## Solutions for Calculus

Problem 1E:

Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c),...Problem 2E:

Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c),...Problem 3E:

Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c),...Problem 4E:

Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c),...Problem 6E:

Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point...Problem 7E:

Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point...Problem 9E:

Sketching a Graph by Point Plotting In Exercises 514, sketch the graph of the equation by point...Problem 10E:

Sketching a Graph by Point Plotting In Exercises 716, sketch the graph of the equation by point...Problem 11E:

Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point...Problem 15E:

Approximating Solution Points Using Technology In Exercises 17 and 18, use a graphing utility to...Problem 16E:

Approximating Solution Points Using Technology In Exercises 17 and 18, use a graphing utility to...Problem 43E:

Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any Intercepts and test for...Problem 45E:

Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any Intercepts and test for...Problem 49E:

Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any Intercepts and test for...Problem 50E:

Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any Intercepts and test for...Problem 52E:

Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any Intercepts and test for...Problem 54E:

Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any Intercepts and test for...Problem 55E:

Using Intercepts and Symmetry to Sketch a Graph In Exercises 3956, find any intercepts and test for...Problem 57E:

Finding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of...Problem 58E:

Finding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of...Problem 60E:

Finding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of...Problem 62E:

Finding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of...Problem 63E:

Finding Points of Intersection Using Technology In Exercises 63-66, use a graphing utility to find...Problem 66E:

Finding Points of Intersection Using Technology In Exercises 6366, use a graphing utility to find...Problem 67E:

Modeling Data The table shows the Gross Domestic Product, or GDP (in trillions of dollars), for...Problem 68E:

Modeling Data The table shows the numbers of cellular phone subscribers (in millions) in the United...Problem 69E:

Break-Even Point Find the sales necessary to break even (R = C) when the cost C of producing x units...Problem 70E:

Copper Wire The resistance y in ohms of 1000 feet of solid copper wire at 77F can be approximated by...Problem 71E:

Using Solution Points For what values of k does the graph of y = kx3 pass through the point? (a)...Problem 72E:

Using Solution Points For what values of k does the graph of y2=4kx pass through the point? (a)...Problem 73E:

WRITING ABOUT CONCEPTS Writing Equations In Exercises 73 and 74, write an equation whose graph has...Problem 74E:

EXPLORING CONCEPTS Using Intercepts Write an equation whose graph has intercepts at x=32,x=4andx=52....Problem 76E:

HOW DO YOU SEE IT? Use the graphs of the two equations to answer the questions below (a) What are...Problem 77E:

True or False ? In Exercises 75-78, determine whether the statement is true or false. If it is...# Browse All Chapters of This Textbook

Chapter P - Preparation For CalculusChapter P.1 - Graphs And ModelsChapter P.2 - Linear Models And Rates Of ChangeChapter P.3 - Functions And Their GraphsChapter P.4 - Review Of Trigonometric FunctionsChapter 1 - Limits And Their PropertiesChapter 1.1 - A Preview Of CalculusChapter 1.2 - Finding Limits Graphically And NumericallyChapter 1.3 - Evaluating Limits AnalyticallyChapter 1.4 - Continuity And One-sided Limits

Chapter 1.5 - Infinite LimitsChapter 2 - DifferentiationChapter 2.1 - The Derivative And The Tangent Line ProblemChapter 2.2 - Basic Differentiation Rules And Rates Of ChangeChapter 2.3 - Product And Quotient Rules And Higher-order DerivativesChapter 2.4 - The Chain RuleChapter 2.5 - Implicit DifferentiationChapter 2.6 - Related RatesChapter 3 - Applications Of DifferentiationChapter 3.1 - Extrema On An IntervalChapter 3.2 - Rolle’s Theorem And The Mean Value TheoremChapter 3.3 - Increasing And Decreasing Functions And The First Derivative TestChapter 3.4 - Concavity And The Second Derivative TestChapter 3.5 - Limits At InfinityChapter 3.6 - A Summary Of Curve SketchingChapter 3.7 - Optimization ProblemsChapter 3.8 - Newton’s MethodChapter 3.9 - DifferentialsChapter 4 - IntegrationChapter 4.1 - Antiderivatives And Indefinite IntegrationChapter 4.2 - AreaChapter 4.3 - Riemann Sums And Definite IntegralsChapter 4.4 - The Fundamental Theorem Of CalculusChapter 4.5 - Integration By SubstitutionChapter 4.6 - Numerical IntegrationChapter 5 - Logarithmic, Exponential, And Other Transcendental FunctionsChapter 5.1 - The Natural Logarithmic Function: DifferentiationChapter 5.2 - The Natural Logarithmic Function: IntegrationChapter 5.3 - Inverse FunctionsChapter 5.4 - Exponential Functions: Differentiation And IntegrationChapter 5.5 - Bases Other Than E And ApplicationsChapter 5.6 - Inverse Trigonometric Functions: DifferentiationChapter 5.7 - Inverse Trigonometric Functions: IntegrationChapter 5.8 - Hyperbolic FunctionsChapter 6 - Differential EquationsChapter 6.1 - Slope Fields And Euler’s MethodChapter 6.2 - Differential Equations: Growth And DecayChapter 6.3 - Separation Of Variables And The Logistic EquationChapter 6.4 - First-order Linear Differential EquationsChapter 7 - Applications Of IntegrationChapter 7.1 - Area Of A Region Between Two CurvesChapter 7.2 - Volume: The Disk MethodChapter 7.3 - Volume: The Shell MethodChapter 7.4 - Arc Length And Surfaces Of RevolutionChapter 7.5 - WorkChapter 7.6 - Moments, Centers Of Mass, And CentroidsChapter 7.7 - Fluid Pressure And Fluid ForceChapter 8 - Integration Techniques, L’hopital’s Rule, And Improper IntegralsChapter 8.1 - Basic Integration RulesChapter 8.2 - Integration By PartsChapter 8.3 - Trigonometric IntegralsChapter 8.4 - Trigonometric SubstitutionChapter 8.5 - Partial FractionsChapter 8.6 - Integration By Tables And Other Integration TechniquesChapter 8.7 - Indeterminate Forms And L’hopital’s RuleChapter 8.8 - Improper IntegralsChapter 9 - Infinite SeriesChapter 9.1 - SequencesChapter 9.2 - Series And ConvergenceChapter 9.3 - The Integral Test And P-seriesChapter 9.4 - Comparisons Of SeriesChapter 9.5 - Alternating SeriesChapter 9.6 - The Ratio And Root TestsChapter 9.7 - Taylor Polynomials And ApproximationsChapter 9.8 - Power SeriesChapter 9.9 - Representation Of Functions By Power SeriesChapter 9.10 - Taylor And Maclaurin SeriesChapter 10 - Conics, Parametric Equations, And Polar CoordinatesChapter 10.1 - Conics And CalculusChapter 10.2 - Plane Curves And Parametric EquationsChapter 10.3 - Parametric Equations And CalculusChapter 10.4 - Polar Coordinates And Polar GraphsChapter 10.5 - Area And Arc Length In Polar CoordinatesChapter 10.6 - Polar Equations Of Conics And Kepler’s LawsChapter 11 - Vectors And The Geometry Of SpaceChapter 11.1 - Vectors In The PlaneChapter 11.2 - Space Coordinates And Vectors In SpaceChapter 11.3 - The Dot Product Of Two VectorsChapter 11.4 - The Cross Product Of Two Vectors In SpaceChapter 11.5 - Lines And Planes In SpaceChapter 11.6 - Surfaces In SpaceChapter 11.7 - Cylindrical And Spherical CoordinatesChapter 12 - Vector-valued FunctionsChapter 12.1 - Vector-valued FunctionsChapter 12.2 - Differentiation And Integration Of Vector-valued FunctionsChapter 12.3 - Velocity And AccelerationChapter 12.4 - Tangent Vectors And Normal VectorsChapter 12.5 - Arc Length And CurvatureChapter 13 - Functions Of Several VariablesChapter 13.1 - Introduction To Functions Of Several VariablesChapter 13.2 - Limits And ContinuityChapter 13.3 - Partial DerivativesChapter 13.4 - DifferentialsChapter 13.5 - Chain Rules For Functions Of Several VariablesChapter 13.6 - Directional Derivatives And GradientsChapter 13.7 - Tangent Planes And Normal LinesChapter 13.8 - Extrema Of Functions Of Two VariablesChapter 13.9 - Applications Of ExtremaChapter 13.10 - Lagrange MultipliersChapter 14 - Multiple IntegrationChapter 14.1 - Iterated Integrals And Area In The PlaneChapter 14.2 - Double Integrals And VolumeChapter 14.3 - Change Of Variables: Polar CoordinatesChapter 14.4 - Center Of Mass And Moments Of InertiaChapter 14.5 - Surface AreaChapter 14.6 - Triple Integrals And ApplicationsChapter 14.7 - Triple Integrals In Other CoordinatesChapter 14.8 - Change Of Variables: JacobiansChapter 15 - Vector AnalysisChapter 15.1 - Vector FieldsChapter 15.2 - Line IntegralsChapter 15.3 - Conservative Vector Fields And Independence Of PathChapter 15.4 - Green’s TheoremChapter 15.5 - Parametric SurfacesChapter 15.6 - Surface IntegralsChapter 15.7 - Divergence TheoremChapter 15.8 - Stokes’s Theorem

### Book Details

P. PREPARATION FOR CALCULUS. Graphs and Models. Linear Models and Rates of Change. Functions and Their Graphs. Fitting Models to Data. Review Exercises. P.S. Problem Solving. 1. LIMITS AND THEIR PROPERTIES. A Preview of Calculus. Finding Limits Graphically and Numerically. Evaluating Limits Analytically. Continuity and One-Sided Limits. Infinite Limits. Section Project: Graphs and Limits of Trigonometric Functions. Review Exercises. P.S. Problem Solving. 2. DIFFERENTIATION. The Derivative and the Tangent Line Problem. Basic Differentiation Rules and Rates of Change. Product and Quotient Rules and Higher-Order Derivatives. The Chain Rule. Implicit Differentiation. Section Project: Optical Illusions. Related Rates. Review Exercises. P.S. Problem Solving. 3. APPLICATIONS OF DIFFERENTIATION. Extrema on an Interval. Rolle's Theorem and the Mean Value Theorem. Increasing and Decreasing Functions and the First Derivative Test. Section Project: Rainbows. Concavity and the Second Derivative Test. Limits at Infinity. A Summary of Curve Sketching. Optimization Problems. Section Project: Connecticut River. Newton's Method. Differentials. Review Exercises. P.S. Problem Solving. 4. INTEGRATION. Antiderivatives and Indefinite Integration. Area. Riemann Sums and Definite Integrals. The Fundamental Theorem of Calculus. Section Project: Demonstrating the Fundamental Theorem. Integration by Substitution. Numerical Integration. Review Exercises. P.S. Problem Solving. 5. LOGARITHMIC, EXPONENTIAL, AND OTHER TRANSCENDENTAL FUNCTIONS. The Natural Logarithmic Function: Differentiation. The Natural Logarithmic Function: Integration. Inverse Functions. Exponential Functions: Differentiation and Integration. Bases Other than e and Applications. Section Project: Using Graphing Utilities to Estimate Slope. Inverse Trigonometric Functions: Differentiation. Inverse Trigonometric Functions: Integration. Hyperbolic Functions. Section Project: St. Louis Arch. Review Exercises. P.S. Problem Solving. 6. DIFFERENTIAL EQUATIONS. Slope Fields and Euler's Method. Differential Equations: Growth and Decay. Separation of Variables and the Logistic Equation. First-Order Linear Differential Equations. Section Project: Weight Loss. Review Exercises. P.S. Problem Solving. 7. APPLICATIONS OF INTEGRATION. Area of a Region Between Two Curves. Volume: The Disk Method. Volume: The Shell Method. Section Project: Saturn. Arc Length and Surfaces of Revolution. Work. Section Project: Tidal Energy. Moments, Centers of Mass, and Centroids. Fluid Pressure and Fluid Force. Review Exercises. P.S. Problem Solving. 8. INTEGRATION TECHNIQUES, L'HOPITAL'S RULE, AND IMPROPER INTEGRALS. Basic Integration Rules. Integration by Parts. Trigonometric Integrals. Section Project: Power Lines. Trigonometric Substitution. Partial Fractions. Integration by Tables and Other Integration Techniques. Indeterminate Forms and L'Hopital's Rule. Improper Integrals. Review Exercises. P.S. Problem Solving. 9. INFINITE SERIES. Sequences. Series and Convergence. Section Project: Cantor's Disappearing Table. The Integral Test and p-Series. Section Project: The Harmonic Series. Comparisons of Series. Section Project: Solera Method. Alternating Series. The Ratio and Root Tests. Taylor Polynomials and Approximations. Power Series. Representation of Functions by Power Series. Taylor and Maclaurin Series. Review Exercises. P.S. Problem Solving. 10. CONICS, PARAMETRIC EQUATIONS, AND POLAR COORDINATES. Conics and Calculus. Plane Curves and Parametric Equations. Section Project: Cycloids. Parametric Equations and Calculus. Polar Coordinates and Polar Graphs. Section Project: Anamorphic Art. Area and Arc Length in Polar Coordinates. 10.6 Polar Equations of Conics and Kepler's Laws. Review Exercises. P.S. Problem Solving. 11. VECTORS AND THE GEOMETRY OF SPACE. Vectors in the Plane. Space Coordinates and Vectors in Space. The Dot Product of Two Vectors. The Cross Product of Two Vectors in Space. Lines and Planes in Space. Section Project: Distances in Space. Surfaces in Space. Cylindrical and Spherical Coordinates. Review Exercises. P.S. Problem Solving. 12. VECTOR-VALUED FUNCTIONS. Vector-Valued Functions. Section Project: Witch of Agnesi. Differentiation and Integration of Vector-Valued Functions. Velocity and Acceleration. Tangent Vectors and Normal Vectors. Arc Length and Curvature. Review Exercises. P.S. Problem Solving. 13. FUNCTIONS OF SEVERAL VARIABLES. Introduction to Functions of Several Variables. Limits and Continuity. Partial Derivatives. Section Project: Moire Fringes. Differentials. Chain Rules for Functions of Several Variables. Directional Derivatives and Gradients. Tangent Planes and Normal Lines. Section Project: Wildflowers. Extrema of Functions of Two Variables. Applications of Extrema of Functions of Two Variables. Section Project: Building a Pipeline. Lagrange Multipliers. Review Exercises. P.S. Problem Solving. 14. MULTIPLE INTEGRATION. Iterated Integrals and Area in the Plane. Double Integrals and Volume. Change of Variables: Polar Coordinates. Center of Mass and Moments of Inertia. Section Project: Center of Pressure on a Sail. Surface Area. Section Project: Capillary Action. Triple Integrals and Applications. Triple Integrals in Cylindrical and Spherical Coordinates. Section Project: Wrinkled and Bumpy Spheres. Change of Variables: Jacobians. Review Exercises. P.S. Problem Solving. 15. VECTOR ANALYSIS. Vector Fields. Line Integrals. Conservative Vector Fields and Independence of Path. Green's Theorem. Section Project: Hyperbolic and Trigonometric Functions. Parametric Surfaces. Surface Integrals. Section Project: Hyperboloid of One Sheet. Divergence Theorem. Stokes's Theorem. Review Exercises. Section Project: The Planimeter. P.S. Problem Solving. 16. SECOND ORDER DIFFERENTIAL EQUATIONS* ONLINE. Exact First-Order Equations. Second-Order Homogeneous Linear Equations. Second-Order Nonhomogeneous Linear Equations. Series Solutions of Differential Equations. Review Exercises. P.S. Problem Solving. APPENDIX. A. Proofs of Selected Theorems. B. Integration Tables. C. Precalculus Review (Web). C.1 Real Numbers and the Real Number Line. C.2 The Cartesian Plane. C.3 Review of Trigonometric Functions. D. Rotation and the General Second-Degree Equation (Web). E. Complex Numbers (Web). F. Business and Economic Applications (Web).

# Sample Solutions for this Textbook

We offer sample solutions for Calculus homework problems. See examples below:

Chapter P, Problem 1REChapter 1, Problem 1REChapter 2, Problem 1REChapter 3, Problem 1REChapter 4, Problem 1REChapter 5, Problem 1REChapter 6, Problem 1REChapter 7, Problem 1REGiven: The integral is ∫xx2−36dx. Formula used: The power formula, ∫undu=un+1n+1+C Calculation:...

Given: The provided sequence is an=5n. Calculation: Consider the expression: an=5n …… (1) To find...Chapter 10, Problem 1REChapter 11, Problem 1REChapter 12, Problem 1REGiven: The function f(x,y)=3x2y. Calculation: Consider the function, f(x,y)=3x2y. So,...Given: The integral is ∫02xxy3dy. Formula used: The rule of integration: ∫xndx=xn+1n+1+C, where C is...Chapter 15, Problem 1RE

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