the following is f (x)? (a) e x csc 2 (x) (b) xe x 1 csc(x) cot(x) e x cot(x) e x csc 2 (x) (d) xe x 1 cot(x) e x csc(x) cot(x) (e) e x cot(x) e x csc(x) cot(x) 10. (5 points) Use the table to compute d dx (a) -3 [f (g(x))] x=1 x f(x) f (x) g(x) g (x) (b) -1 1 (d) 2 (e) 3

8 Math 121 Exam II - Page 8 of 8 11/28/ (5 points) Let f(x) = x x2. Which of the following is f? (a) 2x x (b) 2x x2 x 2 x x2 1 (d) (e) (2x ln x + x)x x2 2x ln x + x x x2 12. (5 points) Let f(x) = 3 x 1. Which of the following is an equation of the line tangent to the graph of f at x = 0? (a) y = 0 (b) y = x y = x (d) y = 3x (e) y = (ln 3)x

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Math Exam 2-11/17/2014 Math 121 - Exam 2-11/17/2014 Name: Section: Section Class,Times Day Instructor Section Class,Times Day Instructor 1 09:00%AM%(%09:50%AM M%T%W%%F% Li,%Huilan 15 11:00%AM%(%11:50%AM M%T%W%%F% Perlstadt,%Marci