In Exercises 112, find dy/dx. 5xy 1
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0.1
The Real Number Line and Order
0.2
Absolute Value and Distance on the Real Number Line
0.3
Exponents and Radicals
0.4
Factoring Polynomials
0.5
Fractions and Rationalization
1
Functions, Graphs, and Limits
1.1
The Cartesian Plane and the Distance Formula
1.2
Graphs of Equations
1.3
Lines in the Plane and Slope
1.4
Functions
1.5
Limits
1.6
Continuity
2
Differentiation
2.1
The Derivative and the Slope of a Graph
2.2
Some Rules for Differentiation
2.3
Rates of Change: Velocity and Marginals
2.4
The Product and Quotient Rules
2.5
The Chain Rule
2.6
Higher-Order Derivatives
2.7
Implicit Differentiation
2.8
Related Rates
3
Applications of the Derivative
3.1
Increasing and Decreasing Functions
3.2
Extrema and the First-Derivative Test
3.3
Concavity and the Second-Derivative Test
3.4
Optimization Problems
3.5
Business and Economics Applications
3.6
Asymptotes
3.7
Curve Sketching: A Summary
3.8
Differentials and Marginal Analysis
4
Exponential and Logarithmic Functions
4.1
Exponential Functions
4.2
Natural Exponential Functions
4.3
Derivatives of Exponential Functions
4.4
Logarithmic Functions
4.5
Derivatives of Logarithmic Functions
4.6
Exponential Growth and Decay
5
Integration and Its Applications
5.1
Antiderivatives and Indefinite Integrals
5.2
The General Power Rule
5.3
Exponential and Logarithmic Integrals
5.4
Area and the Fundamental Theorem of Calculus
5.5
The Area of a Region Bounded by Two Graphs
5.6
The Definite Integral as the Limit of a Sum 3
5.7
Volumes of Solids of Revolution
6
Techniques of Integration
6.1
Integration by Substitution
6.2
Integration by Parts and Present Value
6.3
Partial Fractions and Logistic Growth
6.4
Integration Tables and Completing the Square
6.5
Numerical Integration
6.6
Improper Integrals
7
Functions of Several Variables
7.1
The Three-Dimensional Coordinate System
7.2
Surfaces in Space
7.3
Functions of Several Variables
7.4
Partial Derivatives
7.5
Extrema of Functions of Two Variables
7.6
Lagrange Multipliers
7.7
Least Squares Regression Analysis
7.8
Double Integrals and Area in the Plane
7.9
Applications of Double Integrals
Textbook Solutions for Brief Calculus: An Applied Approach
Chapter 2.7 Problem 32
Question
In Exercises 3134, find implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating at the point. 9x2 16y x 2 144 2
Solution
The first step in solving 2.7 problem number 32 trying to solve the problem we have to refer to the textbook question: In Exercises 3134, find implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating at the point. 9x2 16y x 2 144 2
From the textbook chapter Implicit Differentiation you will find a few key concepts needed to solve this.
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full solution
Title
Brief Calculus: An Applied Approach 7
Author
Ron Larson, Bruce H. Edwards
ISBN
9780618547197
Answer: In Exercises 3134, find implicitly and explicitly (the explicit functions are
Chapter 2.7 textbook questions
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Chapter 2: Problem 1 Brief Calculus: An Applied Approach 7
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Chapter 2: Problem 2 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. 1 2x 5xy 1 2 y 6x
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Chapter 2: Problem 3 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. y 0 x 1 2 1 x2
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Chapter 2: Problem 4 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. 4x2y 3 y 0
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Chapter 2: Problem 5 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. x2y2 4y 1
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Chapter 2: Problem 6 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. xy2 4xy 10
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Chapter 2: Problem 7 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. 4y2 xy 2
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Chapter 2: Problem 8 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. 2xy3 x2y 2
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Chapter 2: Problem 9 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. 2y x y2 3 5
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Chapter 2: Problem 10 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. xy y2 y x 1
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Chapter 2: Problem 11 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. x y 2x y 1 x
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Chapter 2: Problem 12 Brief Calculus: An Applied Approach 7
In Exercises 112, find dy/dx. 2x y x 5y 1 x
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Chapter 2: Problem 13 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 0, 7 2 y2 49 dy
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Chapter 2: Problem 14 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 4, 0 2 y2 16 x
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Chapter 2: Problem 15 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. y xy 4 5, 1 x 4
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Chapter 2: Problem 16 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 2, 1 2 y3 3 y
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Chapter 2: Problem 17 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 0, 2 3 xy y2 4 x
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Chapter 2: Problem 18 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 2, 1 2y y2x 2 x
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Chapter 2: Problem 19 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 0, 0 3y3 y x x
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Chapter 2: Problem 20 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 1, 1 3 y3 2xy x
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Chapter 2: Problem 21 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 16, 25 12 y12 9 x
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Chapter 2: Problem 22 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. xy x 2y 4, 1 x
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Chapter 2: Problem 23 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x 8, 1 23 y23 5 x
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Chapter 2: Problem 24 Brief Calculus: An Applied Approach 7
In Exercises 1324, find by implicit differentiation and evaluate the derivative at the given point. x y 1, 1 3 x3 y3 x 8,
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Chapter 2: Problem 25 Brief Calculus: An Applied Approach 7
In Exercises 2530, find the slope of the graph at the given point. 3x 2 2y 1 0 2 2y 5 0 x
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Chapter 2: Problem 26 Brief Calculus: An Applied Approach 7
In Exercises 2530, find the slope of the graph at the given point. 4x 3x 2 2y 1 0 2
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Chapter 2: Problem 27 Brief Calculus: An Applied Approach 7
In Exercises 2530, find the slope of the graph at the given point. x 2 4 2 y2 4
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Chapter 2: Problem 28 Brief Calculus: An Applied Approach 7
In Exercises 2530, find the slope of the graph at the given point. 4x2 y x 2 4
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Chapter 2: Problem 29 Brief Calculus: An Applied Approach 7
In Exercises 2530, find the slope of the graph at the given point. 4x2 + 9y2 = 36
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Chapter 2: Problem 30 Brief Calculus: An Applied Approach 7
In Exercises 2530, find the slope of the graph at the given point.x2 - y3 = 0
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Chapter 2: Problem 31 Brief Calculus: An Applied Approach 7
In Exercises 3134, find implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating at the point. x 2 144 2 y2 25
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Chapter 2: Problem 32 Brief Calculus: An Applied Approach 7
In Exercises 3134, find implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating at the point. 9x2 16y x 2 144 2
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Chapter 2: Problem 33 Brief Calculus: An Applied Approach 7
In Exercises 3134, find implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating at the point. x y 2 7 2 1 0
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Chapter 2: Problem 34 Brief Calculus: An Applied Approach 7
In Exercises 3134, find implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating at the point. 4y2 x x y 2 7
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Chapter 2: Problem 35 Brief Calculus: An Applied Approach 7
In Exercises 3540, find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. x 5, 12 12, 5 2 y2 169 x (
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Chapter 2: Problem 36 Brief Calculus: An Applied Approach 7
In Exercises 3540, find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. x 0, 3 2, 5 2 y2 9 x 5,
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Chapter 2: Problem 37 Brief Calculus: An Applied Approach 7
In Exercises 3540, find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. y 1, 5 1, 5 2 5x3 x 0,
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Chapter 2: Problem 38 Brief Calculus: An Applied Approach 7
In Exercises 3540, find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. 4xy x 1, 1 5, 1 2 5 y 1,
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Chapter 2: Problem 39 Brief Calculus: An Applied Approach 7
In Exercises 3540, find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. x 0, 2 2, 0 3 y3 8 4xy
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Chapter 2: Problem 40 Brief Calculus: An Applied Approach 7
In Exercises 3540, find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. y 2, 2 2, 2 2 x3 4 x x 0
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Chapter 2: Problem 41 Brief Calculus: An Applied Approach 7
Demand In Exercises 41 44, find the rate of change of with respect to p. p 0.006x x 0 4 0.02x2 10, p
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Chapter 2: Problem 42 Brief Calculus: An Applied Approach 7
Demand In Exercises 41 44, find the rate of change of with respect to p. p 0.002x x 0 4 0.01x2 5, p
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Chapter 2: Problem 43 Brief Calculus: An Applied Approach 7
Demand In Exercises 41 44, find the rate of change of with respect to p. p 0 < x 200 200 x 2x ,
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Chapter 2: Problem 44 Brief Calculus: An Applied Approach 7
Demand In Exercises 41 44, find the rate of change of with respect to p. p 0 < x 500 500 x 2x ,
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Chapter 2: Problem 45 Brief Calculus: An Applied Approach 7
Production Let represent the units of labor and the capital invested in a manufacturing process. When 135,540 units are produced, the relationship between labor and capital can be modeled by (a) Find the rate of change of with respect to when and (b) The model used in the problem is called the CobbDouglas production function. Graph the model on a graphing utility and describe the relationship between labor and capital.
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Chapter 2: Problem 46 Brief Calculus: An Applied Approach 7
Health: U.S. AIDS Epidemic The numbers (in millions) of cases y of AIDS reported in the years 1994 to 2001 can be modeled by where corresponds to 1994. (Source: U.S. Centers for Disease Control and Prevention) (a) Use a graphing utility to graph the model and describe the results. (b) Use the graph to determine the year during which the number of reported cases decreasing most rapidly. (c) Complete the table to confirm your estimate.
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