MTH 161 TEACH Difference Quotient Worked Examples

What does f(2) mean if f(x) = 5x?

Put 2 in the spot where x is: 5 times 2 = 10. Whatever is in the parentheses goes where the x is.

What does f(x + h) mean if f(x) = 5x?

Put the WHOLE (x + h) where x is: 5(x + h) = 5x + 5h.

What is the difference quotient in plain words?

It is the slope between two points that are h apart: change in output divided by change in input, (f(x+h) - f(x)) / h. It is rise over run.

Why is the run just h?

The two inputs are x and x + h. Their difference is (x + h) - x = h.

What are the 3 moves of the difference quotient?

1) Plug in x + h. 2) Subtract f(x), with parentheses around it. 3) Divide by h.

Worked example f(x) = 5x: what is the answer and how?

f(x+h) = 5x + 5h. Subtract 5x: 5h. Divide by h: 5. The slope of the line y = 5x is 5.

Worked example f(x) = 3x + 2?

f(x+h) = 3x + 3h + 2. Subtract (3x + 2): 3h. Divide by h: 3. Linear functions always give their slope.

Why put parentheses around f(x) when subtracting?

The minus sign must flip EVERY sign in f(x). For (3x + 2) it becomes -3x - 2, not just -3x.

How do you expand (x + h)^2?

(x + h)(x + h) = x^2 + 2xh + h^2. It is NOT x^2 + h^2. The middle term 2xh is the most missed part.

Worked example f(x) = x^2?

f(x+h) = x^2 + 2xh + h^2. Subtract x^2: 2xh + h^2. Divide by h: 2x + h.

Self check after subtracting?

Every term with no h in it must cancel. If one is left over, you made a sign or expansion mistake.

Last step: how do you cancel the h?

Every term on top has an h. Divide EVERY term by h (or factor h out and cancel it). 2xh + h^2 over h = 2x + h.

Most common last-step mistake?

Dividing only the first term by h. You must divide every term.

Worked example f(x) = x^2 - 4x?

f(x+h) = x^2 + 2xh + h^2 - 4x - 4h. Subtract (x^2 - 4x): 2xh + h^2 - 4h. Divide by h: 2x + h - 4.

Worked example f(x) = 2x^2?

f(x+h) = 2(x^2 + 2xh + h^2) = 2x^2 + 4xh + 2h^2. Subtract 2x^2: 4xh + 2h^2. Divide by h: 4x + 2h.

What does the answer 2x + h mean for f(x) = x^2?

It is the slope between x and x + h. As h gets tiny the slope near x is 2x, which is the idea behind derivatives in calculus.

What is the difference between f(a) and f(x) = k?

f(a): you are given the input, find the output. f(x) = k: you are given the output k, find the input.