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2.6 - 1
Marginals and Differentials
2.6
OBJECTIVE
• Find marginal cost, revenue, and profit.
• Find ∆y and dy.
• Use differentials for approximations.
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2.6 Marginals and Differentials
DEFINITIONS:
Let C(x), R(x), and P(x) represent, respectively, the
total cost, revenue, and profit from the production and
sale of x items.
The marginal cost at x, given by C(x), is the
approximate cost of the (x + 1)th item:
C(x) ≈ C(x + 1) – C(x), or C(x + 1) ≈ C(x) + C(x).
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2.6 Marginals and Differentials
DEFINITIONS (concluded):
The marginal revenue at x, given by R(x), is the
approximate revenue from the (x + 1)th item:
R(x) ≈ R(x + 1) – R(x), or R(x + 1) ≈ R(x) + R(x).
The marginal profit at x, given by P(x), is the
approximate profit from the (x + 1)th item:
P(x) ≈ P(x + 1) – P(x), or P(x + 1) ≈ P(x) + P(x).
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2.6 Marginals and Differentials
Example 1: Given
C(x) = 62x 2 + 27,500
and
R(x) = x 3 - 12x 2 + 40x + 10
find each of the following:
a) Total profit, P(x).
b) Total cost, revenue, and profit from the
production and sale of 50 units of the product.
c) The marginal cost, revenue, and profit when 50
units are produced and sold.
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2.6 Marginals and Differentials
Example 1 (continued):
a)
P( x) 
P( x) 
P( x) 
R( x)  C ( x)
x3  12 x 2  40 x  10  (62 x 2  27,500)
3
2
x  74 x  40 x  10
b)
C (50)  62(50)2  27,500
R (50)  (50)3  12(50) 2  40(50)  10
P (50)  (50)3  74(50)2  40(50)  10
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 $182,500
 $97,010
 $85,490
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2.6 Marginals and Differentials
Example 1 (concluded):
c) C( x) 
C(50) 
R( x) 
R(50) 
124x
124(50)
 $6200
3x 2  24 x  40
3(50)2  24(50)  40  $6340
P( x)  3x 2  148 x  40
P(50)  3(50)2  148(50)  40  $140
So, when 50 units have been made, the approximate
cost of the 51st unit will be $6200, and the approximate
revenue from the sale of the 51st unit will be $6340 for
an approximate profit on the 51st unit of $140.
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2.6 Marginals and Differentials
Example 2: For y = x 2 , x = 4, and ∆x = 0.1, find ∆y.
y 
f ( x  x)  f ( x)
y 
y 
y 
y 
(4  0.1)2  (4) 2
4.12  42
16.81  16
0.81
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2.6 Marginals and Differentials
Quick Check 1
For y  2 x 4  x, x  2 , and x  0.05 , find y.
y  f ( x  x)  f ( x)
y  [2(2  0.05)4  (2  0.05)]  [2(2)4  2]
y  [2(1.95)4  1.95]  [2(2)4  2]
y  [2(14.45900625)  1.95]  [2(16)  2]
y  30.8680125  34
y  3.1319875
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2.6 Marginals and Differentials
For f a continuous, differentiable function, and
small ∆x.
Dy
f ¢(x) »
and Dy » f ¢(x) × Dx.
Dx
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2.6 Marginals and Differentials
Example 3: Approximate 27 using Dy » f ¢(x)× Dx.
Let f (x) = x and x equal a number close to 27 and
A number for which the square root is easy to
compute. So, here let x = 25 with ∆x = 2. Then,
f ( x) 
y

y

y

1
and
2 x
f (25)  2
1
2
2 25
0.2
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2.6 Marginals and Differentials
Example 3 (concluded):
Now we can approximate
27.
27

25  y
27

5  y
27

5  0.2
27

5.2
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2.6 Marginals and Differentials
Quick Check 2
Approximate 98 using y  f ( x)x . To five decimal places,
98  9.89949 . How close is your approximation?
Let f ( x)  x and x equal a number close to 98 and a number for
which the square root is easy to compute. So here let x  100,
with x  2 . Then,
1
and
f ( x) 
2 x
y  f (100)  2
1
y 
 2
2 100
y  0.1
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2.6 Marginals and Differentials
Quick Check 2 Concluded
Now we can approximate 98 :
98  100  y
98  10  y
98  10  0.1
98  9.9
This is within 0.001 of the actual value of 98.
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2.6 Marginals and Differentials
DEFINITION:
For y = f (x), we define
dx, called the differential of x, by dx = ∆x
and
dy, called the differential of y, by dy = f  (x)dx.
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2.6 Marginals and Differentials
Example 4: For y = x(4 - x)3
a) Find dy.
b) Find dy when x = 5 and dx = 0.2.
a)
dy
 x  3(4  x) 2  1  (4  x)3  1
dx
dy
 (4  x) 2 (3x  4  x)
dx
dy
(4  x)2 (4 x  4)

dx
dy

4(4  x) 2 ( x  1)
dx
dy

4(4  x)2 ( x  1)dx
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2.6 Marginals and Differentials
Example 4 (concluded):
2

4(4

5)
(4  1)  0.2

dy
b)
dy 
3.2
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2.6 Marginals and Differentials
Section Summary
• If C ( x) represents the cost for producing x items, then marginal
cost C ( x) is its derivative, and C ( x)  C ( x  1)  C ( x) . Thus, the
cost to produce the ( x  1)st can be approximated by
C ( x  1)  C ( x)  C ( x).
• If R( x) represents the revenue from selling x items, then marginal
revenue R( x) is its derivative, and R( x)  R( x  1)  R( x). Thus, the
revenue from the ( x  1)st item can be approximated by
R( x  1)  R( x)  R( x).
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2.6 Marginals and Differentials
Section Summary Continued
• If P( x) represents profit from selling x items, then marginal profit
a P( x) is its derivative, and P( x)  P( x  1)  P( x) . Thus, the profit
from the ( x  1)st item can be approximated by P( x  1)  P( x)  P( x).
• In general, profit = revenue – cost, or P( x)  R( x)  C ( x).
• In delta notation, x  ( x  h)  x  h , and y  f ( x  h)  f ( x).
For small values of x, we have y  f ( x) which is equivalent to
x
y  f ( x)x.
dy
• The differential of x is dx  x . Since
 f ( x) , we have
dx
general,
, and
dy  f .(In
x)dx
dy
 the
y approximation can be very
close for sufficiently small . dx
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