Math 105, Assignment 3
Due 2015-02-25, 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 will lose a mark if you do not.
1. Use the definition of integral and the mid point rule to to show that
Z b
b3
x2 dx = .
3
0
You cannot use fundamental theorem to solve this.
2. Evaluate the following sum.
2i
n
X
ie n
lim
n→∞
i=1
n2
Hint: Find a suitable function f and limits a, b and write the above in the form
Z b
f (x)dx
a
and apply fundamental theorem.
3. Find the derivative of
Z
ex
2
log(t2 )dt
log x
4. Find the regions where f (x) is increasing
Z
f (x) =
0
x2
1−t
dt
1+t
5. Evaluate the following:
Z
x
dx
(a)
x+1
Z
(b)
x2 (x + 1)3/2 dx
Z
(c)
3
√
9 − x2 dx (Hint: use geometry)
−3
2015
Z
sin(x2015 ) − x2015 dx (Hint: don’t use FTC)
(d)
−2015
Z
(e)
(log x)2 dx
1
Z
π2
(f)
√
sin( x)dx
0
Z
(g)
t
es cos(t − s)ds
0
2