Math 151 Week in Review
Monday Nov. 15, 2010
Instructor: Jenn Whitfield
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
4. f (x) =
2x2
x−4
5. f (x) = |x2 − 2x|
6. f (x) = (x2 − x)1/3
Section 5.1
1. Sketch a graph satisfying:
• Domain: All real numbers
• f (−1) = −2, f (0) = 0, f (2) = 3
• f ′ (x) < 0 for x < −1 and x > 2
• f ′ (x) > 0 if −1 < x < 2
• f ′′ (x) > 0 if x < 0 and f ′′ (x) < 0 if
x>0
2. Given the graph of f ′ (x) below, find intervals where f (x) increases/decreases, has local extrema, intervals of concavity, and inflection points. Given that f (x) is continuous and f (a) = 0, sketch a possible graph of
f (x)
7. Determine the absolute extrema for g(x) =
2x3 + 3x2 − 12x + 4 on the interval:
(a) [−4, 2]
(b) [0, 2]
8. Suppose that the population (in thousands)
of a certain kind of insect after t months is
given by the formula P (t) = 3t + sin(4t) +
100. Determine the minimum and maximum population in the first 4 months.
9. Suppose that the amount of money in a
bank account after t years is given by A(t) =
t2
2000 − 10te5− 8 . Determine the minimum
and maximum amount of money in the account during the first 10 years that it is
open.
10. Determine the absolute extrema for Q(y) =
3y(y + 4)2/3 on [−5, −1].
11. Sketch a graph of a function that has x = 2
as a critical number, but has no local extrema.
Section 5.2
12. Sketch a graph of a function that is everywhere continuous function, has a local maximum at x = 2, but is not differentiable at
x = 2.
Given
• The critical numbers are where f ′ (c) = 0
or f ′ (c) does not exist.
• If f ′ (x) > 0 on an interval, then f (x) is
increasing.
• If f ′ (x) < 0 on an interval, then f (x) is
decreasing.
• If f ′′ (x) > 0 on an interval, then f is
concave upward on that interval.
• If f ′′ (x) < 0 on an interval, then f is
concave downward on that interval.
For the following functions, identify all critical
values. Be sure to classify each critical value as
a local max, local min, or neither.
3. f (x) = x4 + 4x3 + 2
Section 5.3
For the following functions, determine where the
given functions are increasing, decreasing, concave up, and concave down. Identify any local
extrema and points of inflections then sketch a
graph of the function.
13. f (x) = 3x5 − 5x3 + 3.
−x − 1
x
, given f ′ (x) =
,
2
(x − 1)
(x − 1)3
2x + 4
.
and f ′′ (x) =
(x − 1)4
14. f (x) =
15. g(t) = t(6 − t)2/3 , given g′ (t) =
and g′′ (t) =
10t − 72
.
9(6 − t)4/3
18 − 5t
1
3(6 − t) 3
,