Download Exam 1

APPM 1360
Exam 1
Fall 2015
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name, (2) 1360/Exam 1,
(3) lecture number/instructor name and (4) FALL 2015 on the front of your bluebook. Also make a grading
table with room for 4 problems and a total score. Start each problem on a new page. Box your answers. A
correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work
may receive partial credit. Justify your answers, show all work.
1. The following parts are not related:
Z ⇡/4
(a)(10 pts) Evaluate the integral
x sec2 (x) dx
0
(b)(10 pts) Evaluate the integral using the method of partial fractions:
Z
x2
(x
x+2
dx
1)(x2 + 1)
(c)(5 pts) Suppose f (x) is a di↵erentiable function defined on the interval 0 < a  x  b and suppose af (a) = bf (b),
Z b
Z b
prove that
f (x)dx =
xf 0 (x) dx.
a
a
2. Determine if the integral converges or diverges, find the value of the integral if it converges, show all work:
Z 4
Z 1
ln(x)
dx
p dx
(a)(6 pts)
(b)(6 pts)
dx
(x + 3)4
x
0
1
Determine if the integral converges or diverges, justify your answer:
Z 1
Z 1
sin4 (x) + e x
sec2 (x)
p dx
(c)(6 pts)
dx
(d)(6
pts)
2
x
x x
1
0
(Fun fact! You may or may not be interested to know that 1 < e < 3.)
3. The following problems are not related:
(a)(8 pts) For the given integrals, write down
substitution required to solve the
Z p the appropriate trigonometric
Z
dx
2
integral, do not solve the integrals: (i)
4x
25 dx
(ii)
2
x + 4x + 13
Z
(b)(8 pts) Find the antiderivative:
sin2 (✓) d✓ (Give your final answer in terms of the reference angle ✓.)
Z 1 p
1 x2
(c)(9 pts) Solve the following integral:
dx
x2
1/2
4. The following problems are not related:
(a)(8 pts) Find the area of the region enclosed by f (x) = x2 and g(x) = x2 ln(x), and the lines x = 1 and x = e.
Z 1
(b)(8 pts) Suppose we approximate
sin(⇡x) dx ⇡ T4 , find a bound for |ET |, the absolute error of this approxi0
mation without evaluating the integral or finding the approximation.
(c)(10 pts) The area between the curves y = x + 1 and y = x
2
x + 1 is given by the integral
Z
2
(2x
0
x2 )dx.
Suppose we approximate this area using the Midpoint Rule with n partitions. What is the minimum value of n
required to guarantee an error of at most 1/600? (Do not find the approximation.)
FORMULA SHEET ON THE OTHER SIDE
FORMULA SHEET/Exam 1 (1360 F’15)
Some identities
cos(2x) = cos2 (x) sin2 (x)
sin(2x) = 2 sin(x) cos(x)
2 cos2 (x) = 1 + cos(2x)
2 sin2 (x) = 1 cos(2x)
2 cosh2 (x) = cosh(2x) + 1
2 sinh2 (x) = cosh(2x) 1
Inverse Trigonometric Integral Identities
Z
du
p
= sin 1 (u/a) + C, u2 < a2
2
2
a
u
Z
du
1
= tan 1 (u/a) + C
2
2
a
Z a +u
du
1
p
= sec 1 |u/a| + C, u2 > a2
a
u u2 a2
Inverse Hyperbolic-Trig Integral Identities
Z
du
p
= sinh 1 (u/a) + C, a > 0
2
2
a +u
Z
du
p
= cosh 1 (u/a) + C, u > a > 0
2
2
u
a
Z
du
1
= tanh 1 (u/a) + C, if u2 < a2
a 2 u2
a
Midpoint Rule
Z
b
f (x)dx ⇡
a
where
x[f (x̄1 ) + f (x̄2 ) + · · · + f (x̄n )]
x =
b
K(b a)3
|EM | 
.
24n2
a
n
and x̄i =
xi
+ xi
and
2
1
Trapezoidal Rule
Z
b
a
x
[f (x0 ) + 2f (x1 ) + · · · + 2f (xn
2
K(b a)3
b a
x=
and |ET | 
.
n
12n2
f (x)dx ⇡
where
1)
+ f (xn )]