Math 115 Second in-class exam: answer key
1. For each of the following functions, nd the derivative: (5 points) f (t) = (1 + et ) ln t. Use the product rule: f (t) = (et ) ln t + (1 + et )(1/t) = et ln t + (5 points) g(x) = ln(ln x). Use the chain rule for ln: g (x) =
7
1 + et . t
d (ln x) dx
ln x
=
1/x 1 = . ln x x ln x
(5 points) h(x) = e5x +2x . Use the chain rule for exponential functions: h (x) = d 7 7 (5x7 + 2x)e5x +2x = (35x6 + 2)e5x +2x . dx
2. (10 points) Find the derivative of the following function, using logarithmic dierentiation: 7 (x2 5) y= . (x3 + 4)15 (x2 5) = ln(x2 5)7 ln(x3 + 4)15 ln y = ln (x3 + 4)15 = 7 ln(x2 5) 15 ln(x3 + 4). Then taking
d dx 7
of both sides, and using the chain rule for ln:
1 dy d d =7 ln(x2 5 15 ln(x3 + 4) y dx dx dx 2x 3x2 =7 15 x2 5 x3 + 4 14x 45x2 3 . = 2 x 5 x +4 1
Finally, multiply both sides by y to get: dy =y dx 14x 45x2 3 x2 5 x + 4
7
(x2 5) = (x3 + 4)15 3. For the function f (x) = x3 9x2 :
14x 45x2 3 x2 5 x + 4
.
(10 points) Find the intervals where f is increasing and the intervals where f is decreasing. f (x) = 3x2 18x = 3x(x 6), so f (x) = 0 when x = 0 or x = 6. We need to check test points in the intervals (∞, 0), (0, 6) and (6, ∞). f (1) = 21 > 0, so f is increasing on (∞, 0). f (1) = 15 0, so f is increasing again on (6, ∞). (5 points) Identify the relative extrema of f . Using the First Derivative test and part (a), it follows that f (0) = 0 is a relative maximum and f (6) = 108 is a relative minimum. Since 0 and 6 are the only critical points, there are no other relative extrema. (10 points) Find the intervals where the graph of f is concave down and the intervals where the graph of f is concave up. f (x) = 6x 18 = 6(x 3), which is zero when x = 3, so we need to check test points in the intervals (∞, 3) and (3, ∞). f (0) = 18 0, so the graph of f is concave up on the interval (3, ∞). (5 points) Identify the inection points of f . f (3) = 54 is an inection point of f (and the only one), since it is the place where the graph of f changes from convave down to concave up.

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