Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeMidterm 2Thursday, 02/28/08• Complete the following problems. You may use any result from class you like, but ifyou cite a theorem be sure to verify the hypotheses are satisfied.• This is a closed-book, closed-notes exam. No calculators or other electronic aids willbe permitted.• In order to receive full credit, you must show all of your work and justify your answers.Your answer should be cle arly labeled.• It is OK to leave your work unsimplified.• If you need extra room, use the back sides of each page. If you need more paper, usethe extra at the front of the classroom and staple it to your exam. Include all scratchwork with the test. Do not unstaple or detach pages from this exam.• Please sign the following:“On my honor, I have neither given nor received any aid on this exam-ination. I have furthermore abided by all other aspec ts of the honor codewith respect to this examination.”Name:Signature:1. (/6 points)2. (/8 points)3. (/4 points)4. (/24 points)5. (/5 points)6. (/8 points)7. (/5 points)8. (/40 points)Total. (/100 points)11. (6 points) Determine whether each statement is true or false. If the statement istrue, cite your reasoning. If it is false, provide an example showing thestatement to be false.(a) If a function is differentiable at x = 0 it must be continuous at x = 0.(b) If f(x) satisfies f0(0) = 0, then f(x) has a local maximum or a local minimum atx = 0.22. (8 points) Use the definition of the derivative (i.e. the limit definition) to compute:ddx(3x2+ 15).33. (4 points) The graphs of three f unctions are drawn in the top row. The graphs of theirderivatives are drawn in the bottom row. Match each function with its derivative. Youdo not need to justify your answer, but an incorrect answer without justification willnot receive credit.The graphs of functions.The graphs of the derivatives of the functions above.44. Let f(x) = x4+ 2x3− 89(a) (8 points) Compute f0(x) and f00(x). You do not have to use the limit definitionof the derivative here.(b) (4 points) Find all the critical points of f(x).(c) (4 points) Find the intervals on which f(x) is increasing and the intervals onwhich f(x) is decreasing.5(d) (4 points) Find the intervals where f(x) is concave up and the intervals wheref(x) is concave down.(e) (4 points) From your list of critical points, determine which are actually localmaxima and which are actually local minima (for each, be sure to justify why itis a max/min or neither).65. (5 points) Sketch the graph of a single function f(x) with the following properties:• f0(0) = 0,• f0(1) does not exist,• f0(x) > 0 when x < −2 and when x > 1,• f0(x) < 0 when 0 < x < 1/2,• f00(x) > 0 when x < −1,• f00(x) < 0 when −1 < x < 1/2.76. (8 points) Find the equation of the tangent line to x3− y3= −6xy at (3, −3).7. (5 points) Suppose h(x) = 3f(x)g(x) + 2f(x), where f(x) and g(x) are differentiablefunctions. Find h0(1) when f(1) = 2, g(1) = 3, f0(1) = 1, and g0(1) = −1.88. Compute the following derivatives. You do not have to use the definition of the deriva-tive. If you can “do them in your head” instead of showing every step that is up toyou (though if you get it wrong we cannot give you partial credit.)(a) (8 points) Let f(x) = (7x3+ 2x)90. Find f0(x).(b) (8 points) Let f(x) = x8− 8x. Find f0(x).9(c) (8 points) Differentiate (tan(x)ex+ x).(d) (8 points) Findddx(3√x(ln(x))).(e) (8 points)