<p> MATH 0120 FINAL EXAM</p><p>1. (10 points ) State the definition of the derivative , then use the definition to find the 1 derivative of f (x)  5x  2</p><p> x 2  4x  5 2. (8 points) Find lim using appropriate notation and describe what is x1 x 1 happening on the graph at x = -1. M0120 Final Exam Spring 211-4 Dr. B K Michael 2</p><p>3. (20 points) Given f (x)  x3  9x2  24x 1 f (x)  3x 2 18x  24  3(x  4)(x  2) f (x)  6x 18  6(x  3) and f (2)  21, f (3)  19, f (4)  17, a. Find the critical values and make a sign diagram for the first and second derivative and determine: </p><p> f is increasing on the interval ______</p><p> f is decreasing on the interval ______</p><p> f is concave up on the interval ______</p><p> f is concave down on the interval ______</p><p> f has a relative maximum point(s) at ______</p><p> f has a relative minimum point(s) at ______</p><p> f has an inflection point(s) at ______</p><p> b. Sketch the graph of the function, labeling all relative extreme points and inflection points. M0120 Final Exam Spring 211-4 Dr. B K Michael 3 4. (18 pts) Find the derivative. You do not need to simplify. a. f (x)  (x 2  4x)  3x 2 1</p><p>2x 4 1 b. g(x)  2x3  x</p><p>5. (10 points) Find the equation of the tangent line at x = -1 and y = 2 for x 2  y 2  xy  7 M0120 Final Exam Spring 211-4 Dr. B K Michael 4</p><p>6. (10 points) Find f (t) for f (t)  3t 2 ln(t)</p><p>7. (10 points) The national debt( in billions of dollars ) of a certain country in t years is given by N(t)  20  e0.05t . Find the instantaneous rate of change and the relative rate of change at t = 0. Include and explain the units in your answer. M0120 Final Exam Spring 211-4 Dr. B K Michael 5</p><p>8. (12 points) Find all critical values and the relative extreme values of the function: f (x, y)  3xy  2x2  2y2 14x  7y  5 and determine whether they are relative minima, relative maxima, or saddle points. M0120 Final Exam Spring 211-4 Dr. B K Michael 6</p><p>9. (12 points ) Use Lagrange multipliers to minimize the function f (x, y)  2x2  3y2  2xy , subject to the constraint 2x  y  18 . M0120 Final Exam Spring 211-4 Dr. B K Michael 7</p><p>10. (30 points) Find the following integrals a.  x3 ln x 2dx</p><p> x2 1 b. dx  x3  3x 12</p><p> e2 2 c. dx e x(ln x)3 M0120 Final Exam Spring 211-4 Dr. B K Michael 8</p><p>11. ( 10 points) Find the value of 0 2 xex dx </p><p>12. (10 points ) Sketch the area bounded y  x 2  2x  3 and y  2  x  x 2 , then find the bounded area, you do not need to simplify the arithmetic. M0120 Final Exam Spring 211-4 Dr. B K Michael 9</p><p>13. (10 points) Create a piecewise function for the given graph. One of the function pieces is labeled for you. Then find the area bounded by the figure shown below. Set up only (note the points (5,3); (10,3); (20,0) )</p><p> y  0.03(x 10)2  3</p><p>14. (10 points) A company’s marginal cost is MC(x)  x 2 (9  7 x )dx , where x is the number of units. Find the cost function C(x) if the cost to produce 1 unit is 145 dollars. What are the fixed costs of the company? M0120 Final Exam Spring 211-4 Dr. B K Michael 10</p><p>15. (10 points) Find the dimensions and the maximum volume of a package with a square base, if the length plus the girth is to be 84 inches. </p><p>16. (10 points) The demand function for a product is D(x)  500  3x dollars, where x is the number of units of the product. The supply function is S(x)  0.02x 2 dollars. At the equilibrium, the market price is $200. a) What is the market demand? </p><p> b) Find and indicate on the graph the producer’s surplus. Label the intersection. M0120 Final Exam Spring 211-4 Dr. B K Michael 11 (Bonus 8 points) 17. An automobile dealer is selling cars at a price of $20,000. The demand function is D( p)  2(15  .001p)2 , where p is the price of a car. Determine the elasticity of demand and should the dealer raise or lower the price to increase revenue?</p>