MIDTERM 1 FOR CALCULUS
Time: 08:10–10:00, Thursday, Apr. 17, 2003 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1. (1) [5%] Find the radian measure of 750◦.
(2) [5%] We know that d sin tdt = cos t and d cos tdt = − sin t. Use quotient rule to check that d tan tdt = sec2t.
2. (1) [5%] For f = e2x2+3y2+4z2, find fy(1, −1, 1).
(2) [5%] Rewrite the integral R2
0
R4
y2f (x, y) dx dy so that x is the outer variable (i.e., change the order of the integration)
3. (1) [5%] Find the least square line for the points: (−2, 12), (0, 10), (2, 6), (4, 0), and (6, −3).
(2) [5%] A rectangle is measured to have length x and width y, but each measurement may be in error by 1%. Estimate the percentage error in calculating the area.
4. (1) [5%] Find the limit limx→0 sin x ex−1. (2) [5%] Evaluate the integralR2
0 1
x2+4dx.
5. (1) [5%] Evaluate the integralR
cot t dt.
(2) [5%] Evaluate the integralR
t sin t dt.
6. Find the relative extreme values of the function f (x, y) = 16xy − x4− 2y2. (You have to use D-test to characterize the critical points).
7. Find the volume of the solid bounded by the graphs of z = x + y, z = 0, x = 0, x = 3, y = x, and y = 0.
8. A boatyard estimates that sales during week x of the year will be S(x) = 25 − 20 cosπx
26
thousand dollars, where x = 0 corresponds to the beginning of the year.
(1) Graph the sales function on the interval [0, 52].
(2) Find the total sales during the first half of the year.
9. A boatyard builds 18-foot and 22-foot sailboats. Each 18-foot boat costs $ 3000 to build, each 22-foot boat costs $ 5000 to build, and the company’s fixed costs are $ 6000. The price function for the 18-foot boats is p(x) = 7000 − 20x, and that for the 22-foot boat is q(y) = 8000 − 30y (both in dollars), where x and y are the numbers of 18-foot and 22-foot boats, respectively.
(1) Find the company’s cost function C(x, y).
(2) Find the company’s revenue function R(x, y).
(3) Find the company’s profit function P (x, y).
(4) Find the quantities and prices that maximize profit. Also find the maximum profit.
10. An open-top box with a square base and two perpendicular dividers, as shown in the diagram, is to have a volume of 288 cubic inches. Use Lagrange multipliers to find the dimensions that require the least amount of material.
1
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