Unformatted text preview: Methods of Calculus MAC3233
November 2, 1990
(1) (20 pts.) Diﬀerentiate the following functions.
(a) e3−x + e4
(b) e2/x + e−4x
(c) x3 e−2x
(d) ex ln 3 + e3 ln x
(2) (20 pts.) Diﬀerentiate the following functions.
(a) ln(e5x + 1)
(b) (ln x)4 + ln(x4 )
ln(x + 5)
(d) ln x3 (x + 2)2
(3) (15 points) Find the coordinates of each relative extreme point of the function f (x) = 4(x + 4)e−x/4
and determine if it is a relative maximum point or a relative minimum point.
(4) (15 points) Five hundred grams of a radioactive substance with decay constant 0.017 (if time is
measured in years) is buried in the ground. (a) How much will remain after 23 years? (b) Find the
half-life of this radioactive substance.
(5) (15 pts.) The growth rate of a certain bacteria culture is proportional to its size. If the culture
doubles in size every 13 minutes, how long will it take for the culture to increase 11-fold?
(6) (15 pts.) Solve the following equations for x. (a) ex −1 e4−4x = 1 (b) ln [ln(x + 3)] = 0 (c) ln x2 = 10 1 ...
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