MATH 151 Honors, FALL SEMESTER 2011
FINAL EXAMINATION
Name (print):
Instructor’s name: Yasskin
Signature:
Section No:
Part 1 – Multiple Choice (15 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.
Evaluate
a.
b.
c.
2.
3.
2
x
6x
4
0
5
2
6
The limit
a.
b.
c.
d.
e.
x
x2
2
3
d.
e.
3
lim x
f
f
f
f
f
lim
42
h 0
32 where f x
2 where f x
4 where f x
2 where f x
2 where f x
h 3
h
c.
d.
e.
can be interpreted as which of the following?
4x 3
4x 3
x3
12x 2
x4
Find the line tangent to y
a.
b.
32
sinh x at x
1. Its y-intercept is
0
e
2
e
1
e
1
2e
1
4.
The function f x
shape:
x2
x2
a.
5.
c.
Find the absolute minimum value of f x
a.
b.
c.
d.
e.
3
2
0. 806
4
0. 785
sin x
2
2. Near x
2, the graph has the
d.
6
x on the interval 0,
2
.
1. 570
1
2
3
2
4
1
2
2
3
2
4
1
2
2
If $1000 is invested at 2% annual interest compounded continuously, how long will it take for the
principal to reach $1000e $2718. 28?
a.
b.
c.
d.
e.
2
6 has a vertical asymptote at x
4
b.
NOTE:
6.
5x
4x
5 years
10 years
25 years
50 years
100 years
7.
A rectangular field is surrounded by a fence and divided into
W
3 pens by 2 additional fences parallel to one side. If the total
area is 72 m 2 find the minimum total length of fence.
a. 48
b. 50
c. 52
d. 60
L
e. 66
8.
Let f x be a differentiable function, and suppose f 1
5 and f x
7 for all values of x. Use
the Mean Value Theorem to determine how small f 4 can possibly be.
a.
b.
c.
d.
e.
9.
26
26
20
35
Not enough information.
When you prove lim 3x
x 4
a.
b.
c.
d.
e.
5
7, and pick
0, the largest possible
is
3
2
2
3
3
10. When you use a left Riemann sum with 3 intervals of equal length to approximate ln 7
7
1
you discover:
HINT: Draw a picture.
a.
ln 7
b.
ln 7
c.
ln 7
d.
ln 7
e.
ln 7
46
15
46
15
11
6
11
6
23
15
/4
11. Evaluate
cos 3x dx
/4
a.
b.
c.
d.
e.
2
3 2
0
2
3
2
2
12. If I
a.
b.
c.
d.
e.
4
0
2
0
2
2
0
5
0
0
2
2
2 x dx, which of the following is FALSE?
2
3 2 x dx
I
3I
32
2 x
2
dx
2 x dx
2
2 x dx
2
I
5
2
2
2 x dx
I
I
1 dx
x
d
dx
13. Compute
a.
b.
c.
d.
e.
x3
sin t 2 dt
sin x 4 2x sin x 6 3x 2
cos x 2 2x cos x 3 3x 2
sin x 4
sin x 6
cos x 2
cos x 3
4
cos x 2x cos x 6 3x
14. Recall:
Then
a.
b.
c.
d.
e.
x2
sinh x
ex
cosh x cosh 2x
e
2
x
cosh x
ex
e
2
.
sinh x sinh 2x
sinh 3x
sinh x
cosh 3x
cosh x
cosh x
15. Find the intervals of concavity of the function f x
a.
b.
c.
d.
e.
x
12 12
x2
1
.
Concave up: 2,
Concave down:
, 2
2, 2
Concave up:
, 2
2,
Concave down: 2, 2
Concave up: 2, 2
Concave down:
, 2
2,
Concave up:
,
Concave down: nowhere
Concave up: nowhere
Concave down:
,
5
Part 2 – Work Out Problems (5 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.
1
12 12 x 2 . (See problem #15.)
Be sure to label all maxima, minima, inflection points and asymptotes.
Be careful with intervals of increase, decrease and concavity.
16. (6 points) Graph the function f x
y
1.0
0.5
-5
-4
-3
-2
-1
1
2
3
4
5
x
-0.5
-1.0
17. (8 points) Let g x be the inverse function of f x
6
1 for x
ln x
0. Find g 1 and g 1 .
18. (10 points) The pressure and volume of the gas in an engine cylinder are related by PV 3/2
C
for some constant C. Currently, the pressure is P 15 lb/in and the volume is V 4 in but is
increasing at the rate of dV
2 in 3 /min. Find C and the current value of dP . Is the pressure
dt
dt
currently increasing or decreasing?
2
3
19. (6 points) Find a parametric equation for the line perpendicular to the parametric curve
rt
t 2 , t 3 at the point 4, 8 .
7
20.
y
(12 points) A rectangle is inscribed in the upper half of
2
y2
the ellipse x
1 with its base on the x-axis.
16
9
Find the maximum area of such a rectangle.
4
2
-4
-2
2
-2
-4
Question
1-15
/60
16
/6
17
/8
18
/10
19
/6
20
/12
Total
Name (print):
8
Points/Max
/102
Section No:
4
x