Problem

Find the difference quotient for the function \( f(x) = 3x^2 - 2x + 1 \)

Answer

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Answer

Finally, divide by \(h\) to get the difference quotient: \( \frac{f(x + h) - f(x)}{h} = \frac{6xh + 3h^2 - 2h}{h} = 6x + 3h - 2\)

Steps

Step 1 ：The difference quotient formula is \( \frac{f(x + h) - f(x)}{h} \)

Step 2 ：First, plug \(x + h\) into the function: \(f(x + h) = 3(x + h)^2 - 2(x + h) + 1 = 3x^2 + 6xh + 3h^2 - 2x - 2h + 1\)

Step 3 ：Then, subtract \(f(x)\) from \(f(x + h)\): \(f(x + h) - f(x) = 3x^2 + 6xh + 3h^2 - 2x - 2h + 1 - (3x^2 - 2x + 1) = 6xh + 3h^2 - 2h\)

Step 4 ：Finally, divide by \(h\) to get the difference quotient: \( \frac{f(x + h) - f(x)}{h} = \frac{6xh + 3h^2 - 2h}{h} = 6x + 3h - 2\)

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