MAT 21B Final Exam, Wednesday, March 20, 2013
Last Name:
First Name:
Student ID #:
Discussion Section Time: (Circle 3pm, 4pm, 5pm, 6pm, or 7pm)
Name of Left Neighbor:
Name of Right Neighbor:
If you are next to the aisle or wall, then please write “aisle” or
“wall” appropriately as your left or right neighbor.
• Read each problem carefully.
•
Write every step of your reasoning clearly.
• Usually, a better strategy is to solve the easiest problem first.
• This is a closedbook exam. You may not use the textbook, crib sheets, notes, or any other outside
material. Do not bring your own scratch paper. Do not bring blue books.
• No calculators/laptop computers/cell phones are allowed for the exam. The exam is to test your basic
understanding of the material.
• Everyone works on their own exams. Any suspicions of collaboration, copying, or otherwise violating
the Student Code of Conduct will be forwarded to the Student Judicial Board.
Problem #
Score
1
(10 pts)
2
(10 pts)
3
(10 pts)
4
(10 pts)
5
(10 pts)
6
(10 pts)
7
(10 pts)
8
(10 pts)
9
(10 pts)
10
(10 pts)
Total
(100 pts)
2
Problem 1
(10 pts)
Consider the following simple definite integral:
Z
1
0
x
2
d
x
=
1
3
.
(a)
(5 pts) Find the numerical approximation of the above integral using the
trapezoidal rule
,
T
n
,
for arbitrary
n
∈
N
, and
simplify
the resulting expression so that you have a formula for
T
n
in terms of
n
(and no summation symbols or other variables).
(b)
(3 pts) Define the absolute error
E
T
n
:
=
fl
fl
fl
fl
1
3

T
n
fl
fl
fl
fl
. Then, find the
upper bound
of
E
T
n
derived
from the error estimate theorem involving the second derivative of the integrand, and show
that this upper bound is
achievable
in this problem using the result of Part (a).
Z
b
a
f
(
x
)d
x
states that
E
T
n
≤