Final Examination Math 0120 Final ExaminationSampleName (Print) S.S.# . Signature . Instructor (circle one): Lecture time (circle one): Instructions:1. Show your University of Pittsburgh ID if requested.2. Clearly print your name and social security 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_____________________________________________________________________ ProblemPoints Score ProblemPointsScore1 10 6 202 10 7 423 35 8 204 15 9 155 15 10 18Total 2001Final Examination 1. (a) (5 pts.) A function f(x) is continuous at x = c if three conditions are satisfied: i. ii. iii. (b) (5 pts.) Give an example (no graph) of a function that is continuous at x = -1, but not differentiable at x = -1. 2. (10 pts.) .h)x(f)hx(flim)x(f0h Use this definition to find the derivative of f(x) = .x21 2Final Examination 3. (35 pts.) Find .yor)x(f You need not simplify )x(f. (a) 54xxx4ee)x(f2 (b) f(x) =)x6ln(x3 (c) f(x) = 53)xx(  (d) x1x2x3)x(f2 (e) Find .xyxy3x2fory223Final Examination 4. a. (5 pts.) It is found that a person can memorize x words in f(x) = 2x2 – x seconds for .20x0  Find )5(f and interpret it as an instantaneous rate of change with the proper units. b. (10 pts.) Find an equation of the tangent line to y = x at x = 4.5. (15 pts.) A chocolate cake dealer can sell 12 chocolate cakes per week at a price of $20 each. He estimates that each $4 price decrease will result in 3 more sales per week. If the chocolate cakes cost him $12 each to make, what price should he charge to maximize his profit? How many will he sell at that price and what is the maximum profit?4Final Examination 6. (20 pts.) )24x9x(xx24x9x)x(f223; );4x)(2x(324x18x3)x(f2 and )3x(618x6)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) Sketch the graph of y =f(x) by hand, labeling all relative extreme points and inflection points. 5Final Examination 7. (42 pts.) Find the following integrals: (a)dx)exx(x235e4  (b) xx1dx (c) 121)ex(dx (d) )x6x)(2x(32dx (e)  xxedx (f) dxx)]x[ln(26Final Examination 8. Set up, but do not evaluate, definite integrals for: (a) (10 pts.) The area bounded by the curves y =1 - x2 and y = 1 - x. (b) (10 pts.) The consumers’ surplus for the demand function d(x) = 200 – 0.5 x at the demand level x = 300. 9. (15 pts.) Find all critical points of f(x.y) =y4x12xy32 and classify each as a relative maximum, relative minimum, or saddle point.7Final Examination 10. (18 pts.) Use the method of Lagrange multipliers to maximize and minimize f(x,y) = 4xy subject to .32yx22 (Both extreme values exist.)