Question 2.8.4: Finding the Difference Quotient Let ƒ(x) = 2x² - 3x. Find an...
Finding the Difference Quotient
Let ƒ(x) = 2x² – 3x. Find and simplify the expression for the difference quotient,
\frac{ƒ(x + h) – ƒ(x)}{h}.
We use a three-step process.
Step 1 Find the first term in the numerator, ƒ(x + h). Replace x in ƒ(x) with x + h.
ƒ(x + h)
= 2(x + h)² – 3(x + h) ƒ(x)= 2x² – 3x
Step 2 Find the entire numerator, ƒ(x + h) – ƒ(x).
ƒ(x + h) – ƒ(x)
= [2(x + h)² – 3(x + h)] – (2x² – 3x) Substitute.
= 2(x² + 2xh + h²) – 3(x + h) – (2x² – 3x) Square x + h.
= 2x² + 4xh + 2h² – 3x – 3h – 2x² + 3x Distributive property
= 4xh + 2h² – 3h Combine like terms.
Step 3 Find the difference quotient by dividing by h.
\frac{ƒ(x + h) – ƒ(x)}{h}
= \frac{4xh + 2h² – 3h }{h} Substitute 4xh + 2h² – 3h for
ƒ(x + h) – ƒ(x), from Step 2.
= \frac{h(4x + 2h – 3)}{ h} Factor out h.
= 4x + 2h – 3 Divide out the common factor.