MATH 101 HOMEWORK 3
Due on Wednesday Sept. 29.
For full credit, show all work. Calculators are not allowed.
1. (4 marks) 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?
2. (6 marks) Evaluate the integrals:
Z 3 2 √
x − x+1
dx,
(a)
x
1
Z
(b) (cos3 x − 2 cos x) sin x dx.
3. (4 marks) 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.
Z
sin x p
4. (6 marks) Let F (x) =
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.
(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, . . . .
(c) Find F 0 (x). What is F 0 (0)?
(d) Find all critical points of F . Determine the intervals where F is increasing and the
intervals where it is decreasing.
(e) Based on the above information, sketch roughly the graph of F (x). (In this problem,
the maximum and minimum values of F , or the concavity information based on F 00 (x),
are not required.
This homework covers Sections 5.4–5.6 of the textbook. Additional recommended exercises:
Section 5.4, 1–38; Section 5.5: 1–46, 49–50; Section 5.6: 1–48. How much practice you
need will depend or your background and your committment to the subject – use your own
judgement.
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