math 2413 - calculus i\nreview for the final exam\n1. if a ball is thrown into the air with a velocity of 58…

Explanation:

Step1: Recall velocity - derivative relation

The velocity $v(t)$ is the derivative of the height function $H(t)$. Given $H(t)=58t - 9t^{2}$, we use the power - rule for differentiation. The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$.

Step2: Differentiate $H(t)$

$v(t)=H^\prime(t)=\frac{d}{dt}(58t - 9t^{2})=\frac{d}{dt}(58t)-\frac{d}{dt}(9t^{2})$. Using the power - rule, $\frac{d}{dt}(58t)=58$ and $\frac{d}{dt}(9t^{2})=9\times2t = 18t$. So, $v(t)=58 - 18t$.

Step3: Evaluate $v(t)$ at $t = 9$

Substitute $t = 9$ into $v(t)$. $v(9)=58-18\times9$. $v(9)=58 - 162$. $v(9)=- 104$ ft/s.

Answer:

$-104$ ft/s