Chapter 1: Differentiation
2.6
Marginals and Differentials
OBJECTIVE 1: Find marginal cost, revenue, and profit.
DEFINITIONS:
DEFINITIONS (continued):
cost, revenue and profit, respectively, from the
approximate revenue from the aB € "bth item:
Let G aBb, VaBb, and T aBb represent the total
production and sale of B items.
The marginal cost at B, given by G aBb, is the
The marginal revenue at B, given by Vw aBb, is the
Vw aBb ¸ VaB € "b • VaBb
or
w
approximate cost of the aB € "b item:
th
G w aBb ¸ G aB € "b • G aBb
or
G aB € "b ¸ G aBb € G w aBb.
The marginal profit at B, given by T w aBb, is the
approximate revenue from the aB € "bth item:
T w aBb ¸ T aB € "b • T aBb
or
1. Given G aBb œ '#B# € #(ß &!!
and V aBb œ B$ • "#B# € %!B € "!,
find each of the following:
a) Total profit, T aBb.
b) Total cost, revenue and profit from the
production and sale of 50 units.
c) The marginal cost, revenue and profit
when the 51st unit is produced and sold.
VaB € "b ¸ VaBb € Vw aBb.
T aB € "b ¸ T aBb € T w aBb.
49
Math 1425.P70
Section 2.6/7
Name _____________________________________________
9/30/2014
Solve the problem.
1) A company estimates that the daily revenue (in dollars) from the sale of x cookies is given by
R(x) = 1540 + 0.02x + 0.0003x 2 .
Currently, the company sells 630 cookies per day. Use marginal revenue to estimate the increase in revenue if
the company increases sales by one cookie per day.
A) $0.65
B) $0.40
C) $65.00
D) $39.80
2) A grocery store estimates that the weekly profit (in dollars) from the production and sale of x cases of soup is
given by
P(x) = -5500 + 9.1x - 0.0015x 2
and currently 1300 cases are produced and sold per week. Use the marginal profit to estimate the increase in
profit if the store produces and sells one additional case of soup per week.
A) $5.52
B) $7.15
C) $5.20
D) $3795.00
3) The weekly profit, in dollars, from the production and sale of x bicycles is given by
P(x) = 90.00x - 0.005x2
Currently, the company produces and sells 900 bicycles per week. Use the marginal profit to estimate the
change in profit if the company produces and sells one more bicycle per week.
A) 90.00 dollars
B) 10.00 dollars
C) 99.00 dollars
D) 81.00 dollars
4) A company finds that when it spends x million dollars on advertising, its profit P, in thousands of dollars,
is
given by
P(x) = 1110 + 25x - 4x 2
Currently the company spends 8 million dollars on advertising. Use the marginal profit to estimate the change
in profit if the company increases its advertising expenditure by one million dollars.
A) 1110 thousand dollars
C) -39 thousand dollars
B) 200 thousand
D) 136 thousand
Section 2.6/7
dollars
dollars
Differentiate implicitly to find the slope of the curve at the given point.
5) y3 + yx2 + x 2 - 3y2 = 0; (-1, 1)
1
3
A) B)
C) -2
2
2
6) x2 + y2 = 1;
A)
A)
(3, 8)
7
8
7) x3 - y3 = 5;
B) -
8
3
C)
3
8
D) -
3
8
(3, 5)
9
25
8) y5 + x 3 = y2 + 12x;
A) 4
D) -1
B)
9
5
C) -
25
9
D)
9
25
B)
12
7
C) -
7
2
D)
12
5
(0, 1)
Math 1425.P70 Section 2.6/7, page 2
Find dy/dx by implicit differentiation.
9) 8y2 - 3x 2 = 7
A)
3x
8y
B)
3x
8
10) xy2 = 4
A)
11)
2x
y
B) -
2y
x
C)
3x 2
16y
D)
C)
x
2y
D) -
C)
x2
8y
D)
8y
x2
C)
1-y
2+x
D)
y+ 1
x+2
6x + 7
16y
y
2x
1 3
x - 4y2 = 15
3
A)
8y
x 2 + 15
B)
x2
4y
12) 2y - x + xy = 4
A) -
1-y
x+2
B) -
1+y
x+2
Math 1425.P70 Section 2.6/7, page 3
For the given demand equation, differentiate implicitly to find dp/dx.
13) 2p 2 + x2 = 1400
A)
dp
-2x
=
dx
p
B)
dp
-x
=
dx
2p
C)
dp
-p
=
dx
2x
D)
dp
-2p
=
dx
x
B)
dp
p+6
=dx
5
C)
dp
p+6
=dx
x+4
D)
dp
6
=dx
x+4
B)
dp
1
=dx
3xp
C)
dp
3x
=dx
p
D)
dp
p
=dx
3
B)
dp
40
=
dx
401
C)
dp
401
=
dx
40
D)
dp 40p
=
dx
401
14) (p + 6)(x + 4) = 29
A)
dp
x+4
=dx
p+6
15) xp 3 = 32
A)
dp
p
=dx
3x
16) 20p 2 - 400p + 613 = x
A)
dp
1
=
dx
40p - 400
Math 1425.P70 Section 2.6/7, page 4
Differentiate implicitly to find d2 y/dx 2 .
17) x + xy + y = 3
A)
y-2
(x + 1) 2
B)
y+1
(x + 1)2
C)
2y + 2
(x + 1)2
D)
B)
2y + 2
(2 + x)2
C)
y+1
(2 + x)2
D)
18) 2y - x + xy = 3
A)
y- 2
(2 + x)2
19) y2 + xy + x2 = 7
A) -
36
(x - 2y)3
B)
20) y2 - x 2 = 9
A)
y - x2
y2
B)
36
(x - 2y)3
y2 - x 2
y3
C) -
C)
36
(x + 2y)3
y2 - x 2
y2
Math 1425.P70 Section 2.6/7, page 5
D)
D)
2y - 2
(x + 1)2
2y - 2
(2 + x)2
36
(x + 2y)3
y - x2
y3
Answer Key
Testname: 1425_SECTIONS2_6_7
1) B
2) C
3) D
4) C
5) C
6) D
7) D
8) A
9) A
10) D
11) C
12) C
13) B
14) C
15) A
16) A
17) C
18) D
19) C
20) B
Math 1425.P70 Section 2.6/7, page 6