MATH 101 HOMEWORK 3 – SOLUTIONS
1. We know that the average value of f (x) = 2x3 − 1 on some interval [a, b] is 1. Prove
that a ≤ 1. What can we say about b? – Done in class.
2. Evaluate the integrals:
Z 3 2 √
Z 3
x − x+1
x2
x1/2
−1/2
−1
(a)
(x − x
+ x )dx = ( −
dx =
+ ln x)|31
x
2
1/2
1
1
=
√
√
9 1
− − 2 3 + 2 + ln 3 − ln 1 = 6 − 2 3 + ln 3.
2 2
Z
(b)
(cos3 x − 2 cos x) sin x dx – Done in class.
3. Find a positive number m such that the area of the bounded planar region enclosed by
the line y = mx and the curve y = x3 which lies in the first quadrant (i.e. x, y ≥ 0) is 18.
√
The line y = mx intersects y = x3 if mx = x3 , ie. x = 0 or x = ± m. So the area in
question is
Z √m
x4 √m
m2
m2
mx2
m2
− |0 =
−
=
.
(mx − x3 )dx =
2
4
2
4
4
0
√
√
This is 18 if m2 = 4 · 18 = 72, m = 72 = 6 2.
Z
sin x
4. Let F (x) =
p
1 + t3 dt.
0
(a) Check that F (kπ) = 0 for k = 0, ±1, ±2, . . . , and that F (x) is periodic with period 2π,
i.e. F (x + 2π) = F (x) for all x.
Z
F (kπ) =
Z
sin(kπ) p
1+
Z
t3
dt =
0
Z
sin(x+2π) p
0
p
1 + t3 dt = 0,
0
1 + t3 dt =
F (x + 2π) =
0
sin x
p
1 + t3 dt = F (x).
0
(b) Check that F (x) > 0 on all intervals (2kπ, (2k + 1)π) and F (x) < 0 on all intervals
((2k + 1)π, (2k + 2)π), k = 0, ±1, ±2, . . . .
√
Since 1 + t3 ≥ 0 for all −1 ≤ t ≤ 1 and = 0 only for t = −1, F (x) is positive if and
only if sin x > 0, ie. x is in (2kπ, (2k + 1)π), k = 0, ±1, ±2, . . . , as required. For the same
reason, F (x) is negative if and only if sin x < 0, ie. on all intervals ((2k + 1)π, (2k + 2)π),
k = 0, ±1, ±2, . . . .
(c) Find F 0 (x). What is F 0 (0)?
1
F 0 (x) =
p
1 + sin3 x · cos x, F 0 (0) = 1.
(d) Find all critical points of F . Determine the intervals where F is increasing and the
intervals where it is decreasing.
0
F (x) is differentiable everywhere, and
p F (x) = 0 when cos x = 0 of sin x = −1, ie. x =
π/2 + kπ, k = 0, ±1, ±2, . . .. Also, 1 + sin3 x ≥ 0, so F 0 (x) has the same sign as cos x.
Thus F (x) is increasing on [−π/2 + 2kπ, π/2 + 2kπ] and decreasing on [π/2 + 2kπ, 3π/2 +
2kπ].
(e) Based on the above information, sketch roughly the graph of F (x).
As announced in class, Questions 1 and 2(b) were dropped from the homework when the
Sept 27 class was cancelled.
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