Download Arrington Sample Math 0120 Examination #2 Sample Name (Print

Arrington Sample
Math 0120
Examination #2
Sample
Name (Print)
PeopleSoft #
Signature
Score
.
.
TA (Circle one)
Instructions:
1. Clearly print your name and PeopleSoft number and sign your name in the space
above.
2. There are 7 problems, each worth the specified number of points, for a total of
100 points. There is also an extra-credit problem worth up to 5 points.
3. Please work each problem in the space provided. Extra space is available on the
back of each exam sheet. Clearly identify the problem for which the space is
required when using the backs of sheets.
4. Show all calculations and display answers clearly. Unjustified answers will receive
no credit.
5. Write neatly and legibly. Cross out any work that you do not wish to be considered
for grading.
6. No calculators, earphones, tables, books, notes, or computers may be used.
All derivatives are to be found by learned methods of calculus.
7. Formulas that may be useful:
Rectangle with length = L and width = W: Perimeter = 2L+ 2W; Area = LW
Square with side =X: Perimeter = 4X; Area = X2
Rectangular box with square base of side = X and height = Y:
Open: Surface Area = X2 + 4XY; Volume = X2Y
Closed: Surface Area = 2X2 + 4XY; Volume = X2Y
Cube with edge or side = X: Volume = X3; Surface Area = 6X2
4
3
Sphere with radius = R: Volume = R 3 ; Surface Area = 4R 2 .
Arrington Sample
1. (24 pts.) Find the derivatives of the following functions (you need not simplify):
(a) f(x) =
e x
ln( 3  x )
ln( 3  x )e x
f’(x) =
(b) f(x) =
 1 
e x

2 x
3x
1
ln( 3  x)2
3
ln( x 4  4)


2
  4x 3 
1
4

f’(x) =
ln( x  4 3 
4


3
 x  4
(c) f(x) = 2x + (log3x)-2 + 3
1
f’(x) = (ln2)2x -2 (log3x)-3 x(ln 3)
Arrington Sample
2. (12 pts.) LMN Company finds that it costs $100 to produce a certain gidget and that fixed costs are
$400 per day. The price function is p(x) = 200 – 5x, where p is the price (in dollars) at which exactly x
gidgets will be sold. Find the number of gidgets LMN should produce and the price it should charge to
maximize profit.
C(x) = 100 x + 400
R(x) = x p(x) = -5x2 + 200x
P(X) = R(X) – C(X) = -5x2 + 100x – 400
P’(x) = -10x + 100.
P’(x) = 0 for x = 10.
P”(x) – 10 <0, so by the second derivative test for absolute extreme values, there is an absolute maximum
value of P(x) at x = 10.
The LMN Company should produce 10 gidgets and charge p(10) = $150 to maximize profit.
3. (15 pts.) A poster is to have 2-inch margins at the top and bottom and 1-inch margins on the sides.
The total area is to be 162 square inches. Find the dimensions that maximize the print area.
Let TA be the total area and PA the print area. Then TA = LW = 162 and PA =(L-4)(W-2) = LW-4W-2L+8.
L=
162
324
324
, so PA(W) =162 -4W + 8 = 170 - 4W , W > 2,
W
W
W
324
2
2
PA’(W) = -4 +
PA”(W) = -
W2
648
W3
. PA’(W) = 0 for 4W = 324, W = 81, W = 9.
, PA”(9) = -
648
93
< 0, so by the second derivative test for absolute extreme values, there
is an absolute maximum value of PA(W) at W = 9 in, L = 18 in.
L
W
Arrington Sample
4. (20 pts.) f(x) = 5x4 – x5 = x4(5 - x). f ( x ) = 20x3 - 5x4 = 5x3(4 -x), and f ( x ) = 60x2 - 20x3= 20x2(3 - x).
(a) Find the critical numbers and the inflection points of f. (b) Construct sign charts for the first and
second derivatives. (c) Find all open intervals of increase and decrease and open intervals on
which the graph is concave up and concave down. (d) Classify each critical point as a relative
maximum, relative minimum or neither. (e) Sketch the graph of y = f(x) by hand, labeling only the
relative extreme points, inflection points and intercepts. Use the factored form to evaluate the function.
34 = 81, 44 = 256.
Be sure to give answer to each part.
(a) The critical numbers are x = 0 and x = 4. The inflection point is (3, 162).
___
+++
___
f’(x)
(b) ____f’<0_____________ 0 ___f”>0______________ 4 ___f’<0____________
+++
+++
___
f”(x)
____f’>0_____________ 0 ___f”>0______________ 3 ___f’<0____________
(c) f is increasing on (0, 4) and decreasing on (-, 0)  (4, ).
The graph is concave up on (-, 0)  (0,3) and concave down on (0, 3).
(d) There is a relative minimum at (0, 0) and a relative maximum at (4, 256)
(e)
(0, 0), (3, 162), and (4, 256) should be labeled
Arrington Sample
dy
5. (13 pts.) 2xy + xy2 = x3 – 2. Find
. Also find an equation of the tangent line at x =2, y = 1.
dx
dy
dy
2x
+ 2y + 2xy
+ y2 = 3x2
dx
dx
2x
dy
dy
+ 2xy
= 3x2 - 2y - y2
dx
dx
(2 x + 2xy)
dy
= 3x2 - 2y - y2
dx
dy
3x 2 - 2y - y 2
=
dx
2x  2xy
dy
12  2  1 9


dx x  2
44
8
y 1
LTan(2,1): y – 1 =
9
9x 5
(x-2) or y =
8
8 4
6. (8 pts.) The resale value R (in dollars) of a certain model car after t years is given by
R(t) = 20,000 e 0.1t . Find the instantaneous and relative rates of change (depreciation) the
moment the car is sold (t = 0). Include units in your answers.
Instantaneous rate of change: R’(t) = 20,000 e 0.1t (-.1)
Relative rate of change:
dollars
dollars
; R’(0) = -2,000
yr
yr
R(0)  2000 1

 10% / yr
R(0) 20,000 yr
7. (8 pts.) A snowball (sphere) is melting so that the radius is decreasing at the rate of 2 inches per hour. Find
how fast the volume of the snowball is decreasing at the moment that the radius is 10 inches long.
Let V be the volume, R the radius of the snowball. Then V =
dV dV dR
 2 in 

 4R 2  

dt dR dt
 hr 
dV
800 in3
 2 in 
 4(10 in)2  

dt R 10 in
hr
 hr 
4
dR
2 in
R 3 and

.
3
dt
hr
Arrington Sample
Extra credit (5 pts.)
There may be a problem on Elasticity of Demand:
(6 pts.) For the demand function D(p) = 175  3p
(a) Find the elasticity of demand, E(p).
p
E(p).= -
3
2 175  3p
pD(p)
3p


D(p)
2(175  3p)
175  3p
(b) Is demand elastic, inelastic, or unitary at p = 50?
E(50).=
150
3
2(175  150)
Demand is elastic at p = 50.