Chapter 3

Gravel is being dumped from a conveyor belt at a rate of 10 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 15 feet high? Recall that the volume of a right circular cone with height \(h\) and radius of the base \(r\) is given by \(V=\frac{1}{2} \pi r^{2} h\) When the pile is 15 feet high, its height is increasing at _______ feet per minute.

An open box is to be made out of a 10 -inch by 14 -inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

For some positive constant \(C\), a patient's temperature change, \(T\), due to a dose, \(D\), of a drug is given by \(T=\left(\frac{C}{2}-\frac{D}{3}\right) D^{2}\). What dosage maximizes the temperature change? The sensitivity of the body to the drug is defined as \(d T / d D\). What dosage maximizes sensitivity?

A street light is at the top of a 12 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of \(8 \mathrm{ft} /\) sec along a straight path. How fast is the tip of her shadow moving when she is 50 feet from the base of the pole? The tip of the shadow is moving at ______ \(\mathrm{ft} / \mathrm{sec}\)

A rectangular storage container with an open top is to have a volume of 22 cubic meters. The length of its base is twice the width. Material for the base costs 14 dollars per square meter. Material for the sides costs 8 dollars per square meter. Find the cost of materials for the cheapest such container.

For each family of functions that depends on one or more parameters, determine the function's absolute maximum and absolute minimum on the given interval. a. \(p(x)=x^{3}-a^{2} x,[0, a](a>0)\) b. \(r(x)=a x e^{-b x},\left[\frac{1}{2 b}, b\right](a, b>0)\) c. \(w(x)=a\left(1-e^{-b x}\right),[b, 3 b](a, b>0)\) d. \(s(x)=\sin (k x),\left[\frac{\pi}{3 k}, \frac{5 \pi}{6 k}\right]\)

Find the inflection points of \(f(x)=2 x^{4}+27 x^{3}-21 x^{2}+15\). (Give your answers as a comma separated list, e.g., \(3,-2 .)\) inflection points = ____ .

Water is leaking out of an inverted conical tank at a rate of \(8600.0 \mathrm{~cm}^{3} / \mathrm{min}\) at the same time that water is being pumped into the tank at a constant rate. The tank has height \(12.0 \mathrm{~m}\) and the the diameter at the top is \(4.0 \mathrm{~m}\). If the water level is rising at a rate of \(24.0 \mathrm{~cm} / \mathrm{min}\) when the height of the water is \(5.0 \mathrm{~m}\), find the rate at which water is being pumped into the tank in cubic centimeters per minute.

For each of the functions described below (each continuous on \([a, b])\), state the location of the function's absolute maximum and absolute minimum on the interval \([a, b],\) or say there is not enough information provided to make a conclusion. Assume that any critical numbers mentioned in the problem statement represent all of the critical numbers the function has in \([a, b] .\) In each case, write one sentence to explain your answer. a. \(f^{\prime}(x) \leq 0\) for all \(x\) in \([a, b]\) b. \(g\) has a critical number at \(c\) such that \(a0\) for \(xc\) c. \(h(a)=h(b)\) and \(h^{\prime \prime}(x)<0\) for all \(x\) in \([a, b]\) d. \(p(a)>0, p(b)<0,\) and for the critical number \(c\) such that \(a0\) for \(x>c\)

Consider the one-parameter family of functions given by \(p(x)=x^{3}-a x^{2},\) where \(a>0\) a. Sketch a plot of a typical member of the family, using the fact that each is a cubic polynomial with a repeated zero at \(x=0\) and another zero at \(x=a\). b. Find all critical numbers of \(p\). c. Compute \(p^{\prime \prime}\) and find all values for which \(p^{\prime \prime}(x)=0 .\) Hence construct a second derivative sign chart for \(p\). d. Describe how the location of the critical numbers and the inflection point of \(p\) change as \(a\) changes. That is, if the value of \(a\) is increased, what happens to the critical numbers and inflection point?

A sailboat is sitting at rest near its dock. A rope attached to the bow of the boat is drawn in over a pulley that stands on a post on the end of the dock that is 5 feet higher than the bow. If the rope is being pulled in at a rate of 2 feet per second, how fast is the boat approaching the dock when the length of rope from bow to pulley is 13 feet?

The top and bottom margins of a poster are \(8 \mathrm{~cm}\) and the side margins are each \(2 \mathrm{~cm}\). If the area of printed material on the poster is fixed at 386 square centimeters, find the dimensions of the poster with the smallest area.

Let \(s(t)=3 \sin \left(2\left(t-\frac{\pi}{6}\right)\right)+5\). Find the exact absolute maximum and minimum of \(s\) on the provided intervals by testing the endpoints and finding and evaluating all relevant critical numbers of \(s\). a. \(\left[\frac{\pi}{6}, \frac{7 \pi}{6}\right]\) c. \([0,2 \pi]\) b. \(\left[0, \frac{\pi}{2}\right]\) d. \(\left[\frac{\pi}{3}, \frac{5 \pi}{6}\right]\)

Let \(q(x)=\frac{e^{-x}}{x-c}\) be a one-parameter family of functions where \(c>0\). a. Explain why \(q\) has a vertical asymptote at \(x=c\). b. Determine \(\lim _{x \rightarrow \infty} q(x)\) and \(\lim _{x \rightarrow-\infty} q(x)\). c. Compute \(q^{\prime}(x)\) and find all critical numbers of \(q\). d. Construct a first derivative sign chart for \(q\) and determine whether each critical number leads to a local minimum, local maximum, or neither for the function \(q\). e. Sketch a typical member of this family of functions with important behaviors clearly labeled.

A rectangle is inscribed with its base on the \(x\) -axis and its upper corners on the parabola \(y=12-x^{2}\). What are the dimensions of such a rectangle with the greatest possible area?

Let \(E(x)=e^{-\frac{(x-m)^{2}}{2 s^{2}}},\) where \(m\) is any real number and \(s\) is a positive real number. a. Compute \(E^{\prime}(x)\) and hence find all critical numbers of \(E\). b. Construct a first derivative sign chart for \(E\) and classify each critical number of the function as a local minimum, local maximum, or neither. c. It can be shown that \(E^{\prime \prime}(x)\) is given by the formula $$ E^{\prime \prime}(x)=e^{-\frac{(x-m)^{2}}{2 s^{2}}}\left(\frac{(x-m)^{2}-s^{2}}{s^{4}}\right) $$ Find all values of \(x\) for which \(E^{\prime \prime}(x)=0\). Find all values of \(x\) for which \(E^{\prime \prime}(x)=0\). d. Determine \(\lim _{x \rightarrow \infty} E(x)\) and \(\lim _{x \rightarrow-\infty} E(x)\). e. Construct a labeled graph of a typical function \(E\) that clearly shows how important points on the graph of \(y=E(x)\) depend on \(m\) and \(s\).

Suppose that \(g\) is a differentiable function and \(g^{\prime}(2)=0 .\) In addition, suppose that on \(1

A baseball diamond is a square with sides 90 feet long. Suppose a baseball player is advancing from second to third base at the rate of 24 feet per second, and an umpire is standing on home plate. Let \(\theta\) be the angle between the third baseline and the line of sight from the umpire to the runner. How fast is \(\theta\) changing when the runner is 30 feet from third base?

A rectangular box with a square bottom and closed top is to be made from two materials. The material for the side costs \(\$ 1.50\) per square foot and the material for the bottom costs \(\$ 3.00\) per square foot. If you are willing to spend \(\$ 15\) on the box, what is the largest volume it can contain? Justify your answer completely using calculus.

Sand is being dumped off a conveyor belt onto a pile in such a way that the pile forms in the shape of a cone whose radius is always equal to its height. Assuming that the sand is being dumped at a rate of 10 cubic feet per minute, how fast is the height of the pile changing when there are 1000 cubic feet on the pile?

A farmer wants to start raising cows, horses, goats, and sheep, and desires to have a rectangular pasture for the animals to graze in. However, no two different kinds of animals can graze together. In order to minimize the amount of fencing she will need, she has decided to enclose a large rectangular area and then divide it into four equally sized pens by adding three segments of fence inside the large rectangle that are parallel to two existing sides. She has decided to purchase \(7500 \mathrm{ft}\) of fencing. What is the maximum possible area that each of the four pens will enclose?

Let \(p\) be a function whose second derivative is \(p^{\prime \prime}(x)=(x+1)(x-2) e^{-x}\). a. Construct a second derivative sign chart for \(p\) and determine all inflection points of \(p\). b. Suppose you also know that \(x=\frac{\sqrt{5}-1}{2}\) is a critical number of \(p\). Does \(p\) have a local minimum, local maximum, or neither at \(x=\frac{\sqrt{5}-1}{2}\) ? Why? c. If the point \(\left(2, \frac{12}{e^{2}}\right)\) lies on the graph of \(y=p(x)\) and \(p^{\prime}(2)=-\frac{5}{e^{2}},\) find the equation of the tangent line to \(y=p(x)\) at the point where \(x=2\). Does the tangent line lie above the curve, below the curve, or neither at this value? Why?

A company is designing propane tanks that are cylindrical with hemispherical ends. Assume that the company wants tanks that will hold 1000 cubic feet of gas, and that the ends are more expensive to make, costing \(\$ 5\) per square foot, while the cylindrical barrel between the ends costs \(\$ 2\) per square foot. Use calculus to determine the minimum cost to construct such a tank.