Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Points Score 3 20 4 30 5 20 6 20 7 20 8 20 9 25 10 25 11 20 Total: 200 Page 1 of 11
Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers. If you need scratch paper, use the back of the previous page. Without fully opening the exam, check that you have pages 1 through 11. Fill in your name, etc. on these first two pages. Show all your work. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don t skip limits or equal signs, etc. Include words to clarify your reasoning. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later. If you have any questions please raise your hand and a proctor will come to you. There is no talking allowed during the exam. You will be given exactly 120 minutes for this exam. This is a practice exam. The actual exam may differ significantly from this practice exam because there are many varieties of problems that can test each concept. I have read and understand the above instructions:. SIGNATURE Score: Page 2 of 11
Multiple Choice. Circle the best answer. No work needed. No partial credit available. 1. Evaluate the following limits: 2x + 4 (a) (5 points) lim x 2 + x 2 4x 12 A. 1/5 B. 1/5 C. D. E. None of the above. (b) (5 points) lim y 16 y 4 y 16 A. 1/4 B. 1/4 C. 1/8 D. DNE E. None of the above. (z 2 + 9) z (c) (5 points) lim z 0 3z A. DNE B. 3 C. 3 D. 1/3 E. None of the above. sin(2t 2 2) (d) (5 points) lim t 1 t + 1 A. 2 B. 2 C. 1 D. DNE E. None of the above. Page 3 of 11
Fill in the Blanks. No work needed. No partial credit available. 2. A ball is thrown upward in the air. Suppose the velocity of the ball, in feet per seconds, is given by where t is in seconds. v(t) = 32 32t, (a) (5 points) The ball reaches its maximum height at t = seconds. (b) (5 points) If the ball is thrown from an initial height of 5 feet, find the precise height function h(t) of the ball is given by: h(t) = (c) (5 points) The maximum height reached by the ball is feet. (d) (5 points) The acceleration of the ball after 1 second is feet/second 2. 3. Given the curve C : 2xy 3y 2 + x 3 = 4 and the point P = (2, 2). (a) (5 points) Find the slope of the curve C at P. Slope of C at P = (b) (5 points) Find an equation of the tangent line to C through P. Extra Work Space. Page 4 of 11
Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 4. Find the following derivatives. (You do NOT need to simplify your answers.) [ ( ) ] 2.5 d 1 (a) (5 points) x 2 x + tan(1) dx (b) (5 points) d [ sec(y) + y ] dy y 2 tan(2y) (c) (5 points) [ ( )] d t + 1 cos dt 5 (d) (5 points) [ d x ] sin(t + 1) dt dx 1 Page 5 of 11
5. Evaluate the following integrals: (a) (5 points) 9 4 (1 + x) 2 dx (b) (5 points) 1 0 t 2/3 1 + 3 t dt (c) (5 points) cos( θ) sin 2 ( θ) θ dθ (d) (5 points) sec 2 (α) (1 + tan α) 3 dα Page 6 of 11
6. (10 points) Using the limit definition of the derivative, find f (x) if f(x) = 2 x + 3 7. (10 points) Consider the function x + 8 if x < 1 f(x) = 3 if x = 1 2x 2 a if x > 1 If possible, find the value of a for which the function f(x) above is continuous at x = 1. If there is NO possible value of a, explain why. You must explain your answer mathematically. Computations with no explanation will receive very little credit. Page 7 of 11
8. Consider the functions f(x) = x 2 and g(x) = 2 x. (a) (5 points) Sketch the graphs of y = f(x) and y = g(x) below on the same set of axes AND shade in the region between (bounded by) these curves. (b) (5 points) Set up BUT DO NOT EVALUATE an integral that will find the area A of the region between the curves y = f(x) and y = g(x). 9. (10 points) Estimate 15 using Linearization. Page 8 of 11
10. Consider the function f(x) (as well as its first two derivatives) given below: f(x) = 1 f (x) = 2x f (x) = 6(x2 1) 3 + x 2 (3 + x 2 ) 2 (3 + x 2 ) 3 (a) (5 points) Find all vertical and horizontal asymptotes of f(x) (if there are any). Be specific about which type of asymptote(s) you have found and which types do not exist. (b) (5 points) Find all x-intercepts and y-intercepts of f. (c) (5 points) Find all critical numbers of f (if there are any), find the x-intervals where f is increasing and decreasing, AND find the x-values where f has local maxima and minima (if there are any). (d) (5 points) Find the x-intervals where f is concave up and concave down AND find the x-values of the inflection points of f (if there are any). (e) (5 points) Combining ALL previous parts, draw an accurate sketch of y = f(x) below. ALSO, label any asymptotes, intercepts, local maxima/minima, and inflection points. Page 9 of 11
11. (15 points) The area of an equilateral triangle is increasing at a rate of 3 square inches per second. At what rate will the height of the triangle increase when the length of the sides is 7? 12. (10 points) Show that the equation x 5 x 2 + 2x + 3 = 0 has at least one real solution. Hint: Quote and use the Intermediate Value Theorem. You have to show why it applies. Page 10 of 11
13. (20 points) A rectangular fence is to be built from BOTH wood AND metal so that opposite parallel sides of the fence are made from the same type of material. The wood costs $10 per foot of fence used and the metal costs $20 per foot of fence used. If you only have $400 to spend on fencing material, what dimensions of the fence will enclose the largest area? Congratulations you are now done with the exam! Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED. When you are completely happy with your work please bring your exam to the front to be handed in. Please have your MSU student ID ready so that is can be checked. Page 11 of 11