A rock is thrown vertically upward from the top of a $250\text{ m}$ cliff with an initial velocity of $15\(\text{ m/s}\)$. Assume only gravity acts on the rock $(g=9.8{\text{ m/s}}^2)$. At what time does the rock reach its maximum height? What is that height?

Given the velocity-time graph of a car moving along a straight road for $t$ in $[0,10]$, what is the displacement function for $t\ge 8$ if the velocity remains constant for $t\ge 8$?

Calculate the midpoint Riemann sum for $f\left(x\right)=3x-2$ on the interval $[1,5]$ with $n=4$ subintervals.

What is a definite integral?

Find $\frac{d}{dx}{\int }_{-1}^x(3t^2-2t)dt$.

Let $g^{\prime }\left(x\right)$ be continuous for all real $x$. What is the average value of $g^{\prime }\left(x\right)$ on the interval $[2,7]$?

Assume $g$$,g^{\prime }$ are continuous on $R$. Determine $\int 4\left(g\right(x\left){)}^m{g}^{\prime }\right(x)dx$ where $m\ne -1$.

Calculate the area in the first quadrant bounded by $y=\frac{x}{3}$ and $y=2-\mid x\mid$.

A fish population in a pond starts at $50$ at $t=0$. After $10$ days, the population is $200$. The carrying capacity is $3200$. After how many days will the population reach half the carrying capacity? Round your answer to nearest whole number.

A population of bacteria in a Petri dish is modeled by $P^{\prime }\left(t\right)+aP\left(t\right)=S$, where $P\left(t\right)$ is the population at time $t$ (in hours), $a$ is the death rate constant, and $S$ is the supply rate. If $S=40$ cells per hour and $a=0.2$ per hour, what is the equilibrium population?

Given $w^{\prime }\left(s\right)=\frac{s}{w}$, and $w\left(0\right)=5$, use Euler's method with $\Delta s=0.25$ to approximate $w\left(1\right)$. The exact solution is $w\left(s\right)=\sqrt{s^2+25}$. What is the error at $s=1$?

A cup of tea is initially at a temperature of ${85}^{\circ }\text{C}$ and cools in a room maintained at ${20}^{\circ }\text{C}$. If the cooling constant is $k=0.03$ ${\text{min}}^{-1}$, after how many minutes will the tea reach ${40}^{\circ }\text{C}$? Round your answer to the nearest whole number.

Find the value of $\frac{10,000!}{9,998!}$ exactly.

Does the series $\sum _{m=0}^{\infty }\frac{2^{m+5}}{6^{m-1}}$ converge?