Sivakumar MATH 152H
Example Sheet 1a
1. Suppose R > 0 is fixed. Evaluate the integral ZR
0pR2−x2dx, by interpreting it as an area.
2. Compute
lim
h→0
1
hZ2+h
2p1 + t3dt.
(Hint: Consider the function F(x) := Rx
0√1 + t3dt, use this function to write the expression in
the question as a difference quotient, and appeal to the first part of the Fundamental Theorem
of Calculus.)
3. Given that
F(x) = Zx
0
x2sin(t2)dt,
compute F0(x).
4. Compute
d2
dx2Zx
0"Zsin t
1p1 + u4du#dt.
5. If
f(x) = Zg(x)
0
dt
√1 + t3and g(x) = Zcos x
01 + sin(t2)dt,
determine the value of f0(π/2).
6. Show that the function
f(x) := Zx
1p1 + t2dt
is one-to-one. Compute (f−1)0(0).
The following pair of questions is taken from the book Basic Analysis: Japanese Grade 11 , trans-
lated and published by the American Mathematical Society.
7. Find a function fand a number asuch that
Zx
1
f(t)dt =x3+ax −5.
8. Find the function fsuch that
f(x) = x+Z2
0
f(t)dt.
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