Note: I had written a different question at first for which the instruc
tions were misleading.
2. Compute the following definite integrals. Simplifying is optional.
•
Z
π
2
/
4
0
1
√
x
e
2 sin
√
x
cos
√
x dx
•
Z
√
π/
3
√
π/
6
x
cos
x
2
dx
3. Find the area of the region bounded by the
y
=
√
x
and
y
=
x
3
.
4. Compute the following indefinite integrals.
100
CHAPTER 6.
QUIZZES AND MIDTERMS
•
Z
(ln
x
)
2
dx
•
Z
t
2
e
t
dt
5. Does the following integral converge or diverge? If it converges, com
pute its value.
Z
∞
e
e
1
x
ln
x
(ln ln
x
)
2
dx.
6. (Extra Credit) Consider the logistic equation
dx
dt
=
rx
(1

x
)
.
Find an explicit formula for
x
in terms of
t
by remembering how we
went from
dy/dx
=
ky
to
y
=
y
(0)
e
kx
.
6.3.2
Midterm 1
1. Suppose a medication has a halflife of 24 hours. A particular patient
takes 50mg every day at the same time for several months. Approx
imately how much of the drug will be in the patient’s system right
before
taking the daily dose?
Use (you don’t have to derive it) an
infinite series formula.
2. Compute the following definite integrals. Simplifying is optional.
•
Z
e
2
1
3
x
(ln
x
)
4
dx
•
Z
3
√
π/
3
0
x
2
tan(
x
3
)
dx
3. Find the area of the region bounded by
x
=

π/
6,
x
=
π/
6,
y
= sec
x
and
y
= cos
x
+ 1.
4. Compute the following indefinite integrals.
6.3.
MIDTERMS
101
•
Z
arctan
x dx
•
Z
1
x
+
x
2
dx
5. Does the following integral converge or diverge? If it converges, com
pute its value.
Z
∞
0
e

x
sin
x dx.
6. (Extra Credit) Show that
∑
∞
k
=1
1
k
2
is less than 2 by comparing it to a
series or integral that you know how to compute exactly. Fun fact, it’s
exactly
π
2
/
6.
6.3.3
Practice Midterm 2
1. Find the seconddegree Taylor approximation to the function
f
(
x
) =
sec
x
centered at
x
=
π/
4, a.k.a. with
a
=
π/
4.
2. (a) Estimate the integral
Z
2
0
sin(
x
)
dx
using the Trapezoid Method with
n
= 5.
(b) Give an upper bound on the error of this approximation.
(c) Compute the integral exactly. What does your estimate say about
the value of cos 2?
3. Solve the following initial value problems.
(a)
dN
dt
=
N
(1

N
)
,
N
(0) = 1
/
2
(b)
ds
dt
=
1
√
t
+ 1
,
s
(0) = 1
102
CHAPTER 6.
QUIZZES AND MIDTERMS
4. Graphically analyze the autonomous differential equation
dL
dt
= 3(
L
+ 1)
3
(
L

1)
2
(3

L
)
in order to find the equilibria and classify their stability.
Sketch the four solutions with
L
(0) =

2
,
0
,
2
,
4.
5. Use the standard Ansatz to solve the following secondorder initial value
problems.
(a)
y
00

y
0

2
y
= 0
,
y
(0) = 0
,
y
0
(0) = 2
(b)
y
00
+
y
0
+
1
4
y
= 0
,
y
(0) = 1
,
y
0
(0) = 1
(c)
y
00

2
y
0
+ 2
y
= 0
,
y
(0) = 1
,
y
0
(0) =

1
6. (Extra Credit) (a) Write down a differential equation with two stable
equilibria, one unstable equilibrium, and two saddle points.
(b) Show that the solution to the differential equation
dy
dx
=
√
y,
y
(0) = 0
is not unique. In other words, find two distinct solutions with the same
initial condition.
Remark:
The fact that the derivative of
√
y
blows up at 0 is what
allows for this strange phenomenon.
6.3.4
Midterm 2
1. Find the thirddegree Taylor approximation to the function
f
(
x
) =
e
x
centered at
x
= 1, a.k.a. with
a
= 1.