MATH 151, Fall 2013
FINAL EXAMINATION - SOLUTIONS
Part 1 – Multiple Choice (14 questions, 4 points each, No Calculators)
Write your name and section number on the ScanTron form.
Mark your responses on the ScanTron form and on the exam itself
1.
A ball whose weight is F
50 Newtons hangs from
two wires, one at angle 30° above horizontal on the left,
and the other at angle 60° above horizontal on the right.
Let T 1 be the tension in the first wire, and T 2 be the
tension in the second wire. Which set of equations can
be used to solve for
a.
b.
c.
d.
e.
3
1 T2
T1
2
2
3
1 T1
T2
2
2
3
1 T2
T1
2
2
3
1 T1
T2
2
2
3
1 T1
T2
2
2
SOLUTION: F g
T1
T2
T 1 cos 30° î
T 2 cos 60° î
and
T1
0
0
0
0
0
3
1 T1
T2
2
2
3
1 T2
T1
2
2
3
1 T1
T2
2
2
3
1 T2
T1
2
2
3
1 T2
T1
2
2
3
T1 î
2
T 1 sin 30°
cos
b.
cos
c.
cos
d.
cos
e.
cos
56
65
4
65
16
65
4
65
56
65
SOLUTION: cos
50
50
50
50
T 2 sin 60°
1 T1
2
3
1 T2 î
T2
2
2
components of the sum.
Find the cosine of the angle between the vectors A
a.
Correct Choice
50
50
The equations are the î and
2.
T2 ?
3, 4
and B
12, 5
Correct Choice
A B
A B
36 20
5 13
16
65
1
3.
x4
Evaluate lim
x
a.
b.
c.
d.
e.
8
4
2
4x 3
Correct Choice
x4
x
x
4
4x
x
4
x4
5
4x 3
5
3
x
5
4
x
4x
4
3
4x
5
3
lim
x
5
x
x4
4x 3
5
x4
4x 3
5
x4
4x 3
5
x4
4x 3
5
8x 3
5
10
x4
x
4
4x
3
4x
x3
f0
0
Find the 2017-th derivative of f x
x
a.
b.
c.
d.
e.
2014! x 2015
2016! x 2017
2015! x 2016
2016! x 2017
2015! x 2016
2x
1
x
1
2017
f
f
c.
d.
e.
4
5
3
5
f1
2x
3
8
2
4
3
f2
4 is between 3 and 12.
12
ln x
x
x
x 2
2016! x 2017
Find the line tangent to the curve y 3
b.
5
3
Correct Choice
SOLUTION: f x
f 5 x
4 3 2x 5
a.
3
x
x
2, 1
1, 0
0, 1
1, 2
Correct Choice
SOLUTION: f x
x2
f
x
xy at
2x
3
f
4
x
3 2x
3
5
4
5
2
Tan Line:
y
f 2
f
2 x
2
2
dy
dx
3 x
5
2
10
2
4
2, 2 . Its y-intercept is
x, y
Correct Choice
SOLUTION: By implicit differentiation:
dy
dy
dy
3y 2
2x y x
12
4 2
dx
dx
dx
2
5
x
3
4x
4x 3
Which interval contains the unique real solution of the equation x 3
a.
b.
c.
d.
6.
5
0
lim
x
5.
4x 3
x
SOLUTION: lim
x
4.
x4
5
dy
dx
dy
dx
6
3x
5
4
5
b
2,2
4
5
3
5
4
7.
Near x
2, the graph of f x
x
2
1/5
has the shape:
b.
a.
c.
d.
Correct Choice
e None of these
SOLUTION:
lim
x
“ 1 ”
0 1/5
1/5
2
x 2
8.
x
2
1/5
“ 1 ”
0 1/5
lim x sin x
x 0 cos x
1
a.
b.
c.
d.
e.
2
1
Correct Choice
0
1
2
SOLUTION: lim x sin x
x 0 cos x
1
9.
lim
x 2
l’H
lim sin x
x 0
x cos x
sin x
l’H
lim cos x
x 0
cos x x sin x
cos x
2
Water is poured onto a tabletop at dV
3 cm 3 / sec in such a way that it forms a circular
dt
puddle of radius r and constant height of h 0. 2 cm. Find the rate that the radius is
increasing when it is r 5 cm.
HINT: The puddle is actually a cylinder. What is its volume?
a.
b.
1
2
2
3
c.
d.
e.
3
2
2
Correct Choice
SOLUTION: V
r2h
0. 2 r 2
dV
dt
0. 4 r dr
dt
10. Find the points of inflection of the function f x
a.
b.
c.
d.
e.
x
x
x
x
x
3
1 x5
20
0. 4 5 dr
dt
1 x4.
3
dr
dt
3
2
0 and x 4 only
4 only
Correct Choice
0 only
2 and x 2 only
4 and x 0 only
1 x4 4 x3
f x
x 3 4x 2 x 2 x 4
0
4
3
Potential inflection points are x 0, 4 but the concavity does not change at x
SOLUTION: f x
0.
3
11. If the initial position and velocity of a car are x 0
at
sin t
7
3
6
7
3
a.
b.
c.
d.
e.
6t find its position at t
sin 1
cos 1
sin 1
sin 1
cos 1
Correct Choice
v0
1
3
xt
is the area under the parabola y
6
8
9
11
13
a.
b.
c.
d.
e.
2
z
x 2 dx
2
x 2 between x
z
d
dz
A z
1
21
24
27
30
36
1 and x
3
z find A 3 .
x 2 dx
2
z 2 by the F.T.C.
2
1
x 2 between x
2
4
x 2 dx
2
2x
1
1 and x
4
x3
3
64
3
8
1
1
3
2
1 x 2 between x
Right Riemann Sum with 3 equal intervals and right endpoints.
14. Approximate the area under the parabola y
38
76
86
120
172
1
b
7
n
3
xi a i x
x1 1 2 3
x2
f x1
1 3 2 10
f x2
1 52
x
b
n
a
7
1 2 2 5
26
f x3
1
3
x3
1
f xi
i 1
6
x
2 f x1
f x2
f x3
2 10
26
50
172
63
3
1 and x
2
1
72
3
A
4.
Correct Choice
SOLUTION: a
4
3t 2
cos t
Correct Choice
SOLUTION: A
a.
b.
c.
d.
e.
vt
11
13. Find the area under the parabola y
a.
b.
c.
d.
e.
C 2
C 3
3
sin t t 3t 3
Correct Choice
SOLUTION: A z
A 3
2 and its accereration is
1.
SOLUTION: v t
cos t 3t 2 C
xt
sin t t 3 3t K
x0
K
x1
sin 1 1 3 3 7 sin 1
12. If A z
3 and v 0
3 2
50
7
27
7 using a
Part 2 – Work Out Problems (3 questions. Points indicated. No Calculators)
Solve each problem in the space provided. Show all your work neatly and concisely, and
indicate your final answer clearly. You will be graded, not merely on the final answer, but also on
the quality and correctness of the work leading up to it.
xe x and g x
15. (10 points) Let f x
SOLUTION: If b
g e then e
fb
ex
xe x and f 1
e1
Further f x
f
1
x
the inverse function of f x . Find g e .
be b . So b
1e 1
1.
1
f 1
2e. So g e
1 .
2e
W
(10 points) A rectangular field is surrounded by a fence and
16.
divided into 3 pens by 2 additional fences parallel to one side.
If the total area is 50 m 2 find the minimum total length of fence.
L
SOLUTION: A
4
100
L2
L2
LW
50
25
L
50
L
W
5
W
F
50
5
4L
10
2W
F
4L
4 5
100
L
2 10
F
4
100
L2
0
40
5
2
xe 2x . Find each of the following. If none
17. (30 points) Analyze the graph of the function f x
or nowhere or undefined, say so.
a.
b.
c.
d.
e.
f.
g.
h.
y-intercepts:
0
y f0
1
x-intercepts:
2
xe 2x
0 when x
horizontal asymptote as x
2
lim xe 2x
lim x 2
x
x
e 2x
k.
6
lim
x
1
1
0
2
e 2x 4x
:
l’H
x
lim
1
2
e 2x 4x
1
0
vertical asymptotes:
none
first derivative:
2
f x
e 2x
critical points:
f x
0
1
1
2
xe 2x
4x 2
1
3e 2
2
e 2x 1
4x
0
intervals where increasing:
Put the critical points x
increasing on
j.
1
:
l’H
horizontal asymptote as x
2
x
lim xe 2x
lim
2
x
x
2x
e
f
i.
0 only
0 f 0
1
1
2
x
1
2
1
on a number line and test the 3 intervals:
1
0 f 1
1, 1
2 2
intervals where decreasing:
decreasing on
, 1
2
4x 2
3e 2
0
3
and
local maxima x and y coordinates :
1 e 1/2
f 1
2
2
1 , 1 e 1/2
local maximum at
2 2
local minima x and y coordinates :
1
1 e 1/2
f
2
2
1 , 1 e 1/2
local minimum at
2
2
1,
2
2
2
2
l.
Roughly graph the function:
5
1
y
-4
-3
-2
-1
1
2
3
4
x
-1
m. second derivative:
f
n.
2
4x 2 e 2x
1
0,
3 ,
4
4x 2
2
3
4
2
intervals where concave up:
Put the inflection points x
f
1
4e 2
concave up on
p.
2
4xe 2x 3
4x
inflection points x coordinate only :
x
o.
2
8xe 2x
x
0,
1
2
0 f
3 ,0
4
3
4
on a number line and test the 4 intervals:
4e 1/2
0 f
3 ,
4
and
1
2
4e 1/2
0 f
1
4e 2
0
3
intervals where concave down:
concave down on
,
3
4
and
0,
3
4
2
7