Final ExamMath 220May 12, 2007Instructions: This exam is worth 200 points. Put one problem on each answer sheet (use theback if necessary), and put your name, your TA’s name, your section number, and the problemnumber on each page. Only sign the honor pledge on the first sheet. Show all of your work, andjustify your answers. Scientific, but not graphing calculators are allowed.1. Compute the following:(a)ddxex2 − x2(b)ddtln(t(t − 1)) + e−2 2. For the function f(x, y, z) = exy+ z2(x + y)2, compute the following:(a)∂f∂x(b)∂f∂z3. Compute the following integrals:(a)Z1x2+2x − 1dx(b)Z203x2+ e2xdx4. (a) A certain radioactive isotope is undergoing exponential decay with decay constant λ =0.05. Find the half-life of the isotope.(b) Elizabeth invests $5, 000 in an account gaining 6% annual interest, compounded contin-uously.i. How many years will it take until the balance in the account is $7, 500?ii. At what rate will the balance be growing at that time ?Turn over the page for the remainder of the questions.15. (a) Let h(x) = x3− 12x + 4.i. Use calculus to find all intervals where h(x) is increasing and all intervals whereh(x) is decreasing.ii. Give any critical points of h(x), and classify each as a relative minimum, a relativemaximum, or neither.(b) Find the equation of the line tangent to the graph of g(x) = 3x2+ 2x − ln(x) at x = 1.6. Samuel wants to build a rectangulargarden in his back yard. The out-side fence of the garden will be woodthat costs $2 per foot. In addition,Samuel wants a brick wall down thecenter of his garden, which will cost $6per foot (see the figure to the right).Samuel has $800 to spend on the fenceand wall; find the maximum total areaof Samuel’s garden. Tell how youknow that your answer is the maxi-mum area.7. (a) Approximate the area under the curve k(x) = ln(x) from x = 1 to x = 3 using a left-handRiemann sum with n = 4.(b) Sketch the bounded region enclosed by the graphs of y = x2− 20 and y = −x2+ 12, andcompute its area.8. A ball is thrown (straight up) into the air with velocity v(t) = −32t + 128 feet per second(where t is measured in seconds).(a) Find the value of t for which the height is maximized.(b) Give a physical interpretation forZ41v(t) dt, and compute its value.9. Let f(x, y) = 2x4− 8xy + 2y4.(a) Use the first derivative test to locate all of the points where f (x, y) could have a relativeminimum or relative maximum.(b) Use the second derivative test to classify each point from part (a) as a relative minimum,a relative maximum, or a saddle point. The following formula may help you apply thesecond derivative test:D(x, y) =∂2f∂x2·∂2f∂y2−∂2f∂x∂y2.10. Use the method of Lagrange multipliers to find the minimum value of the function f(x, y) =y2− x2subject to the c onstraint y = 3 −
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