3.E: Using Derivatives (Exercises)

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3.1: Using Derivatives to Identify Extreme Values

1. Finding critical points and inflection points

2. Finding inflection points

3. Matching graphs of $f,f',f''$

4.

This problem concerns a function about which the following information is known:

$f$ is a differentiable function defined at every real number $x$
$\displaystyle f(0) = -1/2$
$y = f'(x)$ has its graph given at center in Figure 3.1.17

$3_1_Ez1.svg$

Figure 3.1.17. At center, a graph of $y = f'(x)\text{;}$ at left, axes for plotting $y = f(x)\text{;}$ at right, axes for plotting $y = f''(x)\text{.}$

Construct a first derivative sign chart for $f\text{.}$ Clearly identify all critical numbers of $f\text{,}$ where $f$ is increasing and decreasing, and where $f$ has local extrema.
On the right-hand axes, sketch an approximate graph of $y = f''(x)\text{.}$
Construct a second derivative sign chart for $f\text{.}$ Clearly identify where $f$ is concave up and concave down, as well as all inflection points.
On the left-hand axes, sketch a possible graph of $y = f(x)\text{.}$

5.

Suppose that $g$ is a differentiable function and $g'(2) = 0\text{.}$ In addition, suppose that on $1 \lt x\lt 2$ and $2 \lt x \lt 3$ it is known that $g'(x)$ is positive.

Does $g$ have a local maximum, local minimum, or neither at $x = 2\text{?}$ Why?
Suppose that $g''(x)$ exists for every $x$ such that $1 \lt x \lt 3\text{.}$ Reasoning graphically, describe the behavior of $g''(x)$ for $x$-values near $2\text{.}$
Besides being a critical number of $g\text{,}$ what is special about the value $x = 2$ in terms of the behavior of the graph of $g\text{?}$

6.

Suppose that $h$ is a differentiable function whose first derivative is given by the graph in Figure 3.1.18.

How many real number solutions can the equation $h(x) = 0$ have? Why?
If $h(x) = 0$ has two distinct real solutions, what can you say about the signs of the two solutions? Why?
Assume that $\lim_{x \to \infty} h'(x) = 3\text{,}$ as appears to be indicated in Figure 3.1.18. How will the graph of $y = h(x)$ appear as $x \to \infty\text{?}$ Why?
Describe the concavity of $y = h(x)$ as fully as you can from the provided information.

$3_1_Ez3.svg$

Figure 3.1.18. The graph of $y = h'(x)\text{.}$

7.

Let $p$ be a function whose second derivative is $p''(x) = (x+1)(x-2)e^{-x}\text{.}$

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

3.2: Using Derivatives to Describe Families of Functions

1. Drug dosage with a parameter

2. Using the graph of $g'$

3

Consider the one-parameter family of functions given by $p(x) = x^3-ax^2\text{,}$ where $a \gt 0\text{.}$

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 $x = a\text{.}$
Find all critical numbers of $p\text{.}$
Compute $p''$ and find all values for which $p''(x) = 0\text{.}$ Hence construct a second derivative sign chart for $p\text{.}$
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?

4

Let $q(x) = \frac{e^{-x}}{x-c}$ be a one-parameter family of functions where $c \gt 0\text{.}$

Explain why $q$ has a vertical asymptote at $x = c\text{.}$
Determine $\lim_{x \to \infty} q(x)$ and $\lim_{x \to -\infty} q(x)\text{.}$
Compute $q'(x)$ and find all critical numbers of $q\text{.}$
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 $q\text{.}$
Sketch a typical member of this family of functions with important behaviors clearly labeled.

5

Let $E(x) = e^{-\frac{(x-m)^2}{2s^2}}\text{,}$ where $m$ is any real number and $s$ is a positive real number.

Compute $E'(x)$ and hence find all critical numbers of $E\text{.}$
Construct a first derivative sign chart for $E$ and classify each critical number of the function as a local minimum, local maximum, or neither.
It can be shown that $E''(x)$ is given by the formula
\[ E''(x) = e^{-\frac{(x-m)^2}{2s^2}} \left(\frac{(x-m)^2 - s^2}{s^4} \right)\text{.} \nonumber \]
Find all values of $x$ for which $E''(x) = 0\text{.}$
Determine $\lim_{x \to \infty} E(x)$ and $\lim_{x \to -\infty} E(x)\text{.}$
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 $s\text{.}$

3.3: Global Optimization

1

Based on the given information about each function, decide whether the function has global maximum, a global minimum, neither, both, or that it is not possible to say without more information. Assume that each function is twice differentiable and defined for all real numbers, unless noted otherwise. In each case, write one sentence to explain your conclusion.

$f$ is a function such that $f''(x) \lt 0$ for every $x\text{.}$
$g$ is a function with two critical numbers $a$ and $b$ (where $a \lt b$), and $g'(x) \lt 0$ for $x \lt a\text{,}$ $g'(x) \lt 0$ for $a \lt x \lt b\text{,}$ and $g'(x) \gt 0$ for $x \gt b\text{.}$
$h$ is a function with two critical numbers $a$ and $b$ (where $a \lt b$), and $h'(x) \lt 0$ for $x \lt a\text{,}$ $h'(x) \gt 0$ for $a \lt x \lt b\text{,}$ and $h'(x) \lt 0$ for $x \gt b\text{.}$ In addition, $\lim_{x \to \infty} h(x) = 0$ and $\lim_{x \to -\infty} h(x) = 0\text{.}$
$p$ is a function differentiable everywhere except at $x = a$ and $p''(x) \gt 0$ for $x \lt a$ and $p''(x) \lt 0$ for $x \gt a\text{.}$

2

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.

$p(x) = x^3 - a^2x\text{,}$ $[0,a]$ ($a \gt 0$)
$r(x) = axe^{-bx}\text{,}$ $[\frac{1}{2b}, \frac{2}{b}]$ ($a \gt 0, b \gt 1$)
$w(x) = a(1-e^{-bx})\text{,}$ $[b, 3b]$ ($a, b \gt 0$)
$s(x) = \sin(kx)\text{,}$ $\left[\frac{\pi}{3k}, \frac{5\pi}{6k}\right]$ ($k \gt 0$)

3

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 $[a,b]\text{,}$ 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 $[a,b]\text{.}$ In each case, write one sentence to explain your answer.

$f'(x) \le 0$ for all $x$ in $[a,b]$
$g$ has a critical number at $c$ such that $a \lt c\lt b$ and $g'(x) \gt 0$ for $x \lt c$ and $g'(x) \lt 0$ for $x \gt c$
$h(a) = h(b)$ and $h''(x) \lt 0$ for all $x$ in $[a,b]$
$p(a) \gt 0\text{,}$ $p(b) \lt 0\text{,}$ and for the critical number $c$ such that $a \lt c \lt b\text{,}$ $p'(x) \lt 0$ for $x \lt c$ and $p'(x) \gt 0$ for $x \gt c$

4

Let $s(t) = 3\sin(2(t-\frac{\pi}{6})) + 5\text{.}$ 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 $s\text{.}$

$\displaystyle [\frac{\pi}{6}, \frac{7\pi}{6}]$
$\displaystyle [0, \frac{\pi}{2}]$
$\displaystyle [0, 2\pi]$
$\displaystyle [\frac{\pi}{3}, \frac{5\pi}{6}]$

3.4: Applied Optimization

1. Maximizing the volume of a box

2. Minimizing the cost of a container

3. Maximizing area contained by a fence

4. Minimizing the area of a poster

5. Maximizing the area of a rectangle

6

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 top and 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.

7

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 ft of fencing. What is the maximum possible area that each of the four pens will enclose?

8

Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases 100 ft apart. A cable that is stretched from the top of one pole to some point on the ground between the poles, and then to the top of the other pole. What is the minimum possible length of cable required? Justify your answer completely using calculus.

9

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.

3.5: Related Rates

1. Height of a conical pile of gravel

2. Movement of a shadow

3. A leaking conical tank

4

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?

5

A swimming pool is $60$ feet long and $25$ feet wide. Its depth varies uniformly from $3$ feet at the shallow end to $15$ feet at the deep end, as shown in the Figure 3.5.5.

$3_5_Ez3.svg$

Figure 3.5.5. The swimming pool.

Suppose the pool has been emptied and is now being filled with water at a rate of $800$ cubic feet per minute. At what rate is the depth of water (measured at the deepest point of the pool) increasing when it is $5$ feet deep at that end? Over time, describe how the depth of the water will increase: at an increasing rate, at a decreasing rate, or at a constant rate. Explain.

6

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?

7

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?

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