Name: PID:TA: Sec. No: Sec. Time:Math 10A.Midterm Exam 1December 7, 2010Turn off and put away your cell phone .You may use one page of notes, but no boo ks or other assis tance during this exam.You may leave answers in symbolic form, for example√42 or ln(6).Read each question carefully, and answer each question completely.Show all of your work; no credit will be given for unsupported answers.Write your solutions clearly and legibl y; no credit will be given for illegible solutions.If any q ues tion is not clear, ask for clarification.# Points Score1 122 93 64 65 46 87 68 99 5Σ 651. (12 points) Suppose that f and g are functions with f (1) = 2, f′(1) = 3, g(1) = 4 ,g′(1) = 6. Find h′(1) if:(a) h(x) = f(x)g(x)(b) h(x) =ln(x)f(x)(c) h(x) = g( x2)(d) h(x) = arctan(f(x))2. (9 points) Evaluate the given limit or explain why it does not exist.(a) limx→∞3x3− 6xx2− 2x3(b) limx→0|x|x(c) limh→013+h−13h3. (6 points) Suppose that P (t) is the populat ion of Whoville t years after 1990.(a) Explain the practical meaning of the equation P (12) = 5050.(b) Explain the practical meaning of the equation P−1(3800) = 7.(c) What are the units of P′(t)?(d) Explain the practical meaning of the equation P′(10) = 250.4. (6 points) The amount o f caffeine (in milligrams) left in a person’s body t hours af terdrinking a cup of coffee containing 100 milligrams of caffeine is given by the formulaQ(t) = 100(0.84 )t.(a) How many hours after drinking the coffee until only 2 0% of the caffeine is left inthe body? (Please leave your answer in symbolic form, for example ln(6) .)(b) Find Q′(2). (Please leave your answer in symbolic form, for example ln(6).)5. (4 points) On the axes provided below, sketch the graph of a f unction f with thefollowing properties:• f (−4) = f (1) = f(3) = 0• f′(x) > 0 if x < −1 or x > 2• f′(x) < 0 if −1 < x < 2• f′′(x) < 0 if x < 0• f′′(x) > 0 if x > 012345−1−2−3−4−51 2 3 4 5−1−2−3−4−5xy6. (8 points) The graph of the equation 2y2= 3x2+ x3appears below:1234−1−2−3−41 2 3 4−1−2−3−4xy2y2= 3x2+ x3(a) How many points are there on the gr aph at which the tangent line is horizontal?Write the number o f points here and also clearly mark the points on the graph.(b) Use implicit differentiation to finddydx.(c) Use your formula fordydxfrom part (b) to find the exact (x, y) coordinates of thepoints from par t (a).7. (6 points) The graph of the function f(x) = e(x2−1)appears below.1234−11 2−1−2xyy = e(x2−1)(a) Find the equation of the tangent line to the graph of f at the point with x = −1.(b) Will the tangent line approximation to f (x) near x = −1 give an over-estimateor an under-estimate? Briefly explain your answer. (You may want to use thegraph.)8. (9 points) Let g(x) = x4− 4x3.(a) Find all of the critical points of g and classify them as local minima, local maxima,or neither.(b) Find all value(s) of x at which t he graph of g has an inflection point.(c) Find the global maximum and the global minimum values of g(x) (if t hey exist)on the interval (0, ∞).9. (5 po ints) A box with an open top is made from a squarepiece of material by removing equally sized squares fromeach corner and turning up the sides. Find the dimen-sions of the box of largest volume that can be made inthis way if the square piece of material is 6 feet by 6