section 12.1 techniques for finding derivative. constant rule power rule sum and difference rule
TRANSCRIPT
Section 12.1 Techniques for Finding Derivative
Constant Rule
Power Rule
Sum and Difference Rule
0d
cdx
1n ndx n x
dx
( ) ( ) '( ) '( )d
f x g x f x g xdx
Examples
Find y’ for the function.
42
15y x xx
4 2y x x x
Constant Times a Function
Examples:
( ) '( )d
c f x c f xdx
Find dy/dx for the function.
34
5 42 3y x x
x x
5 3 1.5 2 710 3 55
7
xy x x
xx
Marginal Functions
Example: The total cost in dollars incurred each week for manufacturing q refrigerators is given by the total cost function:
C(q) =8000 + 200q – 0.25q2
What is the actual cost incurred for manufacturing the 251st refrigerator?
Marginal Cost
Marginal cost is the cost incurred in producing an additional unit of a certain item given that the plant is already at a certain level of operation.
Mathematically, marginal cost is the rate of change of the total cost function with respect to x
derivative of the cost function. If C(x) is a total cost function, then the derivative C’(x) is called the marginal cost function.
Example
The total cost in dollars incurred each week for manufacturing q refrigerators is given by the total cost function:
C(q) =8000 + 200q – 0.25q2
Find the marginal cost for producing 251 refrigerators.
Marginal Revenue / Profit
If R(x) is a revenue function, then the derivative R’(x) is called the marginal revenue function.
If P(x) is a profit function, then the derivative P’(x) is called the marginal profit function.
Recall R(q) = pq or R(x) = px and P = R - C
Example
The total cost C(q) =8000 + 200q – 0.25q2
Suppose the demand equation for the refrigerators each week is given by
q = 9000 – 5p.
Find the marginal revenue for the production level of
a) 200 units.
b) 400 units.