Final Examination 1Math 0120 Final ExaminationSampleName (Print) PeopleSoft # . Signature .Instructor (circle one): Lecture time (circle one): Instructions:1. Show your University of Pittsburgh ID if requested.2. Clearly print your name and PeopleSoft number and sign your name in the space above. Circle the name of your lecturer and the time of your lecture.3. Work each problem in the space provided. Extra space is available on the back of each exam sheet. Clearly identify the problem for which the space is required when using the backs of sheets.4. Show all calculations and display answers clearly. Unjustified answers will receive no credit.5. Write neatly and legibly. Cross out any work that you do not wish to be considered for grading.6. No tables, books, notes, headphones, calculators, or computers may be used. All derivatives and integrals are to be found by learned methods of calculus.DO NOT WRITE BELOW THIS LINE_____________________________________________________________________ Problem Points Score Problem Points Score1 40 6 152 20 7 203 10 8 154 40 9 125 10 10 18Total 200Final Examination 21. (40 pts.) Find ).x(fDo not simplify. (a) 43xxex1e)x(f2 (b) f(x) = 92)x31( (c) 1xx2x)x(f23 (d) f(x) = ))xln(ln( (e) f(x) = e2x ln(x2 + 7)Final Examination 32. (a) (12 pts.) .h)x(f)hx(flim)x(f0h Use this definition to find the derivative of f(x) = .x b. (5 pts.) Find an equation of the tangent line to f(x) = x at x = 43. (10 pts.).dxdyFind.9yx2yx244Final Examination 44. (40 pts.) Find the following integrals: (a)dx)exx(x2453 (b) 3)2x(x2dx (c) xlnx3dx (d)xexdxFinal Examination 55. (10 pts.) Assume that the demand function for yams is given by D(p) = 5000 – p2 where D(p) is the quantity in pounds of yams and p is the price in dollars of a pound of yams. (a) If the current price of yams is $3 per pound, how many yams will be sold? (b) At $3 per pound, is the demand elastic or inelastic? (c) Is it more accurate to say “People must have their yams and will buy them no matter what the price” or “Yams are a luxury item and people will stop buying them if the price gets too high”?6. (15 pts.) It costs the Pitt Motor Company $10,000 to produce each automobile, and its fixed costs are $16,000 per week. The company’s price function is p(x) = 25,000 – 25x, where p(x) is the price at which exactly x automobiles will be sold in a week. How many automobiles should the company produce each week to maximize its profit? (SHOW YOUR WORK)Final Examination 67. (20 pts.) Given 34x4x)x(f  , do the following: (a) Make a sign diagram for the first derivative of f(x). (b) Make a sign diagram for the second derivative of f(x). (c) State the open intervals on which f(x) is increasing, decreasing, concave up and concave down. (d ) What are the critical points and inflection points? (e) Sketch the graph of y =f(x) by hand, labeling all relative extreme points and inflection points.Final Examination 78. (15 pts.) Find the area between the curves y = 2x and y = x2 from x = -1 to x = 3. (SHOW YOUR WORK).9. (13 pts.) Find all critical point(s) of f(x, y) = 2x3 + 2y3- 12xy + 5 and classify each as a relative maximum, relative minimum, or saddle point.Final Examination 810. (15 pts.) Use the method of Lagrange multipliers to maximize and minimize f(x, y) = 2x + y subjectto the constraintx2 + 2y2= 72. (Both extreme values