Problem 76 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=2 x^{2}$$

Step-by-Step Solution

Answer

The difference quotient simplified is \( 4x + 2h \).

Step 1: Understand the Difference Quotient Formula

Remember the formula for the difference quotient, which is \( \frac{f(x+h)-f(x)}{h} \). This formula represents a slope of the secant line from \( x \) to \( x+h \) for the function \( f(x) \).

Step 2: Substitute the Function into the Difference Quotient

Substitute the function \( f(x) = 2x^{2} \) into the difference quotient. This gives \( \frac{f(x+h) - f(x)}{h} = \frac{2(x+h)^{2} - 2x^{2}}{h} = \frac{2x^{2}+ 4xh +2h^{2} - 2x^{2}}{h} \).

Step 3: Simplify the Difference Quotient

Simplify the difference quotient by canceling out the like terms and simplifying what's left. The \( 2x^{2} \) terms cancel each other out, leaving \( \frac{4xh + 2h^{2}}{h} \). Then, divide each term by \( h \) to finalize the difference quotient. The final difference quotient is \( 4x + 2h \).