University of Connecticut Department of Mathematics Math 1131 Final Exam Review Spring 2013 Name: Instructor Name: TA Name: 4 th February 2010 Section: Discussion Section: Read This First! Please read each question carefully. Show ALL work clearly in the space provided. In order to receive full credit on a problem, solution methods must be complete, logical and understandable. Answers must be clearly labeled in the spaces provided after each question. Please mark out or fully erase any work that you do not want graded. The point value of each question is indicated after its statement. No books or other references are permitted. Calculators are allowed but you must show all your work in order to receive credit on the problem. Grading - For Administrative Use Only Question: 1 2 3 4 5 6 7 8 9 10 11 Total Points: 15 15 10 6 6 5 12 7 6 13 5 100 Score:
Final Review 1. Multiple choice questions. (a) If f(x) = x 2 f(a + h) f(a) 4x, Evaluate the difference quotient. h a) h b) 4a 4h c) 2a 4 d) 2a + h 2 e) none of these ln x (b) Find lim x 1 x 2 1 a) 0 b) 2 c) 1/2 d) 1 e) Does not exist 5 x (c) Find the limit if it exists lim x 5 5 x a) 5 b) 1 c) 1 d) 0 e) Does not exist (d) Find the derivative of the function y = x 3x a) y = 3x 3x (3 ln(x) + 1) b) y = 3(ln(x) + 1) c) y = 3x 3x (ln(x) + 1) d)y = 3x 3x (3 ln(x) + 3) e) y = x x (ln(3x) + 1) x (e) If F (x) = (1 + t 2 )(1 t) dt, then F (x) has a local minimum value at the following 0 x-values: (circle all that apply): a) 1 b) 0 c) 1 d) 2 e) none of these Page 1 of 8
2. Multiple choice questions. (a) For what values of the constants a and b is (1, 6) a point of inflection of the curve f(x) = x 3 + ax 2 + bx + 1? a) a = 3, b = 1 b) a = 3, b = 1 c) a = 3, b = 7 d) a = 3, b = 3 e) a = 3, b = 7 (b) How many points of inflection are on the graph of the function? f(x) = x 3 60x 2 120x 1700 a) 1 b) 2 c) 3 d) 4 e) 5 π/2 (c) Evaluate the integral sin(t)dt π/4 a) 2 b) 1/ 2 c) 3/2 d) 1/ 2 e) 2 (d) A particle is moving with a velocity v(t) = sin(t) cos(t). Find the position s(t) of the particle at time t if s(0) = 0 a) s(t) = cos(t) sin(t) b) s(t) = 1 cos(t) sin(t) c) s(t) = 1 cos 2 (t) d) s(t) = 1 cos(t) + sin(t) e) s(t) = 1 cos(t) sin(t) (e) The graph of f(x) = F (x) is given below. On which of the intervals is F (x) increasing? (Circle all answers that apply.) 3 y = f(x) 2 1 1 2 2 4 6 8 x 10 a) (0, 3) b) (3, 5) c) (8, 10) d) (9, 10) e) ( 1, 10) Page 2 of 8
sin(ln(x)) 3. Evaluate dx using the substitution method. Show all steps of your work. Check x your answer by computing the derivative. (a) Choose your substitution and show all the details of your work. (b) Use your work above to evaluate the integral. [4] (c) Verify your solution by computing the derivative. 4. If you jump out of an airplane and your parachute fails to open, your downward velocity t [6] seconds after the jump is approximated by v(t) = 49(1 e.2t ) meters/second. (a) Write an expression for the distance you fall during the first 2 seconds. (b) Use the Fundamental Theorem to find the exact distance you fall during the first 3 seconds. Page 3 of 8
1 5. (a) Use the Fundamental theorem of calculus to find the derivative of the following (t 2 + 1) 4/5 dt x 5 (b) If f(x)dx = 10, and f(x)dx = 7, find f(x)dx 0 4 0 5 4 6. The half-life of radium-226 is 1590 years. The mass after 1000 years is 50 grams. (a) What was the initial mass? (b) Find a formula for the mass that remains after t years. Page 4 of 8
7. (a) The frame for a kite is to be made from six pieces or wood. The four exterior pieces have [6] been cut with lengths indicated in the figure. To maximize the area of the kite, how long should each diagonal piece be? 3 8 3 8 (b) A man starts walking south at 5 ft/sec from a point P. 20 seconds later, a women starts [6] walking east at 8 ft/sec from the same point P. At what rate are the two people moving apart 30 seconds after the women starts walking? Page 5 of 8
8. State the Chain Rule and then explain carefully how it can be used to calculate the derivative of y = 3 cos(ln(x)) (a) State the Chain Rule (b) Explain how it can be applied to this problem, giving complete details. (c) Use the Chain Rule to compute the derivative 9. Water flows into and out of a storage tank. A graph of the rate of change r(t) of the volume [6] of water in the tank, in liters per day, is shown. If the amount of water in the tank at time t = 0 is 25,000 L, use the Left end points Rule with n = 4 to estimate the amount of water in the tank four days later. ). r 4000 2000 t 2000 1 2 3 4 Page 6 of 8
10. Analyze the following function f(x) = 2x2 x 2 1 (a) f (x) = (b) f (x) = (c) x-intercepts (d) y-intercepts (e) Vertical asymptotes (f) Horizontal asymptotes (g) Critical numbers Page 7 of 8
(h) Intervals of increase (i) Intervals of decrease (j) Local extrema (k) Concavity 11. Find the derivative using the limit definition for f(x) = x 9 [5] Page 8 of 8