MATH 151 Engineering Mathematics I
Week In Review
Fall, 2015, Problem Set 10 (Exam3 Review)
JoungDong Kim
1. What is the domain of f (x) =
x ln(4 − x2 )
√
.
x
2. Solve log2 (x2 − 38) − log2 (5 − x) = 1 for x.
3. Find the equation of the tangent line of f (x) = ln x at x = e.
1
4. Suppose g is the inverse of f . Find g ′ (2) if f (x) =
5. Simplify sin(tan−1 x).
6. Find f ′′ (1) if f (x) = arctan(2x).
2
√
x3 + x2 + x + 1.
7. Find the inverse function of each function.
(a) y = ln(x + 3)
√
(b) y = e
x
(c) y =
1 + ex
1 − ex
(d) y =
10x
10x + 1
3
8. Find
dy
2
if y = (1 + 2x)x
dx
6x − 2x
.
x→0
x
9. Find the limit: lim
4
10. Polonium-210 has a half-life of 140 days.
(a) If a sample has a mass of 200 mg, find a formula for the mass that remains after t days.
(b) Find the mass after 100 days.
(c) When will the mass be reduced to 10 mg?
5
11. Find the limit: lim
x→∞
2x + 3
2x + 5
2x+1
.
1
2
12. Find all critical values for f (x) = x − x 3
3
6
13. If f (x) = x4 − 6x2 + 4,
(a) Find the interval(s) where f (x) is increasing and decreasing.
(b) Find the interval(s) where f (x) is concave up and concave down.
(c) Find the Inflection points of f (x).
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14. Consider f (x) = x ln x + 2x,
(a) Find the domain of f (x).
(b) Find lim+ f (x).
x→0
(c) Find the interval(s) where f (x) is increasing and decreasing and identify the local extrema
of f (x).
(d) Find the interval(s) of concavity of f (x).
8
2
15. Consider the function f (x) = xe−2x ,
(a) Find the interval(s) where f (x) is increasing and decreasing.
(b) Find the local extrema of f (x).
(c) Find the interval(s) of concavity of f (x).
(d) Find the point(s) of inflection for f (x).
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16. Sketch the graph of a function that satisfies the given conditions.
(a) f ′ (−1) = f ′ (1) = 0.
(b) f ′ (x) < 0 if −1 < x < 1.
(c) f ′ (x) > 0 if x < −1 or x > 1.
(d) f (−1) = 4.
(e) f (1) = 0.
(f) f ′′ (x) < 0 if x < 0.
(g) f ′′ (x) > 0 if x > 0.
10
17. Find the absolute maximum and absolute minimum values for f (x) = (x2 − 1)3 on the interval
[−2, 2].
11
18. Find all absolute and local extrema by graphing the function.
(a) f (x) = 1 − x2 ,
(
x2
(b) f (x) =
2 − x2
−3 < x ≤ 2
if − 1 ≤ x < 0
if 0 ≤ x ≤ 1
12
19. Suppose 36 square feet of material is available to make a box with a closed up. The length of the
base is 3 times the width. What are the dimensions of the box that maximize the volume?
13
20. If f ′ (x) = 3 cos x + 5 sin x and f (0) = 4, find f (x).
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