Student number
Name [SURNAME(S), Givenname(s)]
MATH 101, Section 212 (CSP)
Week 2: Marked Homework Assignment
Due: Thu 2011 Jan 20 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Find the following derivatives:
(b)
(c)
Rt
2
−1 (x + 2x + 4) dx.
R
2
dy/dx, where y = πsin x e−t dt.
R x3
g 0 (x), where g(x) = 2x
cos(t2 ) dt
R x3
R
R 3
0
(Hint: 2x cos(t2 ) dt = 2x
cos(t2 ) dt + 0x
(a) f 0 (t), where f (t) =
R
2
2. Find the mimimum value of f (x) = 0x −2x
and justify why the value is a minimum.
1
1+t4
cos(t2 ) dt).
dt. Express your answer as an integral,
3. Evaluate the following integrals.
R
3
(a) −1
(t2 + 5t − 1) dt
R2√
(b) 1 x dx
R
(c) 0π sin s ds
R
(d) 010 e−at dt, where a is a constant (consider a 6= 0 and a = 0 separately)
(e)
R 3/2
0
cos πx dx
4. Suppose a microbe population is changing at a rate of 2000 + t3 individuals per hour,
where t is in hours. Determine the net change in population during the time between
t = 2 and t = 4.
5. Suppose the velocity of a particle moving along a line is cos(πt) m/s, where t is measured in s.
(a) Find the displacement of the particle during the time period 0 ≤ t ≤ 1.
(b) Find the distance the particle travels during the time period 0 ≤ t ≤ 1.
(c) Are the answers to parts (a) and (b) the same? Explain why or why not, in terms
of the motion of the particle.
6. Evaluate the following integrals.
(a)
(b)
(c)
(d)
(e)
(f)
R
1
dx
4−3x
√
R5
x − 3 dx
4
R t2
√
dt
t−1
R 2 x−1
0 1+(x−1)4 dx
R4 s
√
ds.
0
2s+1
R π/(2ω)
−π/(2ω) cos ωt dt,
7. If f is continuous and
where ω > 0 is a constant.
R9
0
f (x) dx = 4, find
R3
0
xf (x2 ) dx.