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Exam_exam_4_(4)

# Exam_exam_4_(4) - 2(25 points ±ind the center of gravity x...

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MIDTERM 4 — MATH 141 — SPRING 2003 — BOYLE Put your name, your TA’s name and the question number on EACH page. Use exactly ONE answer sheet per question (use the back of the sheet if needed). Put a BOX around the Fnal answer to a question. No books, no notes, no calculators. Before handing in your test: on your Frst answer sheet, please copy the pledge, and sign. ———————————————————————————————————– 1. Suppose the position ( x ( t ) , y ( t )) of a particle at time t is given by the rule ( x ( t ) , y ( t )) = (ln( t ) , t 3 ), 1 t 3 . Let C be the curve traversed by the particle. (a) (10 points) Compute the velocity of the particle at time t = 2. (b) (10 points) Express the length L of C as a deFnite integral. Do not evaluate the integral. (c) (10 points) (New problem.) ±ind a parametrization P ( t ) = ( x ( t ) , y ( t )), a t b , which describes a clockwise motion 3 times around the unit circle with velocity constant and equal to 1.
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Unformatted text preview: 2. (25 points) ±ind the center of gravity ( x, y ) of the region R which is the top half of the ellipse x 2 + (1 / 4) y 2 = 1. (You may appeal to symmetry, and you may use the fact that R has area π .) 3. (a) (15 points) The amount of work required to stretch a certain spring 6 cen-timeters from equilibrium is 1000 ergs. How much work is required to stretch the spring an additional 6 centimeters? (b) (10 points) Sketch the curve described in polar coordinates by the equation r = cos(3 θ ). 4. (a) (10 points) Suppose w = 1-2 i and z = 2 e 6 i . Plot w and z in the complex plane. Compute | w | , | z | and | wz | . (b) (3 points) Write an equation which relates the exponential, sine and cosine functions. (c) (3 points) Compute the radius of convergence of the complex power series ∑ ∞ n =0 (1-2 i ) n z n . (d) (4 points) ±ind real numbers a, b such that a + ib = ∞ X n =0 ± 1 2 + i 2 ² n ....
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