Use the given graph of \(f\) to find the following. (a) The open intervals on which \(f\) is increasing. (b) The open intervals on which \(f\) is decreasing. (c) The open intervals on which \(f\) is concave upward. (d) The open intervals on which \(f\) is concave downward. (e) The coordinates of the points of inflection. Equation Transcription: Text Transcription: f f f f f
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Table of Contents
1
Functions and Models
1.1
Four Ways to Represent a Function
1.2
Mathematical Models: A Catalog of Essential Functions
1.3
New Functions from Old Functions
1.4
Exponential Functions
1.5
Inverse Functions and Logarithms
2
Limits and Derivatives
2.1
The Tangent and Velocity Problems
2.2
The Limit of a Function
2.3
Calculating Limits Using the Limit Laws
2.4
The Precise Definition of a Limit
2.5
Continuity
2.6
Limits at Infinity; Horizontal Asymptotes
2.7
Derivatives and Rates of Change
2.8
The Derivative as a Function
3
Differentiation Rules
3.1
Derivatives of Polynomials and Exponential Functions
3.10
Linear Approximations and Differentials
3.11
Hyperbolic Functions
3.2
The Product and Quotient Rules
3.3
Derivatives of Trigonometric Functions
3.4
The Chain Rule
3.5
Implicit Differentiation
3.6
Derivatives of Logarithmic and Inverse Trigonometric Functions
3.7
Rates of Change in the Natural and Social Sciences
3.8
Exponential Growth and Decay
3.9
Related Rates
4
Applications of Differentiation
4.1
Maximum and Minimum Values
4.2
The Mean Value Theorem
4.3
What Derivatives Tell Us about the Shape of a Graph
4.4
Indeterminate Forms and l’Hospital’s Rule
4.5
Summary of Curve Sketching
4.6
Graphing with Calculus and Technology
4.7
Optimization Problems
4.8
Newton’s Method
4.9
Antiderivatives
5
Integrals
5.1
The Area and Distance Problems
5.2
The Definite Integral
5.3
The Fundamental Theorem of Calculus
5.4
Indefinite Integrals and the Net Change Theorem
5.5
The Substitution Rule
6
Applications of Integration
6.1
Areas Between Curves
6.2
Volumes
6.3
Volumes by Cylindrical Shells
6.4
Work
6.5
Average Value of a Function
7
Techniques of Integration
7.1
Integration by Parts
7.2
Trigonometric Integrals
7.3
Trigonometric Substitution
7.4
Integration of Rational Functions by Partial Fractions
7.5
Strategy for Integration
7.6
Integration Using Tables and Technology
7.7
Approximate Integration
7.8
Improper Integrals
8
Further Applications of Integration
8.1
Arc Length
8.2
Area of a Surface of Revolution
8.3
Applications to Physics and Engineering
8.4
Applications to Economics and Biology
8.5
Probability
9
Differential Equations
9.1
Modeling with Differential Equations
9.2
Direction Fields and Euler’s Method
9.3
Separable Equations
9.4
Models for Population Growth
9.5
Linear Equations
9.6
Predator-Prey Systems
10
Parametric Equations and Polar Coordinates
10.1
Curves Defined by Parametric Equations
10.2
Calculus with Parametric Curves
10.3
Polar Coordinates
10.4
Calculus in Polar Coordinates
10.5
Conic Sections
10.6
Conic Sections in Polar Coordinates
11
Sequences, Series, and Power Series
11.1
Sequences
11.10
Taylor and Maclaurin Series
11.11
Applications of Taylor Polynomials
11.2
Series
11.3
The Integral Test and Estimates of Sums
11.4
The Comparison Tests
11.5
Alternating Series and Absolute Convergence
11.6
The Ratio and Root Tests
11.7
Strategy for Testing Series
11.8
Power Series
11.9
Representations of Functions as Power Series
12
Vectors and the Geometry of Space
12.1
Three-Dimensional Coordinate Systems
12.2
Vectors
12.3
The Dot Product
12.4
The Cross Product
12.5
Equations of Lines and Planes
12.6
Cylinders and Quadric Surfaces
13
Vector Functions
13.1
Vector Functions and Space Curves
13.2
Derivatives and Integrals of Vector Functions
13.3
Arc Length and Curvature
13.4
Motion in Space: Velocity and Acceleration
14
Partial Derivatives
14.1
Functions of Several Variables
14.2
Limits and Continuity
14.3
Partial Derivatives
14.4
Tangent Planes and Linear Approximations
14.5
The Chain Rule
14.6
Directional Derivatives and the Gradient Vector
14.7
Maximum and Minimum Values
14.8
Lagrange Multipliers
15
Multiple Integrals
15.1
Double Integrals over Rectangles
15.2
Double Integrals over General Regions
15.3
Double Integrals in Polar Coordinates
15.4
Applications of Double Integrals
15.5
Surface Area
15.6
Triple Integrals
15.7
Triple Integrals in Cylindrical Coordinates
15.8
Triple Integrals in Spherical Coordinates
15.9
Change of Variables in Multiple Integrals
16
Vector Calculus
16.1
Vector Fields
16.10
Applications of Double Integrals
16.2
Line Integrals
16.3
The Fundamental Theorem for Line Integrals
16.4
Green’s Theorem
16.5
Curl and Divergence
16.6
Parametric Surfaces and Their Areas
16.7
Surface Integrals
16.8
Stokes’ Theorem
16.9
The Divergence Theorem
Textbook Solutions for Calculus: Early Transcendentals
Chapter 4.3 Problem 13
Question
Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\).
\(f(x)=\frac{x^{2}-24}{x-5}\)
Solution
Step 1 of 5
The function is given as,
\(f(x)=\frac{x^{2}-24}{x-5}\)
Note that the denominator of the rational function must not be zero. So it is required to exclude the value x = 5 from the domain of the function.
Obtain the first derivative of the function,
\(\begin{aligned}
f^{\prime}(x) & =\frac{d}{d x}\left(\frac{x^{2}-24}{x-5}\right) \\
& =\frac{(x-5) \frac{d}{d x}\left(x^{2}-24\right)-\left(x^{2}-24\right) \frac{d}{d x}(x-5)}{(x-5)^{2}} \\
& =\frac{(x-5)(2 x-0)-\left(x^{2}-24\right)(1-0)}{(x-5)^{2}} \\
& =\frac{2 x^{2}-10 x-x^{2}+24}{(x-5)^{2}} \\
& =\frac{x^{2}-10 x+24}{(x-5)^{2}}
\end{aligned}\)
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full solution
full solution
Title
Calculus: Early Transcendentals 9
Author
Daniel K. Clegg, Saleem Watson, James Stewart
ISBN
9781337613927
?Find the intervals on which \(f\) is increasing or decreasing, and find the local
Chapter 4.3 textbook questions
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Chapter 4: Problem 1 Calculus: Early Transcendentals 9
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Chapter 4: Problem 2 Calculus: Early Transcendentals 9
Use the given graph of \(f\) to find the following. (a) The open intervals on which \(f\) is increasing. (b) The open intervals on which \(f\) is decreasing. (c) The open intervals on which \(f\) is concave upward. (d) The open intervals on which \(f\) is concave downward. (e) The coordinates of the points of inflection. Equation Transcription: Text Transcription: f f f f f
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Chapter 4: Problem 3 Calculus: Early Transcendentals 9
Suppose you are given a formula for a function \(f\). (a) How do you determine where \(f\) is increasing or decreasing? (b) How do you determine where the graph of \(f\) is concave upward or concave downward? (c) How do you locate inflection points? Equation Transcription: Text Transcription: f f f
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Chapter 4: Problem 4 Calculus: Early Transcendentals 9
(a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?
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Chapter 4: Problem 5 Calculus: Early Transcendentals 9
The graph of the derivative \(f^{\prime}\) of a function \(f\) is shown. (a) On what intervals is \(f\) increasing? Decreasing? (b) At what values of \(x\) does \(f\) have a local maximum? Local minimum? Equation Transcription: Text Transcription: f' f f x f
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Chapter 4: Problem 6 Calculus: Early Transcendentals 9
The graph of the derivative \(f^{\prime}\) of a function \(f\) is shown. (a) On what intervals is \(f\) increasing? Decreasing? (b) At what values of \(x\) does \(f\) have a local maximum? Local minimum? Equation Transcription: Text Transcription: f' f f x f
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Chapter 4: Problem 7 Calculus: Early Transcendentals 9
In each part state the \(x\)-coordinates of the inflection points of \(f\). Give reasons for your answers. (a) The curve is the graph of \(f\). (b) The curve is the graph of \(f^{\prime}\). (c) The curve is the graph of \(f^{\prime \prime}\). Equation Transcription: ’ Text Transcription: x-coordinates f f f’ f"
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Chapter 4: Problem 8 Calculus: Early Transcendentals 9
The graph of the first derivative \(f^{\prime}\) of a function \(f\) is shown. (a) On what intervals is \(f\) increasing? Explain. (b) At what values of \(x\) does \(f\) have a local maximum or minimum? Explain. (c) On what intervals is \(f\) concave upward or concave downward? Explain. (d) What are the \(x\)-coordinates of the inflection points of \(f\) ? Why? Equation Transcription: Text Transcription: f' f f x f f x-coordinates f
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Chapter 4: Problem 9 Calculus: Early Transcendentals 9
Find the intervals on which f is increasing or decreasing, and find the local maximum and minimum values of f. \(f(x)=2 x^{3}-15 x^{2}+24 x-5\)
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Chapter 4: Problem 10 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=x^{3}-6 x^{2}-135 x\) Equation Transcription: Text Transcription: f f f(x)=x^3-6x^2-135x
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Chapter 4: Problem 11 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=6 x^{4}-16 x^{3}+1\) Equation Transcription: Text Transcription: f f f(x)=6x^4-16x^3+1
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Chapter 4: Problem 12 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=x^{2 / 3}(x-3)\) Equation Transcription: Text Transcription: f f f(x)=x^2/3(x-3)
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Chapter 4: Problem 13 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=\frac{x^{2}-24}{x-5}\) Equation Transcription: Text Transcription: f f f(x)=x^2 - 24/x - 5
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Chapter 4: Problem 14 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=x+\frac{4}{x^{2}}\) Equation Transcription: Text Transcription: f f f(x)=x+4/x^2
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Chapter 4: Problem 15 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=\sin x+\cos x, \quad 0 \leqslant x \leqslant 2 \pi\) Equation Transcription: ? ? Text Transcription: f f f(x)=sin x+cos x, 0 leqslant x leqslant 2pi
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Chapter 4: Problem 16 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=x^{4} e^{-x}\) Equation Transcription: Text Transcription: f f f(x)=x^4 e^-x
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Chapter 4: Problem 17 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=x^{3}-3 x^{2}-9 x+4\) Equation Transcription: Text Transcription: f f f(x)=x^3-3x^2-9x+4
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Chapter 4: Problem 18 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=2 x^{3}-9 x^{2}+12 x-3\) Equation Transcription: Text Transcription: f f f(x)=2x^3-9x^2+12x-3
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Chapter 4: Problem 19 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=\sin ^{2} x-\cos 2 x, \quad 0 \leqslant x \leqslant \pi\) Equation Transcription: ? ? Text Transcription: f f f(x)=sin^2 x-cos 2x, 0 leqslant x leqslant pi
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Chapter 4: Problem 20 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=\ln (2+\sin x), \quad 0 \leqslant x \leqslant 2 \pi\) Equation Transcription: ? ? Text Transcription: f f f(x)=ln(2+sin x), 0 leqslant x leqslant 2pi
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Chapter 4: Problem 21 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=\ln \left(x^{2}+5\right)\) Equation Transcription: Text Transcription: f f f(x)=ln(x^2+5)
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Chapter 4: Problem 22 Calculus: Early Transcendentals 9
Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=\frac{e^{x}}{e^{x}+2}\) Equation Transcription: Text Transcription: f f f(x)=e^x/e^x + 2
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Chapter 4: Problem 23 Calculus: Early Transcendentals 9
(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f\). (c) Find the intervals of concavity and the inflection points. \(f(x)=x^{4}-2 x^{2}+3\) Equation Transcription: Text Transcription: f f f(x)=x^4-2x^2+3
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Chapter 4: Problem 24 Calculus: Early Transcendentals 9
(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f\). (c) Find the intervals of concavity and the inflection points. \(f(x)=\frac{x}{x^{2}+1}\) Equation Transcription: Text Transcription: f F f(x)=x/x^2 + 1
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Chapter 4: Problem 25 Calculus: Early Transcendentals 9
(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.
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Chapter 4: Problem 26 Calculus: Early Transcendentals 9
(a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. \(f(x)=x^{2} \ln x\)
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Chapter 4: Problem 27 Calculus: Early Transcendentals 9
(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.
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Chapter 4: Problem 28 Calculus: Early Transcendentals 9
(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.
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Chapter 4: Problem 29 Calculus: Early Transcendentals 9
Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? \(f(x)=1+3 x^{2}-2 x^{3}\)
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Chapter 4: Problem 30 Calculus: Early Transcendentals 9
Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer?
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Chapter 4: Problem 31 Calculus: Early Transcendentals 9
Suppose the derivative of a function f is \(f^{\prime}(x)=(x-4)^{2}(x+3)^{7}(x-5)^{8}\) On what interval(s) is f increasing?
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Chapter 4: Problem 32 Calculus: Early Transcendentals 9
(a) Find the critical numbers of . (b) What does the Second Derivative Test tell you about the behavior of at these critical numbers? (c) What does the First Derivative Test tell you?
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Chapter 4: Problem 33 Calculus: Early Transcendentals 9
Suppose is continuous on . (a) If and , what can you say about ? (b) If and , what can you say about ?
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Chapter 4: Problem 34 Calculus: Early Transcendentals 9
Sketch the graph of a function that satisfies all of the given conditions. (a) and for all (b) and for all
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Chapter 4: Problem 35 Calculus: Early Transcendentals 9
Sketch the graph of a function that satisfies all of the given conditions. (a) and for all (b) and for all
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Chapter 4: Problem 36 Calculus: Early Transcendentals 9
Sketch the graph of a function that satisfies all of the given conditions. Vertical asymptote if , if , if if
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Chapter 4: Problem 37 Calculus: Early Transcendentals 9
Sketch the graph of a function that satisfies all of the given conditions. , if or , if or , if if or
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Chapter 4: Problem 38 Calculus: Early Transcendentals 9
Sketch the graph of a function that satisfies all of the given conditions. for all , vertical asymptote , if or , if
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Chapter 4: Problem 39 Calculus: Early Transcendentals 9
Sketch the graph of a function that satisfies all of the given conditions. when , when , when or , for
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Chapter 4: Problem 40 Calculus: Early Transcendentals 9
Sketch the graph of a function that satisfies all of the given conditions.
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Chapter 4: Problem 41 Calculus: Early Transcendentals 9
Sketch the graph of a function that satisfies all of the given conditions. if if , if , has inflection point ,
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Chapter 4: Problem 42 Calculus: Early Transcendentals 9
The graph of a function is shown. At which point(s) are the following true? (a) and are both positive. (b) and are both negative. (c) is negative but is positive.
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Chapter 4: Problem 43 Calculus: Early Transcendentals 9
The graph of the derivative ’ of a continuous function is shown. (a) On what intervals is increasing? Decreasing? (b) At what values of does have a local maximum? Local minimum? (c) On what intervals is concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of .
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Chapter 4: Problem 44 Calculus: Early Transcendentals 9
The graph of the derivative ’ of a continuous function is shown. (a) On what intervals is increasing? Decreasing? (b) At what values of does have a local maximum? Local minimum? (c) On what intervals is concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of .
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Chapter 4: Problem 45 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.
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Chapter 4: Problem 46 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)-(c) to sketch the graph. You may want to check your work with a graphing calculator or computer. \(f(x)=36 x+3 x^{2}-2 x^{3}\)
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Chapter 4: Problem 47 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) - (c) to sketch the graph. You may want to check your work with a graphing calculator or computer. \(f(x)=\frac{1}{2} x^4-4 x^2+3\)
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Chapter 4: Problem 48 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.
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Chapter 4: Problem 49 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)-(c) to sketch the graph. You may want to check your work with a graphing calculator or computer. \(g(t)=3 t^{4}-8 t^{3}+12\)
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Chapter 4: Problem 50 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.
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Chapter 4: Problem 51 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.
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Chapter 4: Problem 52 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) - (c) to sketch the graph. You may want to check your work with a graphing calculator or computer. \(f(x)=\left(x^2-4\right)^3\)
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Chapter 4: Problem 53 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.
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Chapter 4: Problem 54 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.
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Chapter 4: Problem 55 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer. C(x) = x1/3(x + 4)
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Chapter 4: Problem 56 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer. f(x) = ln(x2 + 9)
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Chapter 4: Problem 57 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.
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Chapter 4: Problem 58 Calculus: Early Transcendentals 9
(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.
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Chapter 4: Problem 59 Calculus: Early Transcendentals 9
(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f. \(f(x)=1+\frac{1}{x}-\frac{1}{x^{2}}\)
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Chapter 4: Problem 60 Calculus: Early Transcendentals 9
(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.
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Chapter 4: Problem 61 Calculus: Early Transcendentals 9
(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.
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Chapter 4: Problem 62 Calculus: Early Transcendentals 9
(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.
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Chapter 4: Problem 63 Calculus: Early Transcendentals 9
(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.
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Chapter 4: Problem 64 Calculus: Early Transcendentals 9
(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f. \(f(x)=x-\frac{1}{6} x^{2}-\frac{2}{3} \ln x\)
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Chapter 4: Problem 65 Calculus: Early Transcendentals 9
(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.
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Chapter 4: Problem 66 Calculus: Early Transcendentals 9
(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.
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Chapter 4: Problem 67 Calculus: Early Transcendentals 9
Use the methods of this section to sketch several members of the given family of curves. What do the members have in common? How do they differ from each other?
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Chapter 4: Problem 68 Calculus: Early Transcendentals 9
Use the methods of this section to sketch several members of the given family of curves. What do the members have in common? How do they differ from each other?
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Chapter 4: Problem 69 Calculus: Early Transcendentals 9
(a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of x at which f increases most rapidly. Then find the exact value.
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Chapter 4: Problem 70 Calculus: Early Transcendentals 9
(a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of x at which f increases most rapidly. Then find the exact value. \(f(x)=x^{2} e^{-x}\)
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Chapter 4: Problem 71 Calculus: Early Transcendentals 9
(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates.
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Chapter 4: Problem 72 Calculus: Early Transcendentals 9
(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates.
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Chapter 4: Problem 73 Calculus: Early Transcendentals 9
Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph .
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Chapter 4: Problem 74 Calculus: Early Transcendentals 9
Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph .
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Chapter 4: Problem 75 Calculus: Early Transcendentals 9
A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point.
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Chapter 4: Problem 76 Calculus: Early Transcendentals 9
In an episode of The Simpsons television show, Homer reads from a newspaper and announces “Here’s good news! According to this eye-catching article, SAT scores are declining at a slower rate.” Interpret Homer’s statement in terms of a function and its first and second derivatives.
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Chapter 4: Problem 77 Calculus: Early Transcendentals 9
The president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its first and second derivatives.
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Chapter 4: Problem 78 Calculus: Early Transcendentals 9
Let f(t) be the temperature at time t where you live and suppose that at time t = 3 you feel uncomfortably hot. How do you feel about the given data in each case? (a) (b) (c) (d)
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Chapter 4: Problem 79 Calculus: Early Transcendentals 9
Let K(t) be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, K(8) - K(7) or K(3) - K(2)? Is the graph of K concave upward or concave downward? Why?
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Chapter 4: Problem 80 Calculus: Early Transcendentals 9
Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?
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Chapter 4: Problem 81 Calculus: Early Transcendentals 9
A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) ? Atpe-kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A = 0.01, p = 4, k = 0.07, and t is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. Then graph the drug response curve.
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Chapter 4: Problem 82 Calculus: Early Transcendentals 9
Normal Density Functions The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function. The constant is called the mean and the positive constant is called the standard deviation. For simplicity, let’s scale the function so as to remove the factor and let’s analyze the special case where = 0. So we study the function (a) Find the asymptote, maximum value, and inflection points of f. (b) What role does play in the shape of the curve? (c) Illustrate by graphing four members of this family on the same screen.
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Chapter 4: Problem 83 Calculus: Early Transcendentals 9
Find a cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) that has a local maximum value of 3 at x = -2 and a local minimum value of 0 at x = 1.
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Chapter 4: Problem 84 Calculus: Early Transcendentals 9
For what values of the numbers a and b does the function have the maximum value f(2) = 1?
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Chapter 4: Problem 85 Calculus: Early Transcendentals 9
Show that the curve y = (1 + x) / (1 + x2) has three points of inflection and they all lie on one straight line.
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Chapter 4: Problem 86 Calculus: Early Transcendentals 9
Show that the curves \(y=e^{-x}\) and \(y=-e^{-x}\) touch the curve \(y=e^{-x} \sin x\) at its inflection points.
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Chapter 4: Problem 87 Calculus: Early Transcendentals 9
Show that the inflection points of the curve y = x sin x lie on the curve \(y^{2}\left(x^{2}+4\right)=4 x^{2}\).
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Chapter 4: Problem 88 Calculus: Early Transcendentals 9
Assume that all of the functions are twice differentiable and the second derivatives are never 0. (a) If f and g are concave upward on an interval I, show that f + g is concave upward on I. (b) If f is positive and concave upward on I, show that the function g(x) = [f(x)]2 is concave upward on I.
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Chapter 4: Problem 89 Calculus: Early Transcendentals 9
Assume that all of the functions are twice differentiable and the second derivatives are never 0. (a) If f and g are positive, increasing, concave upward functions on an interval I, show that the product function fg is concave upward on I. (b) Show that part (a) remains true if f and g are both decreasing. (c) Suppose f is increasing and g is decreasing. Show, by giving three examples, that fg may be concave upward, concave downward, or linear. Why doesn’t the argument in parts (a) and (b) work in this case?
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Chapter 4: Problem 90 Calculus: Early Transcendentals 9
Assume that all of the functions are twice differentiable and the second derivatives are never 0. Suppose f and g are both concave upward on (-?, ?). Under what condition on f will the composite function h(x) ? f(g(x)) be concave upward?
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Chapter 4: Problem 91 Calculus: Early Transcendentals 9
Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x1, x2, and x3, show that the x-coordinate of the inflection point is (x1 + x2 + x3)/3
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Chapter 4: Problem 92 Calculus: Early Transcendentals 9
For what values of c does the polynomial P(x) + x4 + cx3 + x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases?
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Chapter 4: Problem 93 Calculus: Early Transcendentals 9
Prove that if (c, f(c)) is a point of inflection of the graph of f and \(f^{\prime \prime}\) exists in an open interval that contains c, then \(f^{\prime \prime}(c)=0\) [Hint. Apply the First Derivative Test and Fermat's Theorem to the function \(g=f^{\prime}\).]
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Chapter 4: Problem 94 Calculus: Early Transcendentals 9
Show that if f(x) = x4, then , but (0, 0) is not an inflection point of the graph of f.
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Chapter 4: Problem 95 Calculus: Early Transcendentals 9
Show that the function g(x) = x | x | has an inflection point at (0, 0) but does not exist.
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Chapter 4: Problem 96 Calculus: Early Transcendentals 9
Suppose that is continuous and = = 0, but . Does f have a local maximum or minimum at c? Does f have a point of inflection at c?
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Chapter 4: Problem 97 Calculus: Early Transcendentals 9
Suppose f is differentiable on an interval I and \(f^{\prime}(x)>0\) for all numbers x in I except for a single number c. Prove that f is increasing on the entire interval I.
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Chapter 4: Problem 98 Calculus: Early Transcendentals 9
For what values of c is the function \(f(x)=c x+\frac{1}{x^{2}+3}\) increasing on \((-\infty, \infty)\) ?
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Chapter 4: Problem 99 Calculus: Early Transcendentals 9
The three cases in the First Derivative Test cover the situations commonly encountered but do not exhaust all possibilities. Consider the functions f, g, and h whose values at 0 are all 0 and, for x 0, (a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that f has neither a local maximum nor a local minimum at 0, g has a local minimum, and h has a local maximum.
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