find the indicated quantities for $y = f(x)=4x^{2}$. (a) simplify $\frac{f(1+delta x)-f(1)}{delta x}$. (b)…

find the indicated quantities for $y = f(x)=4x^{2}$. (a) simplify $\frac{f(1+delta x)-f(1)}{delta x}$. (b) what does the quantity in part (a) approach as $delta x$ approaches 0? (a) $\frac{f(1+delta x)-f(1)}{delta x}=$

find the indicated quantities for $y = f(x)=4x^{2}$. (a) simplify $\frac{f(1+delta x)-f(1)}{delta x}$. (b) what does the quantity in part (a) approach as $delta x$ approaches 0? (a) $\frac{f(1+delta x)-f(1)}{delta x}=$

Answer

Explanation:

Step1: Find (f(1+\Delta x)) and (f(1))

Given (f(x) = 4x^{2}), then (f(1+\Delta x)=4(1 + \Delta x)^{2}=4(1 + 2\Delta x+\Delta x^{2})=4 + 8\Delta x+4\Delta x^{2}), and (f(1)=4\times1^{2}=4).

Step2: Substitute into the difference - quotient formula

(\frac{f(1+\Delta x)-f(1)}{\Delta x}=\frac{(4 + 8\Delta x+4\Delta x^{2})-4}{\Delta x}=\frac{8\Delta x+4\Delta x^{2}}{\Delta x}).

Step3: Simplify the expression

(\frac{8\Delta x+4\Delta x^{2}}{\Delta x}=\frac{\Delta x(8 + 4\Delta x)}{\Delta x}=8 + 4\Delta x).

Step4: Find the limit as (\Delta x\rightarrow0)

We want to find (\lim_{\Delta x\rightarrow0}(8 + 4\Delta x)). As (\Delta x) approaches (0), the value of (4\Delta x) approaches (0). So (\lim_{\Delta x\rightarrow0}(8 + 4\Delta x)=8).

Answer:

(A) (8 + 4\Delta x) (B) (8)