Midterm 2 Exam - Midterm II Laurence Field Math 152,...

Midterm II
Laurence Field
Math 152, Section 31
February 24, 2012
Name:
Instructions: This midterm has 6 questions for a total of 60 points.
The value of each part of each question is stated.
No books, notes or calculators are allowed.
Please show your working.
Partial credit will be awarded to partially correct answers so long as your working is legible.
Question Points Score
1 10
2 10
3 10
4 10
5 10
6 10
Total: 60
i

1. Compute the following values.
(a) (2 points)
Z
e
5
1
1
t
dt
(b) (4 points) The average value of the function f ( x ) =
1
x
2
on the interval [ a, b ], where 0 < a < b .
3
(c) (4 points)
- π
x cos x + 1 dx
1

2. (a) (3 points) What does it mean to say “the function f is one-to-one”?
(b) (3 points) Give the definition and the derivative of the natural logarithm function, L ( x ) = log( x ) =
log
e
( x ) = ln( x ) (using whichever notation you prefer).
(c) (2 points) If f is a continuous function on [ a, b ], which of the following is necessarily true? Circle your
answer. No justification needed.
( i )
Z
b
a
f ( x ) dx ≥
Z
b
a
| f ( x ) | dx
( ii )
Z
b
a
f ( x ) dx =
Z
b
a
| f ( x ) | dx
( iii )
Z
b
a
f ( x ) dx ≤
Z
b
a
|