Unformatted text preview: Methods of Calculus MAC3233
Test 3
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 )
x
(c)
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
halflife 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 11fold?
2
(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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 Calculus, Fermat's theorem

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