question 16 given the function f(x)=1 + 7x², find and simplify the difference quotient. question help: video…

Answer:

$14x + 7h$

Explanation:

Step1: Recall difference - quotient formula

The difference quotient of a function $y = f(x)$ is $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$.

Step2: Find $f(x + h)$

Given $f(x)=1 + 7x^{2}$, then $f(x + h)=1+7(x + h)^{2}=1+7(x^{2}+2xh+h^{2})=1 + 7x^{2}+14xh+7h^{2}$.

Step3: Substitute into difference - quotient formula

$\frac{f(x + h)-f(x)}{h}=\frac{(1 + 7x^{2}+14xh+7h^{2})-(1 + 7x^{2})}{h}$.

Step4: Simplify the numerator

$(1 + 7x^{2}+14xh+7h^{2})-(1 + 7x^{2})=14xh+7h^{2}$.

Step5: Simplify the fraction

$\frac{14xh + 7h^{2}}{h}=\frac{h(14x + 7h)}{h}=14x + 7h$.