(a) What is a one-to-one function? f (b) How can you tell from the graph of a function whether it is one-to-one?
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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
Graphing Calculators and Computers
1.5
Exponential Functions
1.6
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
Tangents, Velocities, and Other Rates of Change
2.8
Derivatives
2.9
The Derivative as a Function
3
Differentiation Rules
3.1
Derivatives of Polynomials and Exponential Functions
3.10
Related Rates
3.11
Linear Approximations and Differentials
3.3
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3.4
Derivatives of Trigonometric Functions
3.5
The Chain Rule
3.6
Implicit Differentiation
3.7
Higher Derivatives
3.8
Derivatives of Logarithmic Functions
3.9
Hyperbolic Functions
4
Applications of Differentiation
4.1
Maximum and Minimum Values
4.10
Antiderivatives
4.2
The Mean Value Theorem
4.3
How Derivatives Affect the Shape of a Graph
4.4
Indeterminate Forms and LHospitals Rule
4.5
Summary of Curve Sketching
4.6
Graphing with Calculus and Calculators
4.7
Optimization Problems
4.8
Applications to Business and Economics
4.9
Newtons Method
5
Integrals
5.1
Areas and Distances
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
5.6
The Logarithm Defined as an Integral
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 Computer Algebra Systems
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 Eulers Method
9.3
Separable Equations
9.4
Exponential Growth and Decay
9.5
The Logistic Equation
9.6
Linear Equations
9.7
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
Areas and Lengths in Polar Coordinates
10.5
Conic Sections
10.6
Conic Sections in Polar Coordinates
11
Infinite Sequences and Series
11.1
Sequences
11.10
Taylor and Maclaurin Series
11.11
The Binomial Series
11.12
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
11.6
Absolute Convergence and 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
12.7
Cylindrical and Spherical Coordinates
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
Iterated Integrals
15.3
Double Integrals over General Regions
15.4
Double Integrals in Polar Coordinates
15.5
Applications of Double Integrals
15.6
Surface Area
15.7
Triple Integrals
15.8
Triple Integrals in Cylindrical and Spherical Coordinates
15.9
Change of Variables in Multiple Integrals
16
Vector Calculus
16.1
Vector Fields
16.2
Line Integrals
16.3
The Fundamental Theorem for Line Integrals
16.4
The Fundamental Theorem for Line Integrals
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
17
Second-Order Differential Equations
17.1
Second-Order Linear Equations
17.2
Nonhomogeneous Linear Equations
17.3
Applications of Second-Order Differential Equations
17.4
Series Solutions
Textbook Solutions for Calculus,
Chapter 1.6 Problem 59
Question
If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after hours is . (See Exercise 25 in Section 1.5.) (a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50,000?
Solution
The first step in solving 1.6 problem number 59 trying to solve the problem we have to refer to the textbook question: If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after hours is . (See Exercise 25 in Section 1.5.) (a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50,000?
From the textbook chapter Inverse Functions and Logarithms you will find a few key concepts needed to solve this.
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full solution
Title
Calculus, 5
Author
James Stewart
ISBN
9780534393397
If a bacteria population starts with 100 bacteria and doubles every three hours, then
Chapter 1.6 textbook questions
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Chapter 1: Problem 1 Calculus, 5
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Chapter 1: Problem 2 Calculus, 5
(a) Suppose is a one-to-one function with domain and range . How is the inverse function defined? What is the domain of ? What is the range of ? (b) If you are given a formula for , how do you find a formula for ? f (c) If you are given the graph of , how do you find the graph of ?
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Chapter 1: Problem 3 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 4 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 5 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 6 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 7 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 8 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 9 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 10 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 11 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 12 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 13 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 14 Calculus, 5
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
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Chapter 1: Problem 15 Calculus, 5
Use a graph to decide whether is one-to-one. f x x 3 x 3 x f
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Chapter 1: Problem 16 Calculus, 5
Use a graph to decide whether is one-to-one. f x x f x x 3 x 3 x f
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Chapter 1: Problem 20 Calculus, 5
The graph of is given. (a) Why is one-to-one? (b) State the domain and range of . (c) Estimate the value of .
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Chapter 1: Problem 21 Calculus, 5
The formula , where , expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?
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Chapter 1: Problem 22 Calculus, 5
In the theory of relativity, the mass of a particle with speed is where is the rest mass of the particle and is the speed of light in a vacuum. Find the inverse function of and explain its meaning.
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Chapter 1: Problem 29 Calculus, 5
Find an explicit formula for and use it to graph , and the line on the same screen. To check your work, see whether the graphs of and are reflections about the line.
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Chapter 1: Problem 30 Calculus, 5
Find an explicit formula for and use it to graph , and the line on the same screen. To check your work, see whether the graphs of and are reflections about the line.
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Chapter 1: Problem 33 Calculus, 5
(a) How is the logarithmic function defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function y loga x if .
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Chapter 1: Problem 34 Calculus, 5
(a) What is the natural logarithm? x (b) What is the common logarithm? (c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.
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Chapter 1: Problem 42 Calculus, 5
Use Formula 10 to evaluate each logarithm correct to six decimal places.
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Chapter 1: Problem 43 Calculus, 5
Use Formula 10 to graph the given functions on a common screen. How are these graphs related? y log1.5 x y ln x y log10 x y log50 x log
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Chapter 1: Problem 44 Calculus, 5
Use Formula 10 to graph the given functions on a common screen. How are these graphs related? y ln x y log10 xy 0x y
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Chapter 1: Problem 45 Calculus, 5
Suppose that the graph of is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches ft?
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Chapter 1: Problem 46 Calculus, 5
Compare the functions and by graphing both and in several viewing rectangles. When does the graph of finally surpass the graph of ?
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Chapter 1: Problem 47 Calculus, 5
Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3.
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Chapter 1: Problem 48 Calculus, 5
Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3.
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Chapter 1: Problem 57 Calculus, 5
Graph the function and explain why it is one-to-one. Then use a computer algebra system to find an explicit expression for . (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)
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Chapter 1: Problem 58 Calculus, 5
(a) If , use a computer algebra system to find an expression for . (b) Use the expression in part (a) to graph , and on the same screen.
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Chapter 1: Problem 59 Calculus, 5
If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after hours is . (See Exercise 25 in Section 1.5.) (a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50,000?
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Chapter 1: Problem 60 Calculus, 5
When a camera flash goes off, the batteries immediately begin to recharge the flashs capacitor, which stores electric charge given by (The maximum charge capacity is and is measured in seconds.) (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of capacity if ?
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Chapter 1: Problem 61 Calculus, 5
Starting with the graph of , find the equation of the graph that results from (a) shifting 3 units upward (b) shifting 3 units to the left (c) reflecting about the x-axis (d) reflecting about the y-axis (e) reflecting about the line (f) reflecting about the x-axis and then about the line (g) reflecting about the y-axis and then about the line (h) shifting 3 units to the left and then reflecting about the line
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Chapter 1: Problem 62 Calculus, 5
(a) If we shift a curve to the left, what happens to its reflection about the line ? In view of this geometric principle, find an expression for the inverse of , where is a one-to-one function. (b) Find an expression for the inverse of , where c 0.
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Chapter 1: Problem 73 Calculus, 5
Graph the given functions on the same screen. How are these graphs related?
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Chapter 1: Problem 74 Calculus, 5
Graph the given functions on the same screen. How are these graphs related?
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Chapter 1: Problem 76 Calculus, 5
(a) Graph the function and explain the appearance of the graph. (b) Graph the function . How do you explain the appearance of this graph?
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