Math 105, Assignment 4
Due 2015-03-16, 4:00 pm
In the following problems you are expected to justify your answers unless stated otherwise.
Answers without any explanation will be given a mark of zero. The assignment needs to be
in my hand before I leave the lecture room or you will be given a zero on the assignment!
Don’t forget to staple your assignment! You may lose a mark if you do not.
1. Evaluate the following:
Z π/3
cos(2x) sin5 (x)
dx
(a)
tan2 x
π/6
Z π/4
tan5 (x)dx
(b)
Z0
y
(c)
dy
2
(y + 4y + 8)3/2
Z 4
t + 3t3 + 6
dt
(d)
t2 + 4t + 3
Z
cos x
(e)
dx
(sin x − 1)2 (sin x + 1)
Z
x
(f)
dx
2
x + 6x + 18
Hint: Let u = sin x
2. Evaluate the following improper integrals:
Z a
1
√
dx, a > 0
(a)
a2 − x 2
0
Z ∞
log x
(b)
dx
x3/2
1
R x2 −t
R∞
2
(c) 0 xe−x e 0 e dt dx
R x2
Hint: Let u = 0 e−t dt and be careful of the limits of integration.
3. (a) Find the approximation using 6 rectangles (n = 6) of
Z 1
2
ex dx
0
by using:
i. Midpoint rule
ii. Trapezoid rule
iii. Simpson’s rule
2
(b) Given that the fourth derivative of f (x) = ex satisfies
|f (4) (x)| ≤ 76e,
for all x ∈ [0, 1], find an upper bound on the absolute error in your approximation
for Simpson’s rule.
1
4. Solve the following initial value problems.
(a)
(b)
dy
dx
dP
dt
= xex
2 −2 log y
, y(0) = 0
= kP (N − P ), P (0) = 1
2