Find the difference quotient of [tex]$f$[/tex]; that is, find [tex]$\frac{f(x+h)-f(x)}{h}, h \neq 0$[/tex], for the following function. Be sure to simplify.
[tex]$f(x)=2 x^2-x-3$[/tex]
[tex]$\frac{f(x+h)-f(x)}{h}= \square$[/tex] (Simplify your answer.)
Answer (2)
First, find f ( x + h ) by substituting x + h into the function.
Then, calculate f ( x + h ) − f ( x ) .
Next, divide the result by h .
Finally, simplify the expression to get the difference quotient: 4 x + 2 h − 1 .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 x 2 − x − 3 and we want to find the difference quotient h f ( x + h ) − f ( x ) for h = 0 . This quotient represents the average rate of change of the function f ( x ) over the interval [ x , x + h ] .
Finding f(x+h) First, we need to find f ( x + h ) . We substitute x + h for x in the expression for f ( x ) :
f ( x + h ) = 2 ( x + h ) 2 − ( x + h ) − 3
Expanding this, we get:
f ( x + h ) = 2 ( x 2 + 2 x h + h 2 ) − x − h − 3
f ( x + h ) = 2 x 2 + 4 x h + 2 h 2 − x − h − 3
Computing f(x+h) - f(x) Now we compute f ( x + h ) − f ( x ) :
f ( x + h ) − f ( x ) = ( 2 x 2 + 4 x h + 2 h 2 − x − h − 3 ) − ( 2 x 2 − x − 3 )
f ( x + h ) − f ( x ) = 2 x 2 + 4 x h + 2 h 2 − x − h − 3 − 2 x 2 + x + 3
f ( x + h ) − f ( x ) = 4 x h + 2 h 2 − h
Dividing by h Next, we divide the result by h :
h f ( x + h ) − f ( x ) = h 4 x h + 2 h 2 − h
We can factor out an h from the numerator:
h f ( x + h ) − f ( x ) = h h ( 4 x + 2 h − 1 )
Simplifying the Expression Finally, we simplify the expression by canceling the h in the numerator and denominator (since h = 0 ):
h f ( x + h ) − f ( x ) = 4 x + 2 h − 1
Final Answer Therefore, the difference quotient is 4 x + 2 h − 1 .
Examples
The difference quotient is used to find the derivative of a function, which represents the instantaneous rate of change of the function at a specific point. For example, if f ( x ) represents the position of an object at time x , then the derivative f ′ ( x ) represents the velocity of the object at time x . The difference quotient approximates this velocity over a small interval h . In economics, if f ( x ) represents the cost of producing x items, then the derivative f ′ ( x ) represents the marginal cost, which is the cost of producing one additional item. The difference quotient can be used to estimate the marginal cost.
The difference quotient of the function f ( x ) = 2 x 2 − x − 3 is found by following the steps of calculating f ( x + h ) , finding the difference, dividing by h , and simplifying. The final result for the difference quotient is 4 x + 2 h − 1 .
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