Question

1. - / 1 points find the derivative of the function using the definition of derivative. f(t)=4.5t² + 5t f(t)= state the domain of the function. (enter your answer using interval notation.) state the domain of its derivative. (enter your answer using interval notation.)

Explanation:

Step1: Recall the definition of the derivative

The definition of the derivative of a function $y = f(t)$ is $f^\prime(t)=\lim_{h
ightarrow0}\frac{f(t + h)-f(t)}{h}$. Given $f(t)=4.5t^{2}+5t$, then $f(t + h)=4.5(t + h)^{2}+5(t + h)=4.5(t^{2}+2th+h^{2})+5t + 5h=4.5t^{2}+9th+4.5h^{2}+5t + 5h$.

Step2: Calculate $f(t + h)-f(t)$

$f(t + h)-f(t)=(4.5t^{2}+9th+4.5h^{2}+5t + 5h)-(4.5t^{2}+5t)=9th+4.5h^{2}+5h$.

Step3: Calculate $\frac{f(t + h)-f(t)}{h}$

$\frac{f(t + h)-f(t)}{h}=\frac{9th+4.5h^{2}+5h}{h}=9t + 4.5h+5$.

Step4: Find the limit as $h

ightarrow0$
$f^\prime(t)=\lim_{h
ightarrow0}(9t + 4.5h+5)=9t + 5$.

For the domain of $f(t)=4.5t^{2}+5t$, since it is a polynomial function, the domain is all real - numbers. In interval notation, it is $(-\infty,\infty)$.

For the domain of $f^\prime(t)=9t + 5$, since it is also a polynomial function, the domain is all real - numbers. In interval notation, it is $(-\infty,\infty)$.

Answer:

$f^\prime(t)=9t + 5$
$(-\infty,\infty)$
$(-\infty,\infty)$