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, integrate to find $F$ as a function of $x$.

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

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

Find $d y / d x$.

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.

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.

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

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.

An object is dropped from a height of 400 feet.

If the air resistance is proportional to the square of the velocity, then $d v / d t=-32+k v^2$, where $-32$ feet per second per second is the acceleration due to gravity and $k$ is a constant. Show that the velocity $v$ as a function of time $t$ is $v(t)=-\sqrt{\frac{32}{k}} \tanh (\sqrt{32 k} t)$ by performing the following integration and simplifying the result.

$$\int \frac{d v}{32-k v^2}=-\int d t$$

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?

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

How far has the car moved when its speed has been reduced to 15 miles per hour?

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

$$F(x)=\int_{\pi / 3}^x \sec t \tan t d t$$

An object is dropped from a height of 400 feet.

Use the result in part (a) to find the position function.

An object is dropped from a height of 400 feet.

Find the velocity of the object as a function of time (neglect air resistance on the object).

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

$$F(x)=\int_{\pi / 3}^x \sec t \tan t dt$$