Find the difference quotient of $f(x)=x^2-4$; that is find $\frac{f(x+h)-f(x)}{h}, h \neq 0$. Be sure to simplify.
The difference quotient is $\square$
Answer (1)
Calculate f ( x + h ) : f ( x + h ) = ( x + h ) 2 − 4 = x 2 + 2 x h + h 2 − 4 .
Calculate f ( x + h ) − f ( x ) : f ( x + h ) − f ( x ) = ( x 2 + 2 x h + h 2 − 4 ) − ( x 2 − 4 ) = 2 x h + h 2 .
Divide by h : h f ( x + h ) − f ( x ) = h 2 x h + h 2 .
Simplify: h 2 x h + h 2 = 2 x + h . The difference quotient is 2 x + h .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 − 4 and we want to find the difference quotient, which is given by the formula h f ( x + h ) − f ( x ) , where 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 into the function f ( x ) = x 2 − 4 :
f ( x + h ) = ( x + h ) 2 − 4
Expanding (x+h)^2 Expanding ( x + h ) 2 , we get: ( x + h ) 2 = x 2 + 2 x h + h 2 So, f ( x + h ) = x 2 + 2 x h + h 2 − 4
Calculating f(x+h) - f(x) Next, we need to find f ( x + h ) − f ( x ) . We have f ( x + h ) = x 2 + 2 x h + h 2 − 4 and f ( x ) = x 2 − 4 , so: f ( x + h ) − f ( x ) = ( x 2 + 2 x h + h 2 − 4 ) − ( x 2 − 4 ) f ( x + h ) − f ( x ) = x 2 + 2 x h + h 2 − 4 − x 2 + 4 f ( x + h ) − f ( x ) = 2 x h + h 2
Calculating the Difference Quotient Now, we need to find h f ( x + h ) − f ( x ) . We have f ( x + h ) − f ( x ) = 2 x h + h 2 , so: h f ( x + h ) − f ( x ) = h 2 x h + h 2 We can factor out an h from the numerator: h 2 x h + h 2 = h h ( 2 x + h ) Since h = 0 , we can cancel the h in the numerator and denominator: h h ( 2 x + h ) = 2 x + h
Final Answer Therefore, the difference quotient is 2 x + h .
Examples
The difference quotient is used to approximate the derivative of a function, which represents the instantaneous rate of change. For example, if f ( x ) represents the position of a car at time x , then the difference quotient h f ( x + h ) − f ( x ) represents the average velocity of the car over the time interval [ x , x + h ] . As h approaches 0, this average velocity approaches the instantaneous velocity at time x . This concept is fundamental in physics and engineering for analyzing motion and rates of change.
