Sketch the graphs of the given functions by determining the appropriate information and points from the first and second derivatives. Use a calculator to check the graph.

$$y = x^3 - 12x + 12$$

Solve the given problems.

The number $n$ of dollars saved by increasing the fuel efficiency of $e$ mi/gal to $e+6$ mi/gal for a car driven $10,000$ mi/year is $n=\frac{195,000}{e(e+6)},$ if the cost of gas is $\$3.25$/ gal. Sketch the graph.

Determine the required values by using Newton’s method.

In earlier example, Find $x$, the side of the square to be removed to form the box, using Newton's method to solve the equation $4x^3-136x^2+1147x-2770=0$. Find $x_2$, using $x_1=4.4$.

Solve the given maximum and minimum problems.

U.S. Postal Service regulations require that the length plus the girth (distance around) of a package not exceed $108$ in. What are the dimensions of the largest rectangular box with square ends that can be mailed?

Find the indicated velocities and accelerations.

A ship is moving around an island on a route described by $y=3x^2-0.2x^3$. If $v_x=1.2$ km/h, find the velocity of the ship where $x=3.5$ km.

Solve the problems in related rates.

A metal sphere is placed in seawater to study the corrosive effect of seawater. If the surface area decreases at $35$ cm$^2$/year due to corrosion, how fast is the radius changing when it is $12$ cm?

Solve the given problems involving tangent and normal lines.

In a video game, airplanes move from left to right along the path described by $y=2+1/x$. They can shoot rockets tangent to the direction of flight at targets on the $x$-axis located at $x=1, 2, 3,$ and $4$. Will a rocket fired from $(1,3)$ hit a target?

Find the indicated velocities and accelerations.

A rocket follows a path given by $y=x-\frac{1}{90}x^3$ (distances in miles). If the horizontal velocity is given by $v_x=x$, find the magnitude and direction of the velocity when the rocket hits the ground (assume level terrain) if time is in minutes.

Determine the required values by using Newton’s method.

Find the negative root of $3x^5-x^4-12x^3+4x^2+12x-4=0$ by choosing $x_1=-1.4$. Then graph the polynomial on a calculator and note where this root appears.

Solve the given problems.

The concentration $C$ (in mg/cm$^3$) of a certain drug in a patient’s bloodstream is $C=\frac{0.15t}{t^2+1}$, where $t$ is the time (in h) after the drug is administered. Sketch the graph. Display the graph on a calculator. What is the significance of the horizontal asymptote?

Solve the given problems by finding the appropriate differential.

If a spacecraft circles the earth at an altitude of $250$ km, how much farther does it travel in one orbit than an airplane that circles the earth at a low altitude? The radius of the earth is $6370$ km.

Sketch the graphs of the given functions by determining the appropriate information and points from the first and second derivatives. Use a calculator to check the graph.

$$y = x^3 + 3x^2 + 3x + 2$$

Solve the given maximum and minimum problems.

A computer is programmed to display a slowly changing right triangle with its hypotenuse always equal to $12.0$ cm. What are the legs of the triangle when it has its maximum area?

Solve the given problems involving tangent and normal lines.

A certain suspension cable with supports on the same level is closely approximated as being parabolic in shape. If the supports are $200$ ft apart and the sag at the center is $30$ ft, what is the equation of the line along which the tension acts (tangentially) at the right support? (Choose the origin of the coordinate system at the lowest point of the cable.)