Difference quotient
- "Computes the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h.
- Only works with a function
Let's apply the difference quotient to the linear function: f(x) = x+5
- There are two parts to the difference quotient, the first step is f(x+h). But, what exactly is this step telling us to do? It's telling us that for everywhere "x" appears in our equation- f(x)=x+5 is this case- we will substitute "x+h" in for "x." Therefore, here is the first part: f(x+h)=x+h+5.
- The second part of the difference quotient is f(x). This part is easy. We simply take the preexisting equation: f(x)=x+5).
- Now we put all of the difference quotient pieces together (ignore the pieces on the left of the equal sign). Don't forget to put everything over "h": (x+h+5 - (x+5))/h
- Now we can begin to solve. Distribute the minus: (x+h+5-x-5)/h
- The positive and negative "x" and "5" will cancel out. We're left with h/h = 1.
- 1 is our solution
Let's apply the difference quotient to parabola f(x) = 3x²
- Part 1: substitute (x+h) for "x" anywhere it appears in the original equation: f(x+h)=3(x+h)². This step for a parabola requires more work than for the line. We must simplify the equation by FOILing (there is an example of FOILing below). We break apart the squared term to FOIL: f(x) = (x+h)(x+h). The first part of our is now: f(x+h) = 3x²+3h²+6xh.
- Next, we take the original equation f(x)=3x². This is the second part of the difference quotient.
- Put the equation together and solve/simplify: (3x²+3h²+6xh-(3x²))/h
(3x²+3h²+6xh-3x²)/h
(3h²+6xh)/h - 3h+6x is the solution
Let's try applying the difference quotient to cubic function: f(x) = 4x³
- Substitute (x+h) in for anywhere "x" appears in the equation: f(x+h) = 4(x+h)³. Like the parabola, we will break apart the cubed term and FOIL two of the terms. Next we will multiply/distribute the last term:
=4(x+h)(x+h)(x+h)
=4(x²+hx+hx+h²)(x+h)
=4(x³+h³+3hx²+3xh²)
=4x³+4h³+12hx²+12xh² - As before, the original equation is still part 2: f(x)=4x³
- Now we put the pieces together: (4x³+4h³+12hx²+12xh²-4x³)/h (4h³+12hx²+12xh²)/h
- Our solution is 4h²+12x²+12xh
Now try applying the difference quotient to these three functions:
- f(x) = 2x+2
- f(x) = x²+3
- f(x) = 2x³