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11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275361

Chapter 2, Problem 4RE

Textbook Problem

Finding the Derivative by the Limit Process **In Exercises 1-4, find the derivative of the function by the limit process.**

Calculus of a Single Variable

Ch. 2.1 - Tangent Line Describe how to find the slope of the...Ch. 2.1 - Notation List four notation alternatives to f(x).Ch. 2.1 - Derivative Describe how to find the derivative of...Ch. 2.1 - Continuity and Differentiability Describe the...Ch. 2.1 - Estimating Slope In Exercises 5 and 6, estimate...Ch. 2.1 - Estimating Slope In Exercises 5 and 6, estimate...Ch. 2.1 - Slopes of Secant Lines In Exercises 7 and 8, use...Ch. 2.1 - Slopes of Secant Lines In Exercises 7 and 8. use...Ch. 2.1 - Finding the Slope of a Tangent Line In Exercises...Ch. 2.1 - Finding the Slope of a Tangent Line In Exercises...

Ch. 2.1 - Finding the Slope of a Tangent Line In Exercises...Ch. 2.1 - Finding the Slope of a Tangent Line In Exercises...Ch. 2.1 - Finding the Slope of a Tangent Line In Exercises...Ch. 2.1 - Finding the Slope of a Tangent Line In Exercises...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding the Derivative by the Limit Process In...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Sketching a Derivative In Exercises 43-48, sketch...Ch. 2.1 - Sketching a Derivative In Exercises 43-48, sketch...Ch. 2.1 - Sketching a Derivative In Exercises 43-48, sketch...Ch. 2.1 - Sketching a Derivative In Exercises 43-48, sketch...Ch. 2.1 - Sketching a Derivative In Exercises 43-48, sketch...Ch. 2.1 - Sketching a Derivative In Exercises 43-48, sketch...Ch. 2.1 - Sketching a Graph Sketch a graph of a function...Ch. 2.1 - Sketching a Graph Sketch a graph of a function...Ch. 2.1 - Domain of the Derivative Do f and f always have...Ch. 2.1 - Symmetry of a Graph A function f is symmetric with...Ch. 2.1 - Using a Tangent Line The tangent line to the graph...Ch. 2.1 - Using a Tangent Line The tangent line to the graph...Ch. 2.1 - Working Backwards In Exercises 55-58, the Unlit...Ch. 2.1 - Working Backwards In Exercises 55-58, the Unlit...Ch. 2.1 - Working Backwards In Exercises 55-58, the Unlit...Ch. 2.1 - Working Backwards In Exercises 55-58, the Unlit...Ch. 2.1 - Writing a Function Using Derivatives In Exercises...Ch. 2.1 - Writing a Function Using Derivatives In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.1 - Graphical Reasoning Use a graphing utility to...Ch. 2.1 - HOW DO YOU SEE IT? The figure shows die graph of...Ch. 2.1 - Graphical Reasoning Consider the function...Ch. 2.1 - Graphical Reasoning Consider the function...Ch. 2.1 - Approximating a Derivative In Exercises 67 and 68,...Ch. 2.1 - Approximating a Derivative In Exercises 67 and 68,...Ch. 2.1 - Using the Alternative Form of the Derivative In...Ch. 2.1 - Using the Alternative Form of the Derivative In...Ch. 2.1 - Using the Alternative Form of the Derivative In...Ch. 2.1 - Using the Alternative Form of the Derivative In...Ch. 2.1 - Using the Alternative Form of the Derivative In...Ch. 2.1 - Using the Alternative Form of the Derivative In...Ch. 2.1 - Using the Alternative Form of the Derivative In...Ch. 2.1 - Using the Alternative Form of the Derivative In...Ch. 2.1 - Determining Differentiability In Exercises 77-80,...Ch. 2.1 - Determining Differentiability In Exercises 77-80,...Ch. 2.1 - Determining Differentiability In Exercises 77-80,...Ch. 2.1 - Determining Differentiability In Exercises 77-80,...Ch. 2.1 - Graphical Analysis In Exercises 81-84, use a...Ch. 2.1 - Graphical Analysis In Exercises 81-84, use a...Ch. 2.1 - Graphical Analysis In Exercises 81-84, use a...Ch. 2.1 - Graphical Analysis In Exercises 81-84, use a...Ch. 2.1 - Determining Differentiability In Exercises 85-88,...Ch. 2.1 - Determining Differentiability In Exercises 85-88,...Ch. 2.1 - Determining Differentiability In Exercises 85-88,...Ch. 2.1 - Determining Differentiability In Exercises 85-88,...Ch. 2.1 - Determining Differentiability In Exercises 89 and...Ch. 2.1 - Determining Differentiability In Exercises 89 and...Ch. 2.1 - Graphical Reasoning A line with slope m passes...Ch. 2.1 - Conjecture Consider the functions f(x)=x2 and...Ch. 2.1 - True or False? In Exercises 93-96, determine...Ch. 2.1 - True or False? In Exercises 93-96, determine...Ch. 2.1 - True or False? In Exercises 93-96, determine...Ch. 2.1 - True or False? In Exercises 93-96. determine...Ch. 2.1 - Differentiability and Continuity Let...Ch. 2.1 - Writing Use a graphing utility to graph the two...Ch. 2.2 - CONCEPT CHECK Constant Rule What is the derivative...Ch. 2.2 - Finding a Derivative Explain how to find the...Ch. 2.2 - Derivatives of Trigonometric Functions What are...Ch. 2.2 - Average Velocity and Velocity Describe the...Ch. 2.2 - Estimating Slope In Exercises 5 and 6, use the...Ch. 2.2 - Estimating Slope In Exercises 5 and 6, use the...Ch. 2.2 - Finding a Derivative In Exercises 7-26. Use the...Ch. 2.2 - Finding a Derivative In Exercises 7-26. Use the...Ch. 2.2 - Finding a Derivative In Exercises 7-26. Use the...Ch. 2.2 - Finding a Derivative In Exercises 7-26. Use the...Ch. 2.2 - Finding a Derivative In Exercises 7-26. 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The...Ch. 2.2 - Exploring a Relationship In Exercises 69-72. The...Ch. 2.2 - Exploring a Relationship In Exercises 69-72. The...Ch. 2.2 - Exploring a Relationship In Exercises 69-72. The...Ch. 2.2 - A Function and Its Derivative In Exercises 73 and...Ch. 2.2 - A Function and Its Derivative In Exercises 73 and...Ch. 2.2 - Sketching a Graph Sketch the graph of a function f...Ch. 2.2 - HOW DO YOU SEE IT? Use the graph of f to answer...Ch. 2.2 - Finding Equations of Tangent Lines Sketch the...Ch. 2.2 - Tangent Lines Show that the graphs of the two...Ch. 2.2 - Horizontal Tangent Line Show that the graph of the...Ch. 2.2 - Tangent Line Show that the graph of the function...Ch. 2.2 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.2 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.2 - Linear Approximation Consider the function...Ch. 2.2 - Linear Approximation Repeat Exercise 83 for the...Ch. 2.2 - True or False? In Exercises 85-90, determine...Ch. 2.2 - True or False? In Exercises 85-90, determine...Ch. 2.2 - True or False? In Exercises 85-90, determine...Ch. 2.2 - True or False? In Exercises 85-90, determine...Ch. 2.2 - True or False? In Exercises 85-90, determine...Ch. 2.2 - True or False? In Exercises 85-90, determine...Ch. 2.2 - Finding Rates of Change In Exercises 91-94, find...Ch. 2.2 - Finding Rates of Change In Exercises 91-94, find...Ch. 2.2 - Finding Rates of Change In Exercises 91-94, find...Ch. 2.2 - Finding Rates of Change In Exercises 91-94, find...Ch. 2.2 - Vertical Motion In Exercises 95 and 96, use the...Ch. 2.2 - Vertical Motion In Exercises 95 and 96, use the...Ch. 2.2 - Vertical Motion In Exercises 97 and 98, use the...Ch. 2.2 - Vertical Motion In Exercises 97 and 98, use the...Ch. 2.2 - Think About It The graph of the position function...Ch. 2.2 - Think About It The graph of the velocity function...Ch. 2.2 - Volume The volume of a cube with sides of length s...Ch. 2.2 - Area The area of a square with sides of length s...Ch. 2.2 - Modeling Data The stopping distance of an...Ch. 2.2 - Fuel Cost A car is driven 15,000 miles a year and...Ch. 2.2 - Velocity Verify that the average velocity over the...Ch. 2.2 - Inventory Management The annual inventory cost C...Ch. 2.2 - Finding an Equation of a Parabola Find an equation...Ch. 2.2 - Proof Lei (a, b) be an arbitrary point on the...Ch. 2.2 - Tangent Line find the equation(s) of the tangent...Ch. 2.2 - Tangent Line Find the equation(s) of the tangent...Ch. 2.2 - Making a Function Differentiable In Exercises 111...Ch. 2.2 - Making a Function Differentiable In Exercises 111...Ch. 2.2 - Determining Differentiability Where are the...Ch. 2.2 - Proof Prove that ddx[cosx]=sinx.Ch. 2.2 - Find all differentiable functions f: such that...Ch. 2.3 - CONCEPT CHECK Product Rule Describe the Product...Ch. 2.3 - CONCEPT CHECK Quotient Rule Describe the Quotient...Ch. 2.3 - CONCEPT CHECK Trigonometric Functions What are the...Ch. 2.3 - CONCEPT CHECK Higher-Order Derivative What is a...Ch. 2.3 - Using the Product Rule In Exercises 5-10. use the...Ch. 2.3 - Using the Product Rule In Exercises 5-10. use the...Ch. 2.3 - Using the Product Rule In Exercises 5-10. use the...Ch. 2.3 - Using the Product Rule In Exercises 5-10. use the...Ch. 2.3 - Using the Product Rule In Exercises 5-10. use the...Ch. 2.3 - Using the Product Rule In Exercises 5-10. use the...Ch. 2.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 2.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 2.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 2.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 2.3 - Using the Quotient Rule In Exercises 11-16, use...Ch. 2.3 - Using the Quotient Rule In Exercises 11-16. use...Ch. 2.3 - Finding and Evaluating a Derivative In Exercises...Ch. 2.3 - Finding and Evaluating a Derivative In Exercises...Ch. 2.3 - Finding and Evaluating a Derivative In Exercises...Ch. 2.3 - Finding and Evaluating a Derivative In Exercises...Ch. 2.3 - Finding and Evaluating a Derivative In Exercises...Ch. 2.3 - Finding and Evaluating a Derivative In Exercises...Ch. 2.3 - Using the Constant Multiple Rule In Exercises...Ch. 2.3 - Using the Constant Multiple Rule In Exercises...Ch. 2.3 - Using the Constant Multiple Rule In Exercises...Ch. 2.3 - Using the Constant Multiple Rule In Exercises...Ch. 2.3 - Using the Constant Multiple Rule In Exercises...Ch. 2.3 - Using the Constant Multiple Rule In Exercises...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative In Exercises 29-40, Find the...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative of a Trigonometric Function...Ch. 2.3 - Finding a Derivative Using Technology In Exercises...Ch. 2.3 - Finding a Derivative Using Technology In Exercises...Ch. 2.3 - Finding the Slope of a Graph In Exercises 59-62,...Ch. 2.3 - Finding the Slope of a Graph In Exercises 59-62,...Ch. 2.3 - Finding the Slope of a Graph In Exercises 59-62,...Ch. 2.3 - Finding the Slope of a Graph In Exercises 59-62,...Ch. 2.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.3 - Famous Curves In Exercises 69-72, find an equation...Ch. 2.3 - Famous Curves In Exercises 69-72, find an equation...Ch. 2.3 - Famous Curves In Exercises 69-72, find an equation...Ch. 2.3 - Famous Curves In Exercises 69-72, find an equation...Ch. 2.3 - Horizontal Tangent Line In Exercises 73-76....Ch. 2.3 - Horizontal Tangent Line In Exercises 73-76....Ch. 2.3 - Horizontal Tangent Line In Exercises 73-76....Ch. 2.3 - Horizontal Tangent Line In Exercises 73-76....Ch. 2.3 - Tangent Lines Find equations of the tangent lines...Ch. 2.3 - Tangent Lines Find equations of the tangent lines...Ch. 2.3 - Exploring a Relationship In Exercises 79 and 80....Ch. 2.3 - Exploring a Relationship In Exercises 79 and 80....Ch. 2.3 - Finding Derivatives In Exercises 81 and 82, use...Ch. 2.3 - Finding Derivatives In Exercises 81 and 82, use...Ch. 2.3 - Area The length of a rectangle is given by 6t+5...Ch. 2.3 - Volume The radius of a right circular cylinder is...Ch. 2.3 - Inventory Replenishment The ordering and...Ch. 2.3 - Population Growth A population of 500 bacteria is...Ch. 2.3 - Proof Prove each differentiation rule (a)...Ch. 2.3 - Rate of Change Determine whether there exist any...Ch. 2.3 - Modeling Data The table shows the national health...Ch. 2.3 - Satellites When satellites observe Earth, they can...Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Second Derivative In Exercises 91-100....Ch. 2.3 - Finding a Higher-Order Derivative In Exercises...Ch. 2.3 - Finding a Higher-Order Derivative In Exercises...Ch. 2.3 - Finding a Higher-Order Derivative In Exercises...Ch. 2.3 - Finding a Higher-Order Derivative In Exercises...Ch. 2.3 - Using Relationships In Exercises 105-108. use the...Ch. 2.3 - Using Relationships In Exercises 105-108. use the...Ch. 2.3 - Using Relationships In Exercises 105-108, use the...Ch. 2.3 - Using Relationships In Exercises 105-108. use the...Ch. 2.3 - Higher-Order Derivatives Polynomials of what...Ch. 2.3 - Differentiation of Piecewise Functions Describe...Ch. 2.3 - Identifying Graphs In Exercises 111 and 112, the...Ch. 2.3 - Identifying Graphs In Exercises 111 and 112, the...Ch. 2.3 - Sketching Graphs In Exercises 113 and 114, the...Ch. 2.3 - Sketching Graphs In Exercises 113 and 114, the...Ch. 2.3 - Sketching a Graph Sketch the graph of a...Ch. 2.3 - Sketching a Graph Sketch the graph of a...Ch. 2.3 - Acceleration The velocity of an object is...Ch. 2.3 - Acceleration The velocity of an automobile...Ch. 2.3 - Stopping Distance A car is traveling at a rate of...Ch. 2.3 - HOW DO YOU SEE IT? The figure shows the graphs of...Ch. 2.3 - Finding a Pattern In Exercises 121 and 122,...Ch. 2.3 - Finding a Pattern In Exercises 121 and 122....Ch. 2.3 - Finding a Pattern Consider the function...Ch. 2.3 - Finding a Pattern Develop a general rule for the...Ch. 2.3 - Finding a Pattern In Exercises 125 and 126, find...Ch. 2.3 - Finding a Pattern In Exercises 125 and 126, find...Ch. 2.3 - Differential Equations In Exercises 127130, verify...Ch. 2.3 - Differential Equations In Exercises 127-130,...Ch. 2.3 - Differential Equations In Exercises 127-130,...Ch. 2.3 - Differential Equations In Exercises 127-130,...Ch. 2.3 - True or False? In Exercises 131-136, determine...Ch. 2.3 - True or False? In Exercises 131-136. determine...Ch. 2.3 - True or False? In Exercises 131-136. determine...Ch. 2.3 - True or False? In Exercises 131-136, determine...Ch. 2.3 - True or False? In Exercises 131-136, determine...Ch. 2.3 - True or False? In Exercises 131-136. determine...Ch. 2.3 - Proof Use the Product Rule twice to prove that if...Ch. 2.3 - Think About It Let f and g be functions whose...Ch. 2.4 - Chain Rule Describe the Chain Rule for the...Ch. 2.4 - General Power Rule What is the difference between...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Decomposition of a Composite Function In Exercises...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - 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Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a DerivativeIn Exercises 934, find the...Ch. 2.4 - Finding a Derivative In Exercises 9-34, find the...Ch. 2.4 - Finding a Derivative of a Trigonometric FunctionIn...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative of a Trigonometric Function...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Finding a Derivative Using Technology In Exercises...Ch. 2.4 - Slope of a Tangent Line In Exercises 61 and 62,...Ch. 2.4 - Slope of a Tangent Line In Exercises 61 and 62,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding the Slope of a Graph In Exercises 63-70,...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.4 - Famous Curves In Exercises 79 and 80, find an...Ch. 2.4 - Famous Curves In Exercises 79 and 80, find an...Ch. 2.4 - Horizontal Tangent Line Determine the point(s) in...Ch. 2.4 - Horizontal Tangent Line Determine the point(s) at...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Finding a Second Derivative In Exercises 83-88,...Ch. 2.4 - Evaluating a Second Derivative In Exercises 89-92,...Ch. 2.4 - Evaluating a Second Derivative In Exercises 89-92,...Ch. 2.4 - Evaluating a Second Derivative In Exercises 89-92,...Ch. 2.4 - Evaluating a Second Derivative In Exercises 89-92,...Ch. 2.4 - Identifying Graphs In Exercises 93 and 94, the...Ch. 2.4 - Identifying Graphs In Exercises 93 and 94, the...Ch. 2.4 - Describing Relationships Describe the relationship...Ch. 2.4 - Comparing Methods Consider the function...Ch. 2.4 - Think about It The table shows some values of the...Ch. 2.4 - Using Relationships Given that g(5)=3,g(5)=6,...Ch. 2.4 - Finding Derivatives In Exercises 99 and 100, the...Ch. 2.4 - Finding Derivatives In Exercises 99 and 100, the...Ch. 2.4 - Doppler Effect The frequency F of a fire truck...Ch. 2.4 - Harmonic Motion The displacement from equilibrium...Ch. 2.4 - Pendulum A 15-centimeter pendulum moves according...Ch. 2.4 - Wave Motion A buoy oscillates in simple harmonic...Ch. 2.4 - Modeling Data The normal daily maximum...Ch. 2.4 - HOW DO YOU SEE IT? The cost C (in dollars) of...Ch. 2.4 - Biology The number N of bacteria in a culture...Ch. 2.4 - Depreciation The value V of a machine t years...Ch. 2.4 - Finding a Pattern Consider the function f(x)=sinx,...Ch. 2.4 - Conjecture Let f be a differentiable function of...Ch. 2.4 - Think About It Let r(x)=f(g(x)) and s(x)=g(f(x)),...Ch. 2.4 - Using Trigonometric Functions (a) Find the...Ch. 2.4 - Even and Odd Functions (a) Show that the...Ch. 2.4 - Proof Let u be a differentiable function of x. Use...Ch. 2.4 - Using Absolute Value In Exercises 115-118. use the...Ch. 2.4 - Using Absolute Value In Exercises 115-118. use the...Ch. 2.4 - Using Absolute Value In Exercises 115-118. use the...Ch. 2.4 - Using Absolute Value In Exercises 115-118. use the...Ch. 2.4 - Linear and Quadratic Approximations The linear and...Ch. 2.4 - Linear and Quadratic Approximations The linear and...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - True or False? In Exercises 121-124. determine...Ch. 2.4 - PUTNAM EXAM CHALLENGE Let...Ch. 2.4 - PUTNAM EXAM CHALLENGE Let k be a fixed positive...Ch. 2.5 - CONCEPT CHECK Explicit and Implicit Functions...Ch. 2.5 - CONCEPT CHECK Implicit Differentiation In your own...Ch. 2.5 - CONCEPT CHECK Implicit Differentiation Explain...Ch. 2.5 - Chain Rule How is the Chain Rule applied when...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding a Derivative In Exercises 5-20, find dy /...Ch. 2.5 - Finding Derivatives Implicitly and Explicitly In...Ch. 2.5 - Finding Derivatives Implicitly and Explicitly In...Ch. 2.5 - Finding Derivatives Implicitly and Explicitly In...Ch. 2.5 - Finding Derivatives Implicitly and Explicitly In...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Finding the Slope of a Graph In Exercises 25-32,...Ch. 2.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 2.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 2.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 2.5 - Famous Curves In Exercises 39-42, find the slope...Ch. 2.5 - Famous Curves In Exercises 43-48, find an equation...Ch. 2.5 - Famous Curves In Exercises 43-48, find an equation...Ch. 2.5 - Famous Curves In Exercises 37-42, find an equation...Ch. 2.5 - Famous Curves In Exercises 37-42, find an equation...Ch. 2.5 - Famous Curves In Exercises 43-48, find an equation...Ch. 2.5 - Famous Curves In Exercises 43-48, find an equation...Ch. 2.5 - EXPLORING CONCEPTS Implicit and Explicit Forms...Ch. 2.5 - EXPLORING CONCEPTS Think About ItExplain why the...Ch. 2.5 - Ellipse (a) Use implicit differentiation to find...Ch. 2.5 - Hyperbola (a) Use implicit differentiation to find...Ch. 2.5 - Determining a Differentiable Function In Exercises...Ch. 2.5 - Determining a Differentiable Function In Exercises...Ch. 2.5 - Finding a Second Derivative In Exercises 49-54....Ch. 2.5 - Finding a Second Derivative In Exercises 49-54....Ch. 2.5 - Finding a Second Derivative In Exercises 49-54,...Ch. 2.5 - Finding a Second Derivative In Exercises, find...Ch. 2.5 - Finding a Second Derivative In Exercises, 49-54....Ch. 2.5 - Finding a Second Derivative In Exercises 49-54,...Ch. 2.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 2.5 - Tangent Lines and Normal Lines In Exercises 63 and...Ch. 2.5 - Tangent Lines and Normal Lines In Exercises 57 and...Ch. 2.5 - Normal Lines Show that the normal line at any...Ch. 2.5 - Circles Two circles of radius 4 are tangent to the...Ch. 2.5 - Vertical and Horizontal Tangent Lines In Exercises...Ch. 2.5 - Vertical and Horizontal Tangent Lines In Exercises...Ch. 2.5 - Orthogonal Trajectories In Exercises 63-66, use a...Ch. 2.5 - Orthogonal Trajectories In Exercises 63-66. use a...Ch. 2.5 - Orthogonal Trajectories In Exercises 63-66, use a...Ch. 2.5 - Orthogonal Trajectories In Exercises 81-84, use a...Ch. 2.5 - Orthogonal Trajectories In Exercises 85 and 86,...Ch. 2.5 - Orthogonal Trajectories In Exercises 85 and 86,...Ch. 2.5 - Orthogonal Trajectories The figure below shows the...Ch. 2.5 - HOW DO YOU SEE IT? Use the graph to answer the...Ch. 2.5 - Finding Equations of Tangent Lines Consider the...Ch. 2.5 - Tangent Lines and Intercepts Let l. be any tangent...Ch. 2.5 - Proof (a) Prove (Theorem 3.3) that d/dxxn=nxn1 for...Ch. 2.5 - Slope Find all points on the circle x2+y2=100...Ch. 2.5 - Tangent Lines Find equations of both tangent lines...Ch. 2.5 - Normals to a Parabola The graph shows die normal...Ch. 2.5 - Normal Lines (a) Find an equation of the normal...Ch. 2.6 - Related-Rate Equation What is a related-rate...Ch. 2.6 - Related Rates In your own words, state the...Ch. 2.6 - Using Related Rates In Exercises 3-6, assume that...Ch. 2.6 - Using Related Rates In Exercises 3-6, assume that...Ch. 2.6 - Using Related Rates In Exercises 3-6, assume that...Ch. 2.6 - Using Related Rates In Exercises 3-6, assume that...Ch. 2.6 - Moving Point In Exercises 7-10, a point is moving...Ch. 2.6 - Moving Point In Exercises 7-10, a point is moving...Ch. 2.6 - Moving Point In Exercises 710, a point is moving...Ch. 2.6 - Moving Point In Exercises 7-10, a point is moving...Ch. 2.6 - Area The radius r of a circle is increasing at a...Ch. 2.6 - Area The length .v of each side of an equilateral...Ch. 2.6 - Volume The radius r of a sphere is increasing at a...Ch. 2.6 - Radius A spherical balloon is inflated with gas at...Ch. 2.6 - Volume All edges of a cube are expanding at a rate...Ch. 2.6 - Surface Area All edges of a cube are expanding at...Ch. 2.6 - Height At a sand and gravel plant, sand is falling...Ch. 2.6 - Height The volume of oil in a cylindrical...Ch. 2.6 - Depth A swimming pool is 12 meters long, 6 meters...Ch. 2.6 - Depth A trough is 12 feet long and 3 feet across...Ch. 2.6 - Moving Ladder A ladder 25 feet long is leaning...Ch. 2.6 - Construction A construction worker pulls a...Ch. 2.6 - Construction A winch at the top of a 12-meter...Ch. 2.6 - Boating A boat is pulled into a dock by means of a...Ch. 2.6 - Air Traffic Control An air traffic controller...Ch. 2.6 - Air Traffic Control An airplane is flying at an...Ch. 2.6 - Sports A baseball diamond has the shape of a...Ch. 2.6 - Sports For (be baseball diamond in Exercise 27,...Ch. 2.6 - Shadow Length A man 6 feet tall walks at a rate of...Ch. 2.6 - Shadow Length Repeal Exercise 29 for a man 6 feet...Ch. 2.6 - Machine Design The endpoints of a movable rod of...Ch. 2.6 - Machine Design Repeat Exercise 31 for a position...Ch. 2.6 - Evaporation As a spherical raindrop falls, it...Ch. 2.6 - HOW DO YOU SEE IT? 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Moving Shadow A sandbag is dropped from a balloon...Ch. 2 - Finding Equations of Circles Consider the graph of...Ch. 2 - Finding Equations of Tangent Lines Graph the two...Ch. 2 - Finding a Polynomial Find a third-degree...Ch. 2 - Finding a Function Find a function of the form...Ch. 2 - Tangent Lines and Normal Lines (a) Find an...Ch. 2 - Finding Polynomials (a) Find the polynomial...Ch. 2 - Famous Curve The graph of the eight curve...Ch. 2 - Famous Curve The graph of the pear-shaped quartic...Ch. 2 - Shadow Length A man 6 feet tall walks at a rate of...Ch. 2 - Moving Point A particle is moving along the graph...Ch. 2 - Projectile Motion An astronaut standing on the...Ch. 2 - Proof Let E be a function satisfying E(0)=E(0)=1....Ch. 2 - Proof Let L be a differentiable function for all...Ch. 2 - Radians and Degrees The fundamental limit...Ch. 2 - Acceleration and Jerk If a is the acceleration of...

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