The table shows the velocities (in miles per hour) of two cars on an entrance ramp to an interstate highway. The time $t$ is in seconds.

$t$	$0$	$5$	$10$	$15$	$20$	$25$	$30$
$v_1$	$0$	$2.5$	$7$	$16$	$29$	$45$	$65$
$v_2$	$0$	$21$	$38$	$51$	$60$	$64$	$65$

Use the regression capabilities of a graphing utility to find quadratic models for the data in part (a).

Find the time required for an object to cool from $300^{\circ} \mathrm{F}$ to $250^{\circ} \mathrm{F}$ by evaluating

$$t=\frac{10}{\ln 2} \int_{250}^{300} \frac{1}{T-100} d T$$

where $t$ is time in minutes.

In this given exercise, find the indefinite integral by the method shown in the earlier examples.

$$\int t\sqrt[3]{t-4}dt$$

In this exercise, use the equation of the tractrix $y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^2-x^2}, \quad a>0$.

Let $L$ be the tangent line to the tractrix at the point $P$. If $L$ intersects the $y$-axis at the point $Q$, show that the distance between $P$ and $Q$ is $a$.

In this given exercise, demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a).

$$F(x)=\int_{-1}^x e^t d t$$

At the instant the traffic light turns green, a car that has been waiting at an intersection starts with a constant acceleration of 6 feet per second per second. At the same instant, a truck traveling with a constant velocity of 30 feet per second passes the car.

How fast will the car be traveling when it passes the truck?

At the instant the traffic light turns green, a car that has been waiting at an intersection starts with a constant acceleration of 6 feet per second per second. At the same instant, a truck traveling with a constant velocity of 30 feet per second passes the car.

How far beyond its starting point will the car pass the truck?

In this given exercise, integrate to find $F$ as a function of $x$.

$$F(x)=\int_{-1}^x e^t d t$$

In this exercise, use the equation of the tractrix $y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^2-x^2}, \quad a>0$.

Find $d y / d x$.

In this exercise, find the indefinite integral by the method shown in the earlier examples.

$$\int \frac{-x}{(x+1)-\sqrt{x+1}} d x$$

In this exercise, use the Trapezoidal Rule and Simpson's Rule with $n=4$, and use the integration capabilities of a graphing utility, to approximate the definite integral.

$$\int_0^{\pi / 2} \sqrt{x} \cos x d x$$

A population of bacteria is changing at a rate of

$$\frac{d P}{d t}=\frac{3000}{1+0.25 t}$$

where $t$ is the time in days. The initial population (when $t=0$ ) is 1000 . Write an equation that gives the population at any time $t$, and find the population when $t=3$ days.

An object is dropped from a height of 400 feet.

Use the result in part (c) to find $\lim _{t \rightarrow \infty} v(t)$ and give its interpretation.

A car traveling at 45 miles per hour is brought to a stop, at constant deceleration, 132 feet from where the brakes are applied.

Draw the real number line from 0 to 132 , and plot the points found in parts (a) and (b). What can you conclude?