Difference quotient

Question

Find and simplify the difference quotient DQ=f(x+h)-f(x)/h for the function f(x)=5/3x

 

Please explain the steps you took to get the answer

Thanks!

Answer 1

Nick,

 

Substitute (x+h) for every x in f(x):  You get f(x+h) = 5/3(x+h) for this example.

 

Now you put it all together and simplify:  (f(x+h)-f(x))/h = (5/3(x+h) - 5/3x)/h = (5/3x +5/3h -5/3x)/h = 5/3h/h = 5/3

 

 

You must put () around both  (  f(x+h) - f(x)  ) like so to make this work.

 

 

If f(x) = x^2    you get  ((x+h)^2 - x^2)/h = (x^2 +2xh +h^2 - x^2)/h = (2xh + h^2)/h = 2x+h

 

In the limit that h approaches 0, you get 2x, which is the derivative (slope function df/dx) of x^2

 

Hope this helped!

 

Charles