What is the difference quotient and how do you find it?

Question

In these examples:

1) f(x) = x² - x + 1, f(2+h) - f(2) / h , h is not equal to 0

2) f(t) = 1/t , f(t) - f(1) / t-1 t is not equal to 1

I really have no idea where to start, please explain this to me!

Answer 1

You will be using this over and over in Calculus.

Let's begin with a straight line:  y = (3/2)x - 4.

The slope is 3/2.  That means that when y increases by 3 (causing a difference of 3 with the previous y value ), x increases by 2 (again causing a difference).

Think of the slope 3/2 as a difference quotient.

In Calculus, you will be looking at difference quotients in attempting to measure the slope of a straight line that lies tangent to a function.  (Lying tangent to a function means the straight line does not "cross over" the function.)  

For example, let f(x) = x^2 + 1 (a parabola that opens up with a low point at (x, f(x)) = (0, 1) ).

How would we calculate the equation for the straight line that lays tangent to f(x) at (x, f(x)) = (1, 2)?

As we already know the line passes through (1, 2), all we need to know to find the equation of the straight line is its slope.

Now, bring in the difference quotient:

[ f(1 + h) - f(1) ] / [(1 + h) - 1] =  [ f(1 + h) - f(1) ] / h

When h is very close to zero, this difference quotient will be very close to the slope of that tangent line.  As h cannot be equal to zero, we will calculate the limit of the difference quotient as h approaches zero.

limit (h to 0) of [ f(1 + h) - f(1) ] / h =

limit (h to 0) of [ ( (1 + h)^2 + 1 ) - 2 ] / h =

limit (h to 0) of [ ( ( h^2 + 2h + 1) + 1 ) - 2 ] / h =

limit (h to 0) of [ h^2 + 2h ] / h =

limit (h to 0) of (h + 2) = 0 + 2 = 2

Note:  Going from line three above to line four, you wiped out all of the simple numbers in the numerator and each term in the numerator in line four has h in it.  This allows you to eliminate the h in the denominator.  (The denominator in the difference quotient WILL ALWAYS ELIMINATE.  If it appears you cannot eliminate it, then an error was made in calculating.  Go back and correct the error.)

That is the slope of the tangent line passing through (1, 2) and

the equation of the line would be:

f(x) - 2 = 2 (x - 1)    or   f(x) = 2x