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Unformatted text preview: Math 101 Exam April 1997  1. Evaluate the following integrals. (a) 793:? d2}.
(b) ff cos3 :2: sin2 :1: dx. (c) £6123 lnxdzr.  2. The temperature in Celsius in a 3 m long rod at a point :1: metres from the left end of the rod is given by the function T(a:) = 1&5. Determine the average temperature in the rod.  3. (a) Sketch the parametric curve given by :c = 6‘ cost and y = e‘ sint
where 0 _<_ t g 21r. (b) Find the arc length of this curve.  4. Let 72 be the plane region bounded by :5 = 0, :1: = l, y = 0, and
y = cv l + :52, where c is a positive constant. (a) Find the volume V; of the solid obtained by revolving R about
the z-axis. (b) Find the volume V2 of the solid obtained by revolving R about
the y-axis. (c) If V1 = V2, what is the value of c?  5. Simpson’s rule can be used to approximate 1n 2, since ln2 = [12 ﬁdx.  6.  7. (a) Use Simpson’s rule with 6 subintervals to approximate ln 2. (b) How many subintervals are required in order to guarantee that
the absolute error is less than 0.00001? Note that if E, is the error using n subintervals, then IEHI g #ﬁa—ﬁ
where K is the maximum absolute value of the fourth derivative of
the function being integrated and a and b are the end points of the
interval. An investor places some money in a mutal fund where the interest
is compounded continuously and where the interest rate ﬂuctuates be-
tween 4% and 8%. Assume that the amount of money B = B(t) in the
account in dollars after t years satisﬁes the differential equation: dB E = (0.06 + 0.02 sm t)B. (a) Solve this differential equation for B as a function of t. (b) If the initial investment is $1000, what will the balance be at the
end of two years? A car travels for 2 hours without stopping. The driver records the
car’s speed in km/ hr every 20 minutes, as indicated in the table below. 6: IA
time in hours 0. 1/3 2/3 1 4/3 5/3 2
speed in km/hr 50 70 80 55 60 80 40 (a) Use the trapezoidal rule to estimate the total distance traveled in
the 2 hours. (b) Use the answer to part (a) to estimate the average speed of the
car during this period. 8 8. The portion of the curve y = J, to the right of the line a: = 2 is
revolved about the x-axis. (a) Express the area A of the resulting surface as an improper integral
and show that A is ﬁnite. (b) Show that 1r 3 A.
 9. The function Si(:z:) is deﬁned by Si(:1:) 2 I: “i” dt. (a) Find the Maclaurin series for Si(:z:). (b) It can be shown that Si(:t:) has an absolute maximum which occurs
at its smallest positive ciritcal point (see the graph of Si(1:) below).
Find this critical point. (c) Use the previous information to find the maximum value of Si(a;)
to within $0.01. ...
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