hw8 defining the derivative (targets l6, d3; §3.1) score: 0/5 answered: 0/5 question 1 use the limit…

Explanation:

Step1: Recall limit - definition of derivative

The limit - definition of the derivative of a function $y = f(x)$ at $x=a$ is $f^\prime(a)=\lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $f(x)=4x^{2}$ and $a = 4$. First, find $f(4 + h)$ and $f(4)$. $f(4 + h)=4(4 + h)^{2}=4(16 + 8h+h^{2})=64+32h + 4h^{2}$, and $f(4)=4\times4^{2}=64$.

Step2: Substitute into the limit - formula

$f^\prime(4)=\lim_{h\rightarrow0}\frac{f(4 + h)-f(4)}{h}=\lim_{h\rightarrow0}\frac{(64 + 32h+4h^{2})-64}{h}$. Simplify the numerator: $\frac{64 + 32h+4h^{2}-64}{h}=\frac{32h + 4h^{2}}{h}$.

Step3: Simplify the fraction

$\frac{32h + 4h^{2}}{h}=\lim_{h\rightarrow0}(32 + 4h)$ (since $h\neq0$ when taking the limit), cancel out the common factor $h$ in the numerator and denominator.

Step4: Evaluate the limit

As $h\rightarrow0$, $\lim_{h\rightarrow0}(32 + 4h)=32$.

Answer:

32