# CAS MA 124 Study Guide - Midterm Guide: Antiderivative, Improper Integral, Indeterminate Form

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10 Oct 2018

School

Department

Course

Professor

MA 124 CALCULUS II C1, Solutions to First Midterm Exam

Prof. Nikola Popovic, February 16, 2006, 08:00am - 09:20am

Problem 1 (15 points).

Determine whether the statements below are true or false. If a statement is true, explain why; if it

is false, give a counter-example.

(a) If fand gare continuous on [a, b], then

Zb

a

[f(x)·g(x)]dx =Zb

a

f(x)dx·Zb

a

g(x)dx.

(b) If fis a continuous, decreasing function on [a, ∞) and limx→∞ f(x)= 0, then Z∞

a

f(x)dx

is convergent.

(c) All continuous functions have antiderivatives.

Solution.

(a) FALSE. Take e.g. f(x)=x,g(x)=x, and [a, b] = [0,1]. Then,

Zb

a

[f(x)·g(x)]dx =Z1

0

x·x dx =Z1

0

x2dx =x3

3

1

0

=1

3,

but

Zb

a

f(x)dx·Zb

a

g(x)dx=Z1

0

x dx·Z1

0

x dx=x2

2

1

0·x2

2

1

0

=1

2·1

2=1

4.

(b) FALSE. Take e.g. f(x)=1

xand a= 1; then, f(x) is continuous on [1,∞), limx→∞ f(x)= 0,

and f(x) is decreasing on [1,∞) (since f0(x)=−1

x2<0), but

Z∞

1

1

xdx =lim

t→∞ Zt

1

1

xdx =lim

t→∞ ln x

t

1

=lim

t→∞ ln t=∞.

Hence, the improper integral diverges.

(c) TRUE. By the Fundamental Theorem of Calculus, Part I, we know that if f(x) is continuous

on [a, b],

g(x)=Zx

a

f(t)dt

is an antiderivative of f, i.e., g0(x)=f(x). (Note that this does not necessarily imply that we

can compute gexplicitly; take e.g. f(x)= ex2. However, even in that case, we know that g

exists.)