Evaluate the integral by making the given substitution.
Read moreTable 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
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
-
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 5.5 Problem 78
Question
Alabama Instruments Company has set up a production line to fx 0 0 a b manufacture a new calculator. The rate of production of these calculators after weeks is (Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week. I
Solution
The first step in solving 5.5 problem number 78 trying to solve the problem we have to refer to the textbook question: Alabama Instruments Company has set up a production line to fx 0 0 a b manufacture a new calculator. The rate of production of these calculators after weeks is (Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week. I
From the textbook chapter The Substitution Rule you will find a few key concepts needed to solve this.
Visible to paid subscribers only
Step 3 of 7)Visible to paid subscribers only
Subscribe to view the
full solution
full solution
Title
Calculus, 5
Author
James Stewart
ISBN
9780534393397
Alabama Instruments Company has set up a production line to fx 0 0 a b manufacture a new
Chapter 5.5 textbook questions
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Chapter 5: Problem 45 Calculus, 5
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take ).
Read more -
Chapter 5: Problem 46 Calculus, 5
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take ).
Read more -
Chapter 5: Problem 47 Calculus, 5
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take ).
Read more -
Chapter 5: Problem 48 Calculus, 5
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take ).
Read more -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Chapter 5: Problem 71 Calculus, 5
Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area.
Read more -
Chapter 5: Problem 72 Calculus, 5
Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area.
Read more -
Chapter 5: Problem 73 Calculus, 5
Evaluate by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.
Read more -
Chapter 5: Problem 74 Calculus, 5
Evaluate by making a substitution and interpreting the resulting integral in terms of an area.
Read more -
-
Chapter 5: Problem 76 Calculus, 5
A bacteria population starts with 400 bacteria and grows at a rate of bacteria per hour. How many bacteria will there be after three hours?
Read more -
Chapter 5: Problem 77 Calculus, 5
Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 Ls. This explains, in part, why the function has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time .t
Read more -
Chapter 5: Problem 78 Calculus, 5
Alabama Instruments Company has set up a production line to fx 0 0 a b manufacture a new calculator. The rate of production of these calculators after weeks is (Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week. I
Read more -
Chapter 5: Problem 79 Calculus, 5
If is continuous and 0 79. f fx dx 10 dx , find 0 y f 2x dx 4 0 .
Read more -
Chapter 5: Problem 80 Calculus, 5
If is continuous and f fx dx 4 y , find xfx 2 y dx 9 0 .
Read more -
Chapter 5: Problem 81 Calculus, 5
If is continuous on , prove that y b a f x dx y a b fx dx f y For the case where and , draw a diagram to interpret this equation geometrically as an equality of areas.
Read more -
Chapter 5: Problem 82 Calculus, 5
If is continuous on , prove that y b a fx c dx y bc ac f x dx f For the case where , draw a diagram to interpret this equation geometrically as an equality of areas.
Read more -
Chapter 5: Problem 83 Calculus, 5
If and are positive numbers, show that y 1 0 x a 1 x b dx y 1 0 x b 1 x a dx a b
Read more -
Chapter 5: Problem 84 Calculus, 5
Use the substitution to show that 0 xfsin x dx 2 y 0 f sin x dx u
Read more -
Chapter 5: Problem 85 Calculus, 5
Use Exercise 84 to evaluate the integral y 0 x sin x 1 cos2 x dx
Read more