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Unformatted text preview:      Math 101 Exam April 1995 where k is a positive constant. If there were initially 90,000 ﬁsh in the
lake, and 40,000 were left after 6 weeks, when will the ﬁsh population
be reduced to 10,000? 1. The plane region lying under the curve y = 1'3 and above the :c-axis,  6- Let [(1') 2 I: Hit—“1L
between :1: = 0 and x = l, is rotated about the x-axis to generate a
solid of revolution. (8) Determine the Maclaurin series for [(22).
(3) Find the volume of the solid. (b) Approximate I(%) to within :l:0.0001.
(b) Find the total area of the surface of the solid. (c) Is your approximation in (b) larger or smaller than the true value of I(=i,-)? Explain. 2. Let L be the length of the part of the curve y = cosz lying between
a: = 0 and x = 21r. Express L as a deﬁnite integral, and use the  7. (8) Sketch the polar curve r = 2sin2(39). Trapezoidal Rule with six equal subintervals to approximate L.
(b) Find the area of the region that is inside the curve in (a). 3. (3) Sketch the region bounded by y = 731:?“ y = 0, a: = 0, and x = 2. DO TWO OF THE FOLLOWING FOUR QUESTIONS (b) Find the y—coordinate of the centroid of this region.  8. A swimming pool has the shape shown in the ﬁgure below. The vertical 4. Let R be the part of the ﬁrst quadrant that lies below the curVe cross-sections of the pool are the bottom halves of circular disks. The
y = arctan 1: and between the lines a: = 0 and 1: = l. distances in feet across the pool are given in the ﬁgure at 2 foot intervals along the 16 foot length of the pool. Use Simpson’s Rule to approximate . (a) Sketch the region R and determine its area. the volume or the pool. (b) Find the volume of the solid obtained by rotating R about the
y-axis. 5. The ﬁsh population in a lake is attacked by adisease at timet = 0, with
the result that the size P(t) of the population at time t Z 0 satisﬁes dP
7F _ 4M7),  9. [10110. [10111. Consider the parametric curve cost __ sint
tk , y - tk ) x: where k is a positive constant. (8) Express the length L of this curve as an improper integral. Simplify
the integrand as much as possible, but do not evaluate the integral. (b) Determine all values of the positive constant k for which the length
L is ﬁnite. Justify your answer. An object of mass m is projected straight upward at time t = 0 with
initial speed 110. While it is going up, the only forces acting on it are
the force of gravity (assumed constant) and a drag, force proportional
to the square of the object’s speed 11(1). It follows that the differential equation of motion is do 2
mm _‘ —(mg + kl} )) where g and k are positive constants. At what time does the object
reach its highest point? Suppose that f’(1:) is continuous on [0,27r]. Show that lim 2" f(::) cos(m:)d1: = 0. n—too 0 ...
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