MTH 132 MLC Practice Final ExamName: PID:Section: Recitation Instructor:DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE.Page Points Score3 204 305 206 207 208 209 2510 2511 20Total: 200Page 1 of 11MTH 132 MLC Practice Final ExamName: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 thegrader to be able to follow your work. Don’t skip limits or equal signs, etc. Includewords to clarify your reasoning.• Do first all of the problems you know how to do immediately. Do not spend toomuch 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 practiceexam because there are many varieties of problems that can test each concept.I have read and understand the above instructions:. SIGNATUREScore:Page 2 of 11MTH 132 MLC Practice Final ExamMultiple Choice. Circle the best answer. No work needed. No partial credit available.1. Evaluate the following limits:(a) (5 points) limx→−2+2x + 4x2− 4x − 12A. 1/5B. −1/5C. ∞D. −∞E. None of the above.(b) (5 points) limy→16√y − 4y − 16A. 1/4B. −1/4C. 1/8D. DNEE. None of the above.(c) (5 points) limz→0−(z2+ 9) |z|3zA. DNEB. −3C. 3D. 1/3E. None of the above.(d) (5 points) limt→−1sin(2t2− 2)t + 1A. −2B. 2C. −1D. DNEE. None of the above.Page 3 of 11MTH 132 MLC Practice Final ExamFill 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 byv(t) = 32 − 32t,where t is in seconds.(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/second2.3. Given the curve C : 2xy − 3y2+ x3= 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 11MTH 132 MLC Practice Final ExamStandard 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.)(a) (5 points)ddx"x2·1√x+ tan(1)−2.5#(b) (5 points)ddysec(y) + yy2− tan(2y)(c) (5 points)ddt"cos rt + 15!#(d) (5 points)ddxZx1sin(t + 1) dtPage 5 of 11MTH 132 MLC Practice Final Exam5. Evaluate the following integrals:(a) (5 points)Z94(1 +√x)2dx(b) (5 points)Z10t2/3q1 +3√t dt(c) (5 points)Zcos(√θ) sin2(√θ)√θdθ(d) (5 points)Zsec2(α)(1 + tan α)3dαPage 6 of 11MTH 132 MLC Practice Final Exam6. (10 points) Using the limit definition of the derivative, find f0(x) iff(x) =2x + 37. (10 points) Consider the functionf(x) =√x + 8 if x < 13 if x = 12x2− a if x > 1If possible, find the value of a for which the function f (x) above is continuous at x = 1. If there is NOpossible value of a, explain why.You must explain your answer mathematically. Computations with no explanation will receive verylittle credit.Page 7 of 11MTH 132 MLC Practice Final Exam8. Consider the functions f(x) = x2and 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 shadein 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 regionbetween the curves y = f(x) and y = g(x).9. (10 points) Estimate√15 using Linearization.Page 8 of 11MTH 132 MLC Practice Final Exam10. Consider the function f(x) (as well as its first two derivatives) given below:f(x) =13 + x2f0(x) =−2x(3 + x2)2f00(x) =6(x2− 1)(3 + x2)3(a) (5 points) Find all vertical and horizontal asymptotes of f(x) (if there are any). Be specific aboutwhich 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 increasingand 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 ofthe 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 11MTH 132 MLC Practice Final Exam11. (15 points) The area of an equilateral triangle is increasing at a rate of 3 square inches per second. Atwhat rate will the height of the triangle increase when the length of the sides is 7?12. (10 points) Show that the equation x5− x2+ 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 11MTH 132 MLC Practice Final Exam13. (20 points) A rectangular fence is to be built from BOTH wood AND metal so that opposite parallelsides of the fence are made from the same type of material. The wood costs $10 per foot of fence usedand the metal costs $20 per foot of fence used. If you only have $400 to spend on fencing material, whatdimensions of the fence will enclose the largest area?Please have your MSU student ID ready so that is can be checked.When you are completely happy with your work please bring your exam to the front to be handed in.Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED .Congratulations you are now done with the exam!Page 11 of