3.2E: Exercises
- Page ID
- 116565
This page is a draft and is under active development.
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In exercises 1 - 5, find \(f'(x)\) for each function.
1) \(f(x)=3x\left(18x^4+\dfrac{13}{x+1}\right)\)
- Answer
- \(f'(x) = 270x^4+\dfrac{39}{(x+1)^2}\)
2) \(f(x)=(x+2)(2x^2−3)\)
3) \(f(x)=\dfrac{4x^3−2x+1}{x^2}\)
- Answer
- \(f'(x) = \dfrac{4x^4+2x^2−2x}{x^4}\)
4) \(f(x)=\dfrac{x^2+4}{x^2−4}\)
5) \(f(x)=\dfrac{x+9}{x^2−7x+1}\)
- Answer
- \(f'(x) = \dfrac{−x^2−18x+64}{(x^2−7x+1)^2}\)
In exercise 6, find the equation of the tangent line \(T(x)\) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.
6) [Technology Required] \(y=\dfrac{2x}{x−1}\) at \((−1,1)\)
In exercises 7 - 10, assume that \(f(x)\) and \(g(x)\) are both differentiable functions for all \(x\). Find the derivative of each of the functions \(h(x)\).
7) \(h(x)=4f(x)+\dfrac{g(x)}{7}\)
8) \(h(x)=x^3f(x)\)
- Answer
- \(h'(x)=3x^2f(x)+x^3f′(x)\)
9) \(h(x)=\dfrac{f(x)g(x)}{2}\)
10) \(h(x)=\dfrac{3f(x)}{g(x)+2}\)
- Answer
- \(h'(x)=\dfrac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}\)
For exercises 11 - 14, assume that \(f(x)\) and \(g(x)\) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.
| \(x\) | 1 | 2 | 3 | 4 |
| \(f(x)\) | 3 | 5 | −2 | 0 |
| \(g(x)\) | 2 | 3 | −4 | 6 |
| \(f′(x)\) | −1 | 7 | 8 | −3 |
| \(g′(x)\) | 4 | 1 | 2 | 9 |
11) Find \(h′(1)\) if \(h(x)=x f(x)+4g(x)\).
12) Find \(h′(2)\) if \(h(x)=\dfrac{f(x)}{g(x)}\).
- Answer
- \(h'(2) =\frac{16}{9}\)
13) Find \(h′(3)\) if \(h(x)=2x+f(x)g(x)\).
14) Find \(h′(4)\) if \(h(x)=\dfrac{1}{x}+\dfrac{g(x)}{f(x)}\).
- Answer
- \(h'(4)\) is undefined.
In exercises 15 - 16, use the following figure to find the indicated derivatives, if they exist.
15) Let \(h(x)=f(x)g(x).\) Find
a) \(h′(1),\)
b) \(h′(3)\), and
c) \(h′(4).\)
- Answer
- a. \(h'(1) = 2\),
b. \(h'(3)\) does not exist,
c. \(h'(4) = 2.5\)
16) Let \(h(x)=\dfrac{f(x)}{g(x)}.\) Find
a) \(h′(1),\)
b) \(h′(3)\), and
c) \(h′(4).\)
17) Find the equation of the tangent line to the graph of \(f(x)=(3x−x^2)(3−x−x^2)\) at \(x=1\).
- Answer
- \(y=−5x+7\)
18) Find the equation of the line passing through the point \(P(3,3)\) and tangent to the graph of \(f(x)=\dfrac{6}{x−1}\).
- Answer
- \(y=−\frac{3}{2}x+\frac{15}{2}\)
19) [Technology Required] A herring swimming along a straight line has traveled \(s(t)=\dfrac{t^2}{t^2+2}\) feet in \(t\)
seconds. Determine the velocity of the herring when it has traveled 3 seconds.
- Answer
- \(\frac{12}{121}\) or 0.0992 ft/s
20) The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function \(P(t)=\dfrac{8t+3}{0.2t^2+1}\), where \(t\) is measured in years.
a. Determine the initial flounder population.
b. Determine \(P′(10)\) and briefly interpret the result.
21) [Technology Required] The concentration of antibiotic in the bloodstream \(t\) hours after being injected is given by the function \(C(t)=\dfrac{2t^2+t}{t^3+50}\), where \(C\) is measured in milligrams per liter of blood.
a. Find the rate of change of \(C(t).\)
b. Determine the rate of change for \(t=8,12,24\),and \(36\).
c. Briefly describe what seems to be occurring as the number of hours increases.
- Answer
- a. \(\dfrac{−2t^4−2t^3+200t+50}{(t^3+50)^2}\)
b. \(−0.02395\) mg/L-hr, \(−0.01344\) mg/L-hr, \(−0.003566\) mg/L-hr, \(−0.001579\) mg/L-hr
c. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.
In exercise 22, the given function represents the position of a particle traveling along a horizontal line.
a. Find the velocity and acceleration functions.
b. Determine the time intervals when the object is slowing down or speeding up.
22) \(s(t)=\dfrac{t}{1+t^2}\)
23) [Technology Required] A culture of bacteria grows in number according to the function \(N(t)=3000(1+\dfrac{4t}{t^2+100})\), where \(t\) is measured in hours.
a. Find the rate of change of the number of bacteria.
b. Find \(N′(0),\; N′(10),\; N′(20)\), and \(N′(30)\).
c. Interpret the results in (b).
d. Find \(N''(0),\; N''(10),\; N''(20),\) and \(N''(30)\). Interpret what the answers imply about the bacteria population growth.
- Answer
- a. \(N′(t)=3000\left(\dfrac{−4t^2+400}{(t^2+100)^2}\right)\)
b. \(120,0,−14.4,−9.6\)
c. The bacteria population increases from time 0 to 10 hours; afterwards, the bacteria population decreases.
d. \(0,−6,0.384,0.432\). The rate at which the bacteria is increasing is decreasing during the first 10 hours. Afterwards, the bacteria population is decreasing at a decreasing rate.
The following problems deal with the Holling type I, II, and III equations. These equations describe the ecological event of growth of a predator population given the amount of prey available for consumption.
24) [Technology Required] The Holling type I equation is described by \(f(x)=ax\), where \(x\) is the amount of prey available and \(a>0\) is the rate at which the predator meets the prey for consumption.
a. Graph the Holling type I equation, given \(a=0.5\).
b. Determine the first derivative of the Holling type I equation and explain physically what the derivative implies.
c. Determine the second derivative of the Holling type I equation and explain physically what the derivative implies.
d. Using the interpretations from b. and c. explain why the Holling type I equation may not be realistic.
- Answer
-
a.
b. \(f′(x)=a\). The more increase in prey, the more growth for predators.
c. \(f''(x)=0\). As the amount of prey increases, the rate at which the predator population growth increases is constant.
d. This equation assumes that if there is more prey, the predator is able to increase consumption linearly. This assumption is unphysical because we would expect there to be some saturation point at which there is too much prey for the predator to consume adequately.
25) [Technology Required] The Holling type II equation is described by \(f(x)=\dfrac{ax}{n+x}\), where \(x\) is the amount of prey available and \(a>0\) is the maximum consumption rate of the predator.
a. Graph the Holling type II equation given \(a=0.5\) and \(n=5\). What are the differences between the Holling type I and II equations?
b. Take the first derivative of the Holling type II equation and interpret the physical meaning of the derivative.
c. Show that \(f(n)=\frac{1}{2}a\) and interpret the meaning of the parameter n.
d. Find and interpret the meaning of the second derivative. What makes the Holling type II function more realistic than the Holling type I function?
26) [Technology Required] The Holling type III equation is described by \(f(x)=\dfrac{ax^2}{n^2+x^2}\), where x is the amount of prey available and \(a>0\) is the maximum consumption rate of the predator.
a. Graph the Holling type III equation given \(a=0.5\) and \(n=5.\) What are the differences between the Holling type II and III equations?
b. Take the first derivative of the Holling type III equation and interpret the physical meaning of the derivative.
c. Find and interpret the meaning of the second derivative (it may help to graph the second derivative).
d. What additional ecological phenomena does the Holling type III function describe compared with the Holling type II function?
- Answer
-
a.
b. \(f′(x)=\dfrac{2axn^2}{(n^2+x^2)^2}\). When the amount of prey increases, the predator growth increases.
c. \(f''(x)=\dfrac{2an^2(n^2−3x^2)}{(n^2+x^2)^3}\). When the amount of prey is extremely small, the rate at which predator growth is increasing is increasing, but when the amount of prey reaches above a certain threshold, the rate at which predator growth is increasing begins to decrease.
d. At lower levels of prey, the prey is more easily able to avoid detection by the predator, so fewer prey individuals are consumed, resulting in less predator growth.
27) [Technology Required] The populations of the snowshoe hare (in thousands) and the lynx (in hundreds) collected over 7 years from 1937 to 1943 are shown in the following table. The snowshoe hare is the primary prey of the lynx.
| Population of snowshoe hare (thousands) | Population of lynx (hundreds) |
| 20 | 10 |
| 5 | 15 |
| 65 | 55 |
| 95 | 60 |
Snowshoe Hare and Lynx PopulationsSource: http://www.biotopics.co.uk/newgcse/predatorprey.html.
a. Graph the data points and determine which Holling-type function fits the data best.
b. Using the meanings of the parameters \(a\) and \(n\), determine values for those parameters by examining a graph of the data. Recall that \(n\) measures what prey value results in the half-maximum of the predator value.
c. Plot the resulting Holling-type I, II, and III functions on top of the data. Was the result from part a. correct?