Math
Math20,
20,Spring
Spring2016
2017——Schaeffer
Schaeffer
Practice Midterm
1B 22nd, 2016)
Midterm
Exam 1 (April
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First/Given Name
Failure to follow the instructions below will constitute a breach of the Stanford
Honor Code:
• You may not use a calculator or any notes or book during the exam.
• You may not access your cell phone or any other electronics during the exam
for any reason.
• You may not sit directly adjacent to any other student.
• You are bound by the Stanford Honor Code, which stipulates among other
things that you may not communicate with anyone other than the instructor
or his designee(s) during the exam, or look at anyone else’s solutions.
• Additionally, you may not discuss or communicate directly or indirectly
the contents of this exam with ANY other students until noon today.
I understand and accept these instructions.
Signature: _______________________________________________________
Remember to show your work and justify your answer if required (additional tips
are on the next page). Present all solutions in as organized a manner as possible.
G OOD LUCK !
Problem
1.
2.
3.
4.
5.
6.
Grade Problem
7.
8.
9.
10.
11.
Total
Grade
—
Here are some tips:
• If you have time, it’s always a good idea to check your work.
• If you get the wrong answer for an integral but show your work, chances are good that we
can award you partial credit.
• DO NOT attempt to estimate any of your answers as decimals. For example, 1
better answer than 0.682, because it is exact.
1
⇡
is a much
• The boxes at the end of each topic are for grading purposes only. Do not touch or look at
these boxes. Pretend they are not there.
• The last page of the exam is blank, and can be used for extra work. If you think it would
help for us to look at this work, you should indicate that CLEARLY on the problem’s page.
Integration table entries you might need:
Z
du
1
u a
=
ln
+C
2
2
u
a
2a
u+a
Z p
a2
u2
1 p
du = u a2
2
Trigonometric identities you might need:
sin2 x + cos2 x = 1
sin2 x =
1
sin2 x = 1
cos(2x)
2
REMEMBER that sinn x means (sin x)n .
u2
⇣u⌘
a2
+ arcsin
+C
2
a
cos2 x
cos2 x =
cos2 x = 1
1 + cos(2x)
2
sin2 x
Unless otherwise specified, no justification or explanation is necessary for problems 1–6.
8 p
< 4 t2 if 0  t < 2
1. Below is the graph of y = f (t) where f (x) =
t + 2 if 2  t < 3
:
3
if 3  t < 5
Let F (x) =
Z
x
f (t) dt.
0
a. What is the value of F (2)? (Hint: Use the picture, not a formula.)
Draw a box around your answer. You do not need to justify your answer.
b. Which of the following quantities is negative? Circle all correct answers.
F (2) + F (3)
F (3)
F (2)
c. On which of the following intervals is F (x) increasing? Circle all correct answers.
(0, 2)
(2, 3)
(3, 5)
d. How many points a 2 [0, 5] are there where F 0 (a) = 0? Circle the correct answer.
i. There aren’t any.
ii. Just one.
iii. Two or three.
iv. Infinitely many.
2. What is the name of the formal mathematical proposition asserting an inverse relationship
between integration and differentiation?
3. Isaac and Gottfried are testing each other’s integration mettle. The first challenge question
in their contest is a rather simple one,
Find an antiderivative for (x + 1)3 .
Here’s how these two intellectual gladiators solved this problem:
ISAAC
Expanded (x + 1)3
and obtained x3 + 3x2 + 3x + 1.
Integrated term-by-term,
and chose C = 0 to obtain
1 4
x + x3 + 32 x2 + x.
4
GOTTFRIED
Substituted uR= x + 1 and du = dx.
Evaluated u3 du = 14 u4 + C.
Re-substituted u = x + 1, expanded
1
(x + 1)4 + C, and chose C = 0 to get
4
1 4
x + x3 + 32 x2 + x + 14 .
4
Their answers are different: Gottfried’s includes an extra term of 14 . How is this possible?
Circle the true statement below.
i. Gottfried is incorrect.
ii. Isaac is incorrect.
iii. They are both correct.
Briefly explain the answer you circled above.
4. Which table entry below would be useful for evaluating the following integral?
Z
Circle the correct answer.
Z
du
p
i.
= ···
u 2 a2
Z
du
p
ii.
= ···
a2 u 2
Z p
iii.
a2 u2 du = · · ·
Z p
iv.
u2 a2 du = · · ·
(7
3x
2x2 )
1/2
dx
(For brevity we have only provided the left-hand side of the table entry.)
(You do NOT have to evaluate the integral above.)
5. Under each of the integrals below, write the letter (A.–G.) corresponding to its solution.
Z
tan u du
________
Z
du
2
u +1
________
Z
p
du
1 u2
________
A. arctan (u) + C
B. ln |sec u| + C
C.
1
3
sec3 u + C
D. tan u + C
E. arcsin u + C
F. arccos u + C
G.
1
2
ln |u2 + 1| + C
6. Fill in the blank:
Z
dx = x3 + C
Z
u2
u
du
+1
________
For problems 7–11, show your work. If you use u-substitution, clearly indicate your choices
of u and du. If you use integration by parts, clearly indicate your choices of u, du, v, dv.
If you need a table entry to evaluate a given integral, please refer to the second page of the
exam or the appropriate answer in Problem 5.
Z
7. Evaluate the integral e3x dx.
Draw a box around your final answer.
8. Evaluate the integral
Z
4x2 + 1
p
dx.
x
Draw a box around your final answer.
9. Evaluate the integral
Z
10. Evaluate the integral
Z
Draw a box around your final answer.
x ln x dx.
4x2
(Hint: What is (2x + 1)2 ?)
dx
+ 4x
3
.
Draw a box around your final answer.
11. Evaluate the integral
Z
sin2 x dx.
Draw a box around your final answer.
(Hint: Start by applying one of the trig identities on the second page.)