CalculusCalculus of a Single Variable11th Edition
HOW DO YOU SEE IT? Use the information in the graph of f below. (a) What is the slope of the tangent line to the graph of f − 1 at the point ( − 1 2 , − 1 ) ? Explain. (b) What is the slope of the tangent line to the graph of f − 1 at the point (1, 2)? Explain.
P Preparation For Calculus1 Limits And Their Properties2 Differentiation3 Applications Of Differentiation4 Integration5 Logarithmic, Exponential, And Other Transcendental Functions6 Differential Equations7 Applications Of Integration8 Integration Techniques And Improper Integrals9 Infinite Series10 Conics, Parametric Equations, And Polar Coordinates
5.1 The Natural Logarithmic Function: Differentiation5.2 The Natural Logarithmic Function: Integration5.3 Inverse Functions5.4 Exponential Functions: Differentiation And Integration5.5 Bases Other Than E And Applications5.6 Indeterminate Forms And L'hôpital's Rule5.7 Inverse Trigonometric Functions: Differentiation5.8 Inverse Trigonometric Functions: Integration5.9 Hyperbolic FunctionsChapter Questions
Problem 100E
Chapter 5.3, Problem 88E
HOW DO YOU SEE IT? Use the information in the graph of f below.
(a) What is the slope of the tangent line to the graph of f − 1 at the point ( − 1 2 , − 1 ) ? Explain.
(b) What is the slope of the tangent line to the graph of f − 1 at the point (1, 2)? Explain.
To calculate: The slope of the tangent line to the graph of f−1 at the point (a) (−12,−1), (b) (1,2).from the graph of function f(x)
Graph of f with information required is given below.
If a function f has its inverse function f−1 then,
f′(x)=1(f−1)′(x) (f−1)′(x)=1f′(x)m(f−1(x))=1m(f(x)).
From the graph given, f is monotonic and therefore one-to-one. Hence its inverse function f−1 exists.
(a) At point (−12,−1), slope of tangent line to th graph of f−1 is,
m(f−1(x))=1m(f(x)),m(f(−1))=1
Chapter 5.3,
Ch. 5.1 - Natural Logarithmic Function Explain why ln x is...Ch. 5.1 - Logarithmic Properties What is the value of n?...Ch. 5.1 - The Number e How is the number e defined?Ch. 5.1 - Differentiation of Logarithmic Functions State the...Ch. 5.1 - Evaluating a Logarithm Using Technology In...Ch. 5.1 - Evaluating a Logarithm Using Technology In...Ch. 5.1 - Evaluating a Logarithm Using Technology In...Ch. 5.1 - Evaluating a Logarithm Using Technology In...Ch. 5.1 - Matching In Exercises 912, match the function with...Ch. 5.1 - Matching In Exercises 912, match the function with...
Ch. 5.1 - Matching In Exercises 912, match the function with...Ch. 5.1 - Matching In Exercises 912, match the function with...Ch. 5.1 - Sketching the Graph In Exercises 13-18, sketch the...Ch. 5.1 - Sketching the Graph In Exercises 13-18, sketch the...Ch. 5.1 - Sketching a GraphIn Exercised 13-18, sketch the...Ch. 5.1 - Sketching the Graph In Exercises 13-18, sketch the...Ch. 5.1 - Sketching the Graph In Exercises 13-18, sketch the...Ch. 5.1 - Sketching the Graph In Exercises 13-18, sketch the...Ch. 5.1 - Using Properties of Logarithms In Exercises 19 and...Ch. 5.1 - Using Properties of Logarithms In Exercises 19 and...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Expanding a Logarithmic Expression In Exercises...Ch. 5.1 - Condensing a Logarithmic Expression In Exercises...Ch. 5.1 - Condensing a Logarithmic Expression In Exercises...Ch. 5.1 - Condensing a Logarithmic Expression In Exercises...Ch. 5.1 - Condensing a Logarithmic Expression In Exercises...Ch. 5.1 - Condensing a Logarithmic Expression In Exercises...Ch. 5.1 - Condensing a Logarithmic Expression In Exercises...Ch. 5.1 - Verifying Properties of Logarithms In Exercises 37...Ch. 5.1 - Verifying Properties of Logarithms In Exercises 37...Ch. 5.1 - Finding a Limit In Exercises 39-42, find the...Ch. 5.1 - Finding a Limit In Exercises 39-42, find the...Ch. 5.1 - Finding a Limit In Exercises 39-42, find the...Ch. 5.1 - Finding a Limit In Exercises 39-42, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding a Derivative In Exercises 43-66, find the...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Logarithmic Differentiation In Exercises 75-80,...Ch. 5.1 - Implicit Differentiation In Exercises 81-84, use...Ch. 5.1 - Implicit Differentiation In Exercises 81-84, use...Ch. 5.1 - Implicit Differentiation In Exercises 81-84, use...Ch. 5.1 - Implicit Differentiation In Exercises 81-84, use...Ch. 5.1 - Differential Equation In Exercises 85 and 86,...Ch. 5.1 - Differential Equation In Exercises 85 and 86,...Ch. 5.1 - Relative Extrema and Points of Inflection In...Ch. 5.1 - Relative Extrema and Points of Inflection In...Ch. 5.1 - Relative Extrema and Points of Inflection In...Ch. 5.1 - Relative Extrema and Points of Inflection In...Ch. 5.1 - Relative Extrema and Points of Inflection In...Ch. 5.1 - Relative Extrema and Points of Inflection In...Ch. 5.1 - Using Newtons Method In Exercises 93 and 94, use...Ch. 5.1 - Using Newtons Method In Exercises 93 and 94, use...Ch. 5.1 - Comparing Functions In Exercises 95 and 96, let f...Ch. 5.1 - Comparing Functions In Exercises 95 and 96, let f...Ch. 5.1 - Think About It Is lnxy=lnxlny a valid property of...Ch. 5.1 - HOW DO YOU SEE IT? The graph shows the temperature...Ch. 5.1 - True or False? In Exercises 99-102, determine...Ch. 5.1 - True or False? In Exercises 99-102, determine...Ch. 5.1 - True or False? In Exercises 99-102, determine...Ch. 5.1 - True or False? In Exercises 99-102, determine...Ch. 5.1 - Home Mortgage The term t (in years) of a $200,000...Ch. 5.1 - Sound Intensity The relationship between the...Ch. 5.1 - Modeling Data The table shows the temperatures T...Ch. 5.1 - Modeling Data The atmospheric pressure decreases...Ch. 5.1 - Tractrix A person walking along a dock drags a...Ch. 5.1 - Prime Number Theorem There are 25 prime numbers...Ch. 5.1 - Conjecture Use a graphing utility to graph f and g...Ch. 5.2 - CONCEPT CHECK Log Rule for Integration Can you use...Ch. 5.2 - CONCEPT CHECK Long Division Explain when to use...Ch. 5.2 - CONCEPT CHECK Guidelines for Integration Describe...Ch. 5.2 - CONCEPT CHECK Trigonometric Functions Integrating...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 5-28,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Finding an Indefinite Integral In Exercises 528,...Ch. 5.2 - Change of Variables In Exercises 2932, find the...Ch. 5.2 - Change of Variables In Exercises 2932, find the...Ch. 5.2 - Change of Variables In Exercises 2932, find the...Ch. 5.2 - Change of Variables In Exercises 2932, find the...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Finding an Indefinite Integral of a Trigonometric...Ch. 5.2 - Differential Equation In Exercises 4346, find the...Ch. 5.2 - Differential Equation In Exercises 4346, find the...Ch. 5.2 - Differential Equation In Exercises 43-46, find the...Ch. 5.2 - Differential Equation In Exercises 43-46, find the...Ch. 5.2 - Finding a Particular Solution In Exercises 47 and...Ch. 5.2 - Finding a Particular Solution In Exercises 47 and...Ch. 5.2 - Slope Field In Exercises 49 and 50, a differential...Ch. 5.2 - Slope Field In Exercises 49 and 50, a differential...Ch. 5.2 - Evaluating a Definite Integral In Exercises 5158,...Ch. 5.2 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.2 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.2 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.2 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.2 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.2 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.2 - Evaluating a Definite Integral In Exercises 51-58,...Ch. 5.2 - Finding an Integral Using Technology In Exercises...Ch. 5.2 - Finding an Integral Using Technology In Exercises...Ch. 5.2 - Finding a Derivative In Exercises 61-64, find F'(...Ch. 5.2 - Finding a Derivative In Exercises 61-64, find F'(...Ch. 5.2 - Finding a Derivative In Exercises 61-64, find F'(...Ch. 5.2 - Finding a Derivative In Exercises 61-64, find F'(...Ch. 5.2 - Area In Exercises 6568, find the area of the given...Ch. 5.2 - Area In Exercises 6568, find the area of the given...Ch. 5.2 - Area In Exercises 6568, find the area of the given...Ch. 5.2 - Area In Exercises 6568, find the area of the given...Ch. 5.2 - Area In Exercises 69-72, find the area of the...Ch. 5.2 - Area In Exercises 69-72, find the area of the...Ch. 5.2 - Area In Exercises 69-72, find the area of the...Ch. 5.2 - Area In Exercises 69-72, find the area of the...Ch. 5.2 - Finding the Average Value of a Function In...Ch. 5.2 - Finding the Average Value of a Function In...Ch. 5.2 - Finding the Average Value of a Function In...Ch. 5.2 - Finding the Average Value of a Function In...Ch. 5.2 - Midpoint Rule In Exercises 77 and 78, use the...Ch. 5.2 - Midpoint Rule In Exercises 77 and 78, use the...Ch. 5.2 - EXPLORING CONCEPTS Approximation In Exercises 79...Ch. 5.2 - EXPLORING CONCEPTS Approximation In Exercises 79...Ch. 5.2 - Napier's Inequality For 0 x y, use the Mean...Ch. 5.2 - Think About It Is the function F(x)=x2x1tdt...Ch. 5.2 - Finding a Value Find a value of x such that...Ch. 5.2 - Finding a Value Find a value of x such that 1x1tdt...Ch. 5.2 - Proof Prove that cotudu=lnsinu+C.Ch. 5.2 - Proof Prove that cscudu=lncscu+cotu+C.Ch. 5.2 - Using Properties of Logarithms and Trigonometric...Ch. 5.2 - Using Properties of Logarithms and Trigonometric...Ch. 5.2 - Using Properties of Logarithms and Trigonometric...Ch. 5.2 - Using Properties of Logarithms and Trigonometric...Ch. 5.2 - Population Growth A population of bacteria P is...Ch. 5.2 - Sales The rate of change in sales S is inversely...Ch. 5.2 - Heat Transfer Find the time required for an object...Ch. 5.2 - Average Price The demand equation for a product is...Ch. 5.2 - Area and Slope Graph the function f(x)=x/(1+x2) on...Ch. 5.2 - HOW DO YOU SEE IT? Use the graph of f shown in the...Ch. 5.2 - True or False? In Exercises 97-100, determine...Ch. 5.2 - True or False? In Exercises 97100, determine...Ch. 5.2 - True or False? In Exercises 97100, determine...Ch. 5.2 - True or False? In Exercises 97100, determine...Ch. 5.2 - PUTNAM EXAM CHALLENGE Suppose that f is a function...Ch. 5.3 - Inverse Functions In your own words, describe what...Ch. 5.3 - Reflective Property of Inverse Functions Describe...Ch. 5.3 - Domain of an Inverse Function The function f has...Ch. 5.3 - Behavior of an Inverse Function The function f is...Ch. 5.3 - Matching In Exercises 5-8, match the graph of the...Ch. 5.3 - Matching In Exercises 5-8, match the graph of the...Ch. 5.3 - Matching In Exercises 5-8, match the graph of the...Ch. 5.3 - Matching In Exercises 5-8, match the graph of the...Ch. 5.3 - Verifying Inverse Functions In Exercises 9-16,...Ch. 5.3 - Verifying Inverse Functions In Exercises 9-16,...Ch. 5.3 - Verifying Inverse Functions In Exercises 9-16,...Ch. 5.3 - Verifying Inverse Functions In Exercises 9-16,...Ch. 5.3 - Verifying Inverse Functions In Exercises 9-16,...Ch. 5.3 - Verifying Inverse Functions In Exercises 9-16,...Ch. 5.3 - Verifying Inverse Functions In Exercises 9-16,...Ch. 5.3 - Verifying Inverse Functions In Exercises 9-16,...Ch. 5.3 - Using the Horizontal Line Test In Exercises 17-24,...Ch. 5.3 - Using the Horizontal Line Test In Exercises 17-24,...Ch. 5.3 - Using the Horizontal Line Test In Exercises 1724,...Ch. 5.3 - Using the Horizontal Line Test In Exercises 17-24,...Ch. 5.3 - Using the Horizontal Line Test In Exercises 17-24,...Ch. 5.3 - Using the Horizontal Line Test In Exercises 17-24,...Ch. 5.3 - Using the Horizontal Line Test In Exercises 17-24,...Ch. 5.3 - Using the Horizontal Line Test In Exercises 17-24,...Ch. 5.3 - Determining Whether a Function Has an Inverse...Ch. 5.3 - Determining Whether a Function Has an Inverse...Ch. 5.3 - Determining Whether a Function Has an Inverse...Ch. 5.3 - Determining Whether a Function Has an Inverse...Ch. 5.3 - Determining Whether a Function Has an Inverse...Ch. 5.3 - Determining Whether a Function Has an Inverse...Ch. 5.3 - Verifying a Function Has an Inverse Function In...Ch. 5.3 - Verifying a Function Has an Inverse Function In...Ch. 5.3 - Verifying a Function Has an Inverse Function In...Ch. 5.3 - Verifying a Function Has an Inverse Function In...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 35-46,...Ch. 5.3 - Finding an Inverse Function In Exercises 47 and...Ch. 5.3 - Finding an Inverse Function In Exercises 47 and...Ch. 5.3 - Cost You need a total of 50 pounds of two...Ch. 5.3 - Temperature The formula C=59(F32), where F459.6,...Ch. 5.3 - Testing Whether a Function Is One-to-One In...Ch. 5.3 - Testing Whether a Function Is One-to-One In...Ch. 5.3 - Testing Whether a Function Is One-to-One In...Ch. 5.3 - Testing Whether a Function Is One-to-One In...Ch. 5.3 - Making a Function One-to-One In Exercises 55-58,...Ch. 5.3 - Making a Function One-to-One In Exercises 55-58,...Ch. 5.3 - Making a Function One-to-One In Exercises 55-58,...Ch. 5.3 - Making a Function One-to-One In Exercises 55-58,...Ch. 5.3 - Think About It In Exercises 59-62, decide whether...Ch. 5.3 - Think About It In Exercises 59-62, decide whether...Ch. 5.3 - Think About It In Exercises 59-62, decide whether...Ch. 5.3 - Think About It In Exercises 59-62, decide whether...Ch. 5.3 - Evaluating the Derivative of an Inverse Function...Ch. 5.3 - Evaluating the Derivative of an Inverse Function...Ch. 5.3 - Evaluating the Derivative of an Inverse Function...Ch. 5.3 - Evaluating the Derivative of an Inverse Function...Ch. 5.3 - Evaluating the Derivative of an Inverse Function...Ch. 5.3 - Evaluating the Derivative of an Inverse Function...Ch. 5.3 - Evaluating the Derivative of an Inverse Function...Ch. 5.3 - Evaluating the Derivative of an Inverse Function...Ch. 5.3 - Using Inverse Functions In Exercises 71-74, (a)...Ch. 5.3 - Using Inverse Functions In Exercises 71-74, (a)...Ch. 5.3 - Using Inverse Functions In Exercises 71-74, (a)...Ch. 5.3 - Using Inverse Functions In Exercises 71-74, (a)...Ch. 5.3 - Using Composite and Inverse Functions In Exercises...Ch. 5.3 - Using Composite and Inverse Functions In Exercises...Ch. 5.3 - Using Composite and Inverse Functions In Exercises...Ch. 5.3 - Using Composite and Inverse Functions In Exercises...Ch. 5.3 - Using Composite and Inverse Functions In Exercises...Ch. 5.3 - Using Composite and Inverse Functions In Exercises...Ch. 5.3 - Using Composite and Inverse Functions In Exercises...Ch. 5.3 - Using Composite and Inverse Functions In Exercises...Ch. 5.3 - Inverse Function Consider the function f(x)=xn...Ch. 5.3 - Think About It Does adding a constant term to a...Ch. 5.3 - Explaining Why a Function Is Not One-to-One In...Ch. 5.3 - Explaining Why a Function Is Not One-to-One In...Ch. 5.3 - Think About It The function f(x)=k(2xx3)...Ch. 5.3 - HOW DO YOU SEE IT? Use the information in the...Ch. 5.3 - True or False? In Exercises 89 and 90, determine...Ch. 5.3 - True or False? In Exercises 89 and 90, determine...Ch. 5.3 - Making a Function One-to-One (a) Show that...Ch. 5.3 - Proof Let f and g be one-to-one functions. Prove...Ch. 5.3 - Proof Prove that if f has an inverse function,...Ch. 5.3 - Proof Prove that if a function has an inverse...Ch. 5.3 - Proof Prove that a function has an inverse...Ch. 5.3 - Using Theorem 5.7 Is the converse of the second...Ch. 5.3 - Derivative of an Inverse Function Show that...Ch. 5.3 - Derivative of an Inverse FunctionShow that...Ch. 5.3 - Inverse Function Let f(x)=x2x1 Show that f is its...Ch. 5.3 - Using a Function Let f(x)=ax+bcx+d (a) Show that f...Ch. 5.3 - Concavity Let f be twice-differentiable and...Ch. 5.4 - Natural Exponential Function of Describe the graph...Ch. 5.4 - A Function and Its Derivative Which of the...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Solving an Exponential or Logarithmic Equation In...Ch. 5.4 - Sketching a Graph In Exercises 19-24, sketch the...Ch. 5.4 - Sketching a Graph In Exercises 19-24, sketch the...Ch. 5.4 - Sketching a Graph In Exercises 19-24, sketch the...Ch. 5.4 - Sketching a Graph In Exercises 19-24, sketch the...Ch. 5.4 - Sketching a Graph In Exercises 19-24, sketch the...Ch. 5.4 - Sketching a Graph In Exercises 19-24, sketch the...Ch. 5.4 - Matching In Exercises 2528, match the equation...Ch. 5.4 - Matching In Exercises 2528, match the equation...Ch. 5.4 - Matching In Exercises 2528, match the equation...Ch. 5.4 - Matching In Exercises 2528, match the equation...Ch. 5.4 - Inverse Functions In Exercises 29-32, illustrate...Ch. 5.4 - Inverse Functions In Exercises 2932, illustrate...Ch. 5.4 - Inverse Functions In Exercises 2932, illustrate...Ch. 5.4 - Inverse Functions In Exercises 2932, illustrate...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding a Derivative In Exercises 33-54, find the...Ch. 5.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.4 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.4 - Implicit Differentiation In Exercises 63 and 64,...Ch. 5.4 - Implicit Differentiation In Exercises 63 and 64,...Ch. 5.4 - Finding the Equation of a Tangent Line In...Ch. 5.4 - Finding the Equation of a Tangent Line In...Ch. 5.4 - Finding a Second Derivative In Exercises 67 and...Ch. 5.4 - Finding a Second Derivative In Exercises 67 and...Ch. 5.4 - Differential Equation In Exercises 69 and 70, show...Ch. 5.4 - Differential Equation In Exercises 69 and 70, show...Ch. 5.4 - Relative Extrema and Points of Inflection In...Ch. 5.4 - Relative Extrema and Points of Inflection In...Ch. 5.4 - Relative Extrema and Points of Inflection In...Ch. 5.4 - Relative Extrema and Points of Inflection In...Ch. 5.4 - Relative Extrema and Points of Inflection In...Ch. 5.4 - Relative Extrema and Points of Inflection In...Ch. 5.4 - Relative Extrema and Points of Inflection In...Ch. 5.4 - Relative Extrema and Points of Inflection In...Ch. 5.4 - Area Find the area of the largest rectangle that...Ch. 5.4 - Area Perform the following steps to find the...Ch. 5.4 - Finding an Equation of a Tangent Line Find point...Ch. 5.4 - HOW DO YOU SEE IT? The figure shows the graphs of...Ch. 5.4 - Depreciation The value V of an item t years after...Ch. 5.4 - Harmonic Motion The displacement from equilibrium...Ch. 5.4 - Atmospheric Pressure A meteorologist measures the...Ch. 5.4 - Modeling Data The table lists the approximate...Ch. 5.4 - Linear and Quadratic Approximation In Exercises 87...Ch. 5.4 - Linear and Quadratic Approximation In Exercises 87...Ch. 5.4 - Stirlings Formula For large values of n,...Ch. 5.4 - Stirling's Formula For large values of n, n! = 1 ...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Finding an Indefinite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Evaluating a Definite Integral In Exercises...Ch. 5.4 - Differential Equation In Exercises 119 and 120,...Ch. 5.4 - Differential Equation In Exercises 119 and 120,...Ch. 5.4 - Differential Equation In Exercises 121 and 122,...Ch. 5.4 - Differential Equation In Exercises 121 and 122,...Ch. 5.4 - Area In Exercises 123-126, find the area of the...Ch. 5.4 - Area In Exercises 123-126, find the area of the...Ch. 5.4 - Area In Exercises 123-126, find the area of the...Ch. 5.4 - Area In Exercises 123-126, find the area of the...Ch. 5.4 - Midpoint Rule In Exercises 127 and 128, use the...Ch. 5.4 - Midpoint Rule In Exercises 127 and 128, use the...Ch. 5.4 - Asymptotes Compare the asymptotes of the natural...Ch. 5.4 - Comparing Graphs Use a graphing utility to graph...Ch. 5.4 - True or False? In Exercises 131-134, determine...Ch. 5.4 - True or False? In Exercises 131-134, determine...Ch. 5.4 - True or False? In Exercises 131-134, determine...Ch. 5.4 - True or False? In Exercises 131-134, determine...Ch. 5.4 - Probability A car battery has an average lifetime...Ch. 5.4 - Probability The median waiting time (in minutes)...Ch. 5.4 - Modeling Data A valve on a storage tank is opened...Ch. 5.4 - Using the Area of a Region Find the value of a...Ch. 5.4 - Analyzing a Graph Consider the function...Ch. 5.4 - Analyzing a Function Let f(x)=lnxx. (a) Graph f on...Ch. 5.4 - Deriving an Inequality Given ex1 for x0, it...Ch. 5.4 - Solving an Equation Find, to three decimal places,...Ch. 5.4 - Analyzing a Graph Consider f(x)=xekx for k0. Find...Ch. 5.4 - Finding the Maximum Rate of Change Verify that the...Ch. 5.4 - PUTNAM EXAM CHALLENGE Let S be a class of...Ch. 5.5 - Derivatives for Bases Other than e What are the...Ch. 5.5 - Integration for Bases Other than e What are two...Ch. 5.5 - Logarithmic Differentiation Explain when it is...Ch. 5.5 - Compound Interest Formulas Explain how to choose...Ch. 5.5 - Evaluating a Logarithmic Expression In Exercises...Ch. 5.5 - Evaluating a Logarithmic Expression In Exercises...Ch. 5.5 - Evaluating a Logarithmic Expression In Exercises...Ch. 5.5 - Evaluating a Logarithmic Expression In Exercises...Ch. 5.5 - Evaluating a Logarithmic Expression In Exercises...Ch. 5.5 - Evaluating a Logarithmic Expression In Exercises...Ch. 5.5 - Exponential and Logarithmic Forms of Equations In...Ch. 5.5 - Exponential and Logarithmic Forms of Equations In...Ch. 5.5 - Exponential and Logarithmic Forms of Equations In...Ch. 5.5 - Exponential and Logarithmic Forms of Equations In...Ch. 5.5 - Sketching a Graph In Exercises 15-20, sketch the...Ch. 5.5 - Sketching a Graph In Exercises 15-20, sketch the...Ch. 5.5 - Sketching a Graph In Exercises 15-20, sketch the...Ch. 5.5 - Sketching a Graph In Exercises 15-20, sketch the...Ch. 5.5 - Sketching a Graph In Exercises 15-20, sketch the...Ch. 5.5 - Sketching a Graph In Exercises 15-20, sketch the...Ch. 5.5 - Solving an Equation In Exercises 21-26, solve for...Ch. 5.5 - Solving an Equation In Exercises 21-26, solve for...Ch. 5.5 - Solving an Equation In Exercises 21-26, solve for...Ch. 5.5 - Solving an Equation In Exercises 21-26, solve for...Ch. 5.5 - Solving an Equation In Exercises 21-26, solve for...Ch. 5.5 - Solving an Equation In Exercises 21-26, solve for...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Solving an Equation In Exercises 27-36, solve the...Ch. 5.5 - Inverse Functions In Exercises 37 and 38,...Ch. 5.5 - Inverse Functions In Exercises 37 and 38,...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding a Derivative In Exercises 39-60, find the...Ch. 5.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.5 - Logarithmic Differentiation In Exercises 65-68,...Ch. 5.5 - Logarithmic Differentiation In Exercises 65-68,...Ch. 5.5 - Logarithmic Differentiation In Exercises 65-68,...Ch. 5.5 - Logarithmic Differentiation In Exercises 65-68,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 69-76,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 69-76,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 69-76,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 69-76,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 69-76,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 69-76,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 69-76,...Ch. 5.5 - Finding an Indefinite Integral In Exercises 69-76,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 77-80,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 77-80,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 77-80,...Ch. 5.5 - Evaluating a Definite Integral In Exercises 77-80,...Ch. 5.5 - Area In Exercises 81 and 82, find the area of the...Ch. 5.5 - Area In Exercises 81 and 82, find the area of the...Ch. 5.5 - Exponential Function What happens to the rate a of...Ch. 5.5 - Logarithmic Function What happens to the rate of...Ch. 5.5 - Analyzing a Logarithmic Equation Consider the...Ch. 5.5 - Comparing Rates of Growth Order the functions...Ch. 5.5 - Inflation When the annual rate of inflation...Ch. 5.5 - Depreciation After t years, the value of a car...Ch. 5.5 - Compound Interest In Exercises 89-92, complete the...Ch. 5.5 - Compound Interest In Exercises 89-92, complete the...Ch. 5.5 - Compound Interest In Exercises 89-92, complete the...Ch. 5.5 - Compound Interest In Exercises 89-92, complete the...Ch. 5.5 - Compound Interest In Exercises 93-96, complete the...Ch. 5.5 - Compound Interest In Exercises 93-96, complete the...Ch. 5.5 - Compound Interest In Exercises 93-96, complete the...Ch. 5.5 - Compound Interest In Exercises 93-96, complete the...Ch. 5.5 - Compound Interest Assume that you can earn 6% on...Ch. 5.5 - Compound Interest Consider a deposit of $100...Ch. 5.5 - Timber Yield The yield V (in millions of cubic...Ch. 5.5 - HOW DO YOU SEE IT? The graph shows the proportion...Ch. 5.5 - Population Growth A lake is stocked with 500 fish,...Ch. 5.5 - Modeling Data The breaking strengths B (in tons)...Ch. 5.5 - Comparing Models The total numbers y of AIDS cases...Ch. 5.5 - An Approximation of e Complete the table to...Ch. 5.5 - Modeling Data In Exercises 105 and 106, find an...Ch. 5.5 - Modeling Data In Exercises 105 and 106, find an...Ch. 5.5 - Using Properties of Exponents In Exercises...Ch. 5.5 - Using Properties of Exponents In Exercises...Ch. 5.5 - Using Properties of Exponents In Exercises...Ch. 5.5 - Using Properties of Exponents In Exercises...Ch. 5.5 - Comparing Functions (a) Show that (23)22(32). (b)...Ch. 5.5 - Finding an inverse Function Let f(x)=ax1ax+1. for...Ch. 5.5 - Logistic Differential Equation Show that solving...Ch. 5.5 - Using Properties of Exponents function f(x)=ax,...Ch. 5.5 - Tangent Lines (a) Determine y' given yx=xy. (b)...Ch. 5.5 - Which is greater (n)n+1 or (n+1)n where n8?Ch. 5.5 - Show that if x is positive, then loge(1+1x)11+xCh. 5.6 - CONCEPT CHECK L' Hopitals Rule Explain the benefit...Ch. 5.6 - CONCEPT CHECK Indeterminate Forms For each limit,...Ch. 5.6 - Numerical and Graphical Analysis In Exercises 3-6,...Ch. 5.6 - Numerical and Graphical Analysis In Exercises 3-6,...Ch. 5.6 - Numerical and Graphical Analysis In Exercises 3-6,...Ch. 5.6 - Numerical and Graphical Analysis In Exercises 3-6,...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Using Two Methods In Exercises 7-14, evaluate the...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 15-42, evaluate...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - Evaluating a Limit In Exercises 43-62, (a)...Ch. 5.6 - EXPLORING CONCEPTS Finding Functions Find...Ch. 5.6 - EXPLORING CONCEPTS Finding Functions Find...Ch. 5.6 - EXPLORING CONCEPTS L'Hopitals Rule Determine which...Ch. 5.6 - HOW DO YOU SEE IT? Use the graph of f to find each...Ch. 5.6 - Numerical Analysis Complete the table to show that...Ch. 5.6 - Numerical Analysis Complete the table to show that...Ch. 5.6 - Comparing Functions In Exercises 69-74, use...Ch. 5.6 - Comparing Functions In Exercises 69-74, use...Ch. 5.6 - Comparing Functions In Exercises 69-74, use...Ch. 5.6 - Comparing Functions In Exercises 69-74, use...Ch. 5.6 - Comparing Functions In Exercises 69-74, use...Ch. 5.6 - Comparing Functions In Exercises 69-74, use...Ch. 5.6 - Asymptotes and Relative Extrema In Exercises...Ch. 5.6 - Asymptotes and Relative Extrema In Exercises...Ch. 5.6 - Asymptotes and Relative Extrema In Exercises...Ch. 5.6 - Asymptotes and Relative Extrema In Exercises...Ch. 5.6 - Think About It In Exercises 79-82, LHpitals Rule...Ch. 5.6 - Think About It In Exercises 79-82, LHpitals Rule...Ch. 5.6 - Think About It In Exercises 79-82, LHpitals Rule...Ch. 5.6 - Think About It In Exercises 79-82, LHpitals Rule...Ch. 5.6 - Analytic and Graphical Analysis In Exercises 83...Ch. 5.6 - Analytic and Graphical Analysis In Exercises 83...Ch. 5.6 - Graphical Analysis In Exercises 85 and 86, graph...Ch. 5.6 - Graphical Analysis In Exercises 85 and 86, graph...Ch. 5.6 - Electric Circuit The diagram shows a simple...Ch. 5.6 - Velocity in a Resisting Medium The velocity v of...Ch. 5.6 - The Gamma Function The Gamma Function T(n) is...Ch. 5.6 - Compound Interest The formula for the amount A in...Ch. 5.6 - Extended Mean Value Theorem In Exercises 91-94,...Ch. 5.6 - Extended Mean Value Theorem In Exercises 91-94,...Ch. 5.6 - Extended Mean Value Theorem In Exercises 91-94,...Ch. 5.6 - Extended Mean Value Theorem In Exercises 91-94,...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - True or False? In Exercises 95-100, determine...Ch. 5.6 - Area Find the limit, as x approaches 0, of the...Ch. 5.6 - Finding a Limit In Section 1.3, a geometric...Ch. 5.6 - Continuous Function In Exercises 103 and 104, find...Ch. 5.6 - Continuous Function In Exercises 103 and 104, find...Ch. 5.6 - Finding Values Find the values of a and b such...Ch. 5.6 - Evaluating a Limit Use a graphing utility to graph...Ch. 5.6 - Finding a Derivative (a) Let f'(x) be continuous....Ch. 5.6 - Finding a Second Derivative Let f"(x) be...Ch. 5.6 - Evaluating a Limit Consider the limit...Ch. 5.6 - Proof Prove that if f(x)0,limxaf(x)=0, and...Ch. 5.6 - Proof Prove that if f(x)0,limxaf(x)=0, and...Ch. 5.6 - Think About It Use two different methods to find...Ch. 5.6 - Indeterminate Forms Show that the indeterminate...Ch. 5.6 - Calculus History In L'Hopitals 1696 calculus...Ch. 5.6 - Finding a Limit Consider the function h(x)=x+sinxx...Ch. 5.6 - Evaluating a Limit Let f(x)=x+xsinx and g(x)=x24....Ch. 5.6 - PUTNAM EXAM CHALLENGE Evaluate limx[1xax1a1]1/x...Ch. 5.7 - Inverse Trigonometric Function Describe the...Ch. 5.7 - Restricted Domain What is a restricted domain? Why...Ch. 5.7 - Inverse Trigonometric Functions Which inverse...Ch. 5.7 - Finding a Derivative What is the missing value?...Ch. 5.7 - Finding Coordinates In Exercises 5 and 6,...Ch. 5.7 - Finding Coordinates In Exercises 5 and 6,...Ch. 5.7 - Evaluating Inverse Trigonometric Functions In...Ch. 5.7 - Evaluating Inverse Trigonometric Functions In...Ch. 5.7 - Evaluating Inverse Trigonometric Functions In...Ch. 5.7 - Evaluating Inverse Trigonometric Functions In...Ch. 5.7 - Evaluating Inverse Trigonometric Functions In...Ch. 5.7 - Evaluating Inverse Trigonometric Functions In...Ch. 5.7 - Evaluating Inverse Trigonometric Functions In...Ch. 5.7 - Evaluating Inverse Trigonometric Functions In...Ch. 5.7 - Approximating Inverse Trigonometric Functions In...Ch. 5.7 - Approximating Inverse Trigonometric Functions In...Ch. 5.7 - Approximating Inverse Trigonometric Functions In...Ch. 5.7 - Approximating Inverse Trigonometric Functions In...Ch. 5.7 - Using a Right Triangle In Exercises 19-24, use the...Ch. 5.7 - Using a Right Triangle In Exercises 19-24, use the...Ch. 5.7 - Using a Right Triangle In Exercises 19-24, use the...Ch. 5.7 - Using a Right Triangle In Exercises 19-24, use the...Ch. 5.7 - Using a Right Triangle In Exercises 19-24, use the...Ch. 5.7 - Using a Right Triangle In Exercises 19-24, use the...Ch. 5.7 - Evaluating an Expression In Exercises 25-28,...Ch. 5.7 - Evaluating an Expression In Exercises 25-28,...Ch. 5.7 - Evaluating an Expression In Exercises 25-28,...Ch. 5.7 - Evaluating an Expression In Exercises 25-28,...Ch. 5.7 - Simplifying an Expression Using a Right Triangle...Ch. 5.7 - Simplifying an Expression Using a Right Triangle...Ch. 5.7 - Simplifying an Expression Using a Right Triangle...Ch. 5.7 - Simplifying an Expression Using a Right Triangle...Ch. 5.7 - Simplifying an Expression Using a Right Triangle...Ch. 5.7 - Simplifying an Expression Using a Right Triangle...Ch. 5.7 - Simplifying an Expression Using a Right Triangle...Ch. 5.7 - Simplifying an Expression Using a Right Triangle...Ch. 5.7 - Solving an Equation In Exercises 37-40, solve the...Ch. 5.7 - Solving an Equation In Exercises 37-40, solve the...Ch. 5.7 - Solving an Equation In Exercises 37-40, solve the...Ch. 5.7 - Solving an Equation In Exercises 37-40, solve the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding a Derivative In Exercises 41-56, find the...Ch. 5.7 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.7 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.7 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.7 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.7 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.7 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.7 - Finding Relative Extrema In Exercises 63-66, find...Ch. 5.7 - Finding Relative Extrema In Exercises 63-66, find...Ch. 5.7 - Finding Relative Extrema In Exercises 63-66, find...Ch. 5.7 - Finding Relative Extrema In Exercises 63-66, find...Ch. 5.7 - Analyzing an Inverse Trigonometric Graph In...Ch. 5.7 - Analyzing an Inverse Trigonometric Graph In...Ch. 5.7 - Analyzing an Inverse Trigonometric Graph In...Ch. 5.7 - Analyzing an Inverse Trigonometric Graph In...Ch. 5.7 - Implicit Differentiation In Exercises 71-74, use...Ch. 5.7 - Implicit Differentiation In Exercises 71-74, use...Ch. 5.7 - Implicit Differentiation In Exercises 71-74, use...Ch. 5.7 - Implicit Differentiation In Exercises 71-74, use...Ch. 5.7 - Finding Values (a) Use a graphing utility to...Ch. 5.7 - HOW DO YOU SEE IT? The graph of g(x) = cos x is...Ch. 5.7 - Inverse Trigonometric Functions Determine whether...Ch. 5.7 - Inverse Trigonometric Functions Determine whether...Ch. 5.7 - Inverse Trigonometric Functions Explain why sin2=0...Ch. 5.7 - Inverse Trigonometric Functions Explain why tan =0...Ch. 5.7 - Verifying Identities In Exercises 81 and 82,...Ch. 5.7 - Verifying Identities In Exercises 81 and 82,...Ch. 5.7 - True or False? In Exercises 83-86, determine...Ch. 5.7 - True or False? In Exercises 83-86, determine...Ch. 5.7 - True or False? In Exercises 83-86, determine...Ch. 5.7 - True or False? In Exercises 83-86, determine...Ch. 5.7 - Angular Rate of Change An airplane flies at an...Ch. 5.7 - Writing Repeat Exercise 87 for an altitude of 3...Ch. 5.7 - Angular Rate of Change In a free-fall experiment,...Ch. 5.7 - Angular Rate of Change A television camera at...Ch. 5.7 - Maximizing an Angle A billboard 85 feet wide is...Ch. 5.7 - Angular Speed A patrol car is parked 50 feet from...Ch. 5.7 - Proof (a) Prove that...Ch. 5.7 - Proof Prove each differentiation formula. (a)...Ch. 5.7 - Describing a Graph Use a graphing utility to graph...Ch. 5.7 - Think About It Use a graphing utility to graph...Ch. 5.7 - Maximizing an Angle In the figure, find the value...Ch. 5.7 - Finding a Distance In the figure, find PR such...Ch. 5.7 - Proof Prove that arcsinx=arctan(x1x2),x1.Ch. 5.7 - Inverse Secant Function Some calculus textbooks...Ch. 5.8 - CONCEPT CHECK Integration Rules Decide whether you...Ch. 5.8 - CONCEPT CHECK Completing the Square In your own...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Finding an Indefinite Integral In Exercises 3-22,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 2334,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Evaluating a Definite Integral In Exercises 23-34,...Ch. 5.8 - Completing the Square In Exercises 35-42, find or...Ch. 5.8 - Completing the Square In Exercises 35-42, find or...Ch. 5.8 - Completing the Square In Exercises 35-42, find or...Ch. 5.8 - Completing the Square In Exercises 35-42, find or...Ch. 5.8 - Completing the Square In Exercises 35-42, find or...Ch. 5.8 - Completing the Square In Exercises 35-42, find or...Ch. 5.8 - Completing the Square In Exercises 35-42, find or...Ch. 5.8 - Completing the Square In Exercises 35-42, find or...Ch. 5.8 - Integration by Substitution In Exercises 43-46,...Ch. 5.8 - Integration by Substitution In Exercises 43-46,...Ch. 5.8 - Integration by Substitution In Exercises 43-46,...Ch. 5.8 - Integration by Substitution In Exercises 43-46,...Ch. 5.8 - Comparing Integration Problems In Exercises 47-50,...Ch. 5.8 - Comparing Integration Problems In Exercises 47-50,...Ch. 5.8 - Comparing Integration Problems In Exercises 47-50,...Ch. 5.8 - Comparing Integration Problems In Exercises 47-50,...Ch. 5.8 - EXPLORING CONCEPTS Comparing Antiderivatives In...Ch. 5.8 - EXPLORING CONCEPTS Comparing Antiderivatives In...Ch. 5.8 - Inverse Trigonometric Functions The antiderivative...Ch. 5.8 - HOW DO YOU SEE IT? Using the graph, which value...Ch. 5.8 - Slope Field In Exercises 55 and 56, a differential...Ch. 5.8 - Slope Field In Exercises 55 and 56, a differential...Ch. 5.8 - Slope Field In Exercises 57-60, use a graphing...Ch. 5.8 - Slope Field In Exercises 5760, use a graphing...Ch. 5.8 - Slope Field In Exercises 57-60, use a graphing...Ch. 5.8 - Slope Field In Exercises 57-60, use a graphing...Ch. 5.8 - Differential Equation In Exercises 61 and 62, find...Ch. 5.8 - Differential Equation In Exercises 61 and 62, find...Ch. 5.8 - Area In Exercises 63-66, find the area of the...Ch. 5.8 - Area In Exercises 63-66, find the area of the...Ch. 5.8 - Area In Exercises 63-66, find the area of the...Ch. 5.8 - Area In Exercises 63-66, find the area of the...Ch. 5.8 - Area (a) Sketch the region whose area is...Ch. 5.8 - Approximating Pi (a) Show that 0141+x2dx= (b)...Ch. 5.8 - Investigation Consider the function...Ch. 5.8 - Comparing Integrals Consider the integral 16xx2dx...Ch. 5.8 - True or False? In Exercises 71 and 72, determine...Ch. 5.8 - True or False? In Exercises 71 and 72, determine...Ch. 5.8 - Verifying an Integration Rule In Exercises 73-75,...Ch. 5.8 - Verifying an Integration Rule In Exercises 73-75,...Ch. 5.8 - Verifying an Integration Rule In Exercises 7375,...Ch. 5.8 - Proof Graph y1=x1+x2,y2=arctanx, and y3=x on [0,...Ch. 5.8 - Numerical Integration (a) Write an integral that...Ch. 5.8 - Vertical Motion An object is projected upward from...Ch. 5.9 - CONCEPT CHECK Hyperbolic Functions Describe how...Ch. 5.9 - CONCEPT CHECK Domains of Hyperbolic Functions...Ch. 5.9 - CONCEPT CHECK Hyperbolic Identities Which...Ch. 5.9 - CONCEPT CHECK Derivatives of Inverse Hyperbolic...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Fuction In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Evaluating a Function In Exercises 5-10, evaluate...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Verifying an Identity In Exercises 11-18, verify...Ch. 5.9 - Finding Values of Hyperbolic Functions In...Ch. 5.9 - Finding Values of Hyperbolic Functions In...Ch. 5.9 - Finding a Limit In Exercises 21-24, find the...Ch. 5.9 - Finding a Limit In Exercises 21-24, find the...Ch. 5.9 - Finding a Limit In Exercises 21-24, find the...Ch. 5.9 - Finding a Limit In Exercises 21-24, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding a Derivative In Exercises 2534, find the...Ch. 5.9 - Finding a Derivative In Exercises 25-34, find the...Ch. 5.9 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.9 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.9 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.9 - Finding an Equation of a Tangent Line In Exercises...Ch. 5.9 - Finding Relative Extrema In Exercises 39-42, find...Ch. 5.9 - Finding Relative Extrema In Exercises 39-42, find...Ch. 5.9 - Finding Relative Extrema In Exercises 39-42, find...Ch. 5.9 - Finding Relative Extrema In Exercises 39-42, find...Ch. 5.9 - Catenary In Exercises 43 and 44, a model for a...Ch. 5.9 - Catenary In Exercises 43 and 44, a model for a...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 45-54,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 55-60,...Ch. 5.9 - EXPLORING CONCEPTS Using a Graph Explain...Ch. 5.9 - EXPLORING CONCEPTS Hyperbolic Functions Use the...Ch. 5.9 - Think About It Verify the results of Exercise 62...Ch. 5.9 - HOW DO YOU SEE IT? Use the graphs of f and g shown...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 6574, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding a Derivative In Exercises 65-74, find the...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 7582,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Finding an Indefinite Integral In Exercises 75-82,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 83-86,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 83-86,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 83-86,...Ch. 5.9 - Evaluating a Definite Integral In Exercises 83-86,...Ch. 5.9 - Differential Equation In Exercises 87 and 88, find...Ch. 5.9 - Differential Equation In Exercises 87 and 88, find...Ch. 5.9 - Area In Exercises 89-92, find the area of the...Ch. 5.9 - Area In Exercises 8992, find the area of the given...Ch. 5.9 - Area In Exercises 89-92, find the area of the...Ch. 5.9 - Area In Exercises 89-92, find the area of the...Ch. 5.9 - Tractrix Consider the equation of a tractrix...Ch. 5.9 - Tractrix Show that the boat in Example 5 is always...Ch. 5.9 - Proof Prove that tanh1x=12ln(1+x1x),1x1Ch. 5.9 - Proof Prove that sinh1t=ln(t+t2+1)Ch. 5.9 - Using a Right Triangle Show that...Ch. 5.9 - Integration Let x0 and b0. Show that...Ch. 5.9 - Proof In Exercises 99-101, prove the...Ch. 5.9 - Proof In Exercises 99-101, prove the...Ch. 5.9 - Proof In Exercises 99-101, prove the...Ch. 5.9 - Verifying a Differentiation Formula In Exercises...Ch. 5.9 - Verifying a Differentiation Formula In Exercises...Ch. 5.9 - Verifying a Differentiation Formula In Exercises...Ch. 5.9 - PUTNAM EXAM CHALLENGE From the vertex (0, c) of...Ch. 5.9 - PUTNAM EXAM CHALLENGE Prove or disprove: there is...Ch. 5 - Sketching a Graph In Exercises 1 and 2, sketch the...Ch. 5 - Sketching a Graph In Exercises 1 and 2, sketch the...Ch. 5 - Using Properties of Logarithms In Exercises 3 and...Ch. 5 - Using Properties of Logarithms In Exercises 3 and...Ch. 5 - Expanding a Logarithmic Expression In Exercises 5...Ch. 5 - Expanding a Logarithmic Expression In Exercises 5...Ch. 5 - Condensing a Logarithmic Expression In Exercises 7...Ch. 5 - Condensing a Logarithmic Expression In Exercises 7...Ch. 5 - Finding a Derivative In Exercises 9-16, find the...Ch. 5 - Finding a Derivative In Exercises 9-16, find the...Ch. 5 - Finding a Derivative In Exercises 9-16, find the...Ch. 5 - Finding a Derivative In Exercises 9-16, find the...Ch. 5 - Finding a Derivative In Exercises 9-16, find the...Ch. 5 - Finding a Derivative In Exercises 9-16, find the...Ch. 5 - Finding a Derivative In Exercises 9-16, find the...Ch. 5 - Finding a Derivative In Exercises 9-16, find the...Ch. 5 - Finding an Equation of a Tangent Line In Exercises...Ch. 5 - Finding an Equation of a Tangent Line In Exercises...Ch. 5 - Logarithmic Differentiation In Exercises 19 and...Ch. 5 - Logarithmic Differentiation In Exercises 19 and...Ch. 5 - Finding an Indefinite Integral In Exercises 21-26,...Ch. 5 - Finding an Indefinite Integral In Exercises 21-26,...Ch. 5 - Finding an Indefinite Integral In Exercises 21-26,...Ch. 5 - Finding an Indefinite Integral In Exercises 21-26,...Ch. 5 - Finding an Indefinite Integral In Exercises 21-26,...Ch. 5 - Finding an Indefinite Integral In Exercises 21-26,...Ch. 5 - Evaluating a Definite Integral In Exercises 27-30,...Ch. 5 - Evaluating a Definite Integral In Exercises 27-30,...Ch. 5 - Evaluating a Definite Integral In Exercises 27-30,...Ch. 5 - Evaluating a Definite Integral In Exercises 27-30,...Ch. 5 - Area In Exercises 31 and 32, find the area of the...Ch. 5 - Area In Exercises 31 and 32, find the area of the...Ch. 5 - Finding an Inverse Function In Exercises 33-38,...Ch. 5 - Finding an Inverse Function In Exercises 33-38,...Ch. 5 - Finding an Inverse Function In Exercises 33-38,...Ch. 5 - Finding an Inverse Function In Exercises 33-38,...Ch. 5 - Finding an Inverse Function In Exercises 33-38,...Ch. 5 - Finding an Inverse Function In Exercises 33-38,...Ch. 5 - Evaluating the Derivative of an Inverse Function...Ch. 5 - Evaluating the Derivative of an Inverse Function...Ch. 5 - Evaluating the Derivative of an Inverse Function...Ch. 5 - Evaluating the Derivative of an Inverse Function...Ch. 5 - Solving an Exponential or Logarithmic Equation In...Ch. 5 - Solving an Exponential or Logarithmic Equation In...Ch. 5 - Solving an Exponential or Logarithmic Equation In...Ch. 5 - Solving an Exponential or Logarithmic Equation In...Ch. 5 - Finding a Derivative In Exercises 47-52, find the...Ch. 5 - Finding a Derivative In Exercises 47-52, find the...Ch. 5 - Finding a Derivative In Exercises 47-52, find the...Ch. 5 - Finding a Derivative In Exercises 47-52, find the...Ch. 5 - Finding a Derivative In Exercises 47-52, find the...Ch. 5 - Finding a Derivative In Exercises 47-52, find the...Ch. 5 - Finding an Equation of a Tangent Line In Exercises...Ch. 5 - Finding an Equation of a Tangent Line In Exercises...Ch. 5 - Finding Extrema and Points of Inflection In...Ch. 5 - Finding Extrema and Points of Inflection In...Ch. 5 - Finding an Indefinite Integral In Exercises 57-60,...Ch. 5 - Finding an Indefinite Integral In Exercises 57-60,...Ch. 5 - Finding an Indefinite Integral In Exercises 57-60,...Ch. 5 - Finding an Indefinite Integral In Exercises 5760,...Ch. 5 - Evaluating a Definite Integral In Exercises 61-64,...Ch. 5 - Evaluating a Definite Integral In Exercises 61-64,...Ch. 5 - Evaluating a Definite Integral In Exercises 61-64,...Ch. 5 - Evaluating a Definite Integral In Exercises 61-64,...Ch. 5 - Area Find the area of the region bounded by the...Ch. 5 - Depreciation The value V of an item t years after...Ch. 5 - Sketching a Graph In Exercises 67 and 68, sketch...Ch. 5 - Sketching a Graph In Exercises 67 and 68, sketch...Ch. 5 - Solving an Equation In Exercises 69-74, solve the...Ch. 5 - Solving an Equation In Exercises 69-74, solve the...Ch. 5 - Solving an Equation In Exercises 69-74, solve the...Ch. 5 - Solving an Equation In Exercises 69-74, solve the...Ch. 5 - Solving an Equation In Exercises 69-74, solve the...Ch. 5 - Solving an Equation In Exercises 69-74, solve the...Ch. 5 - Finding a Derivative In Exercises 75-82, find the...Ch. 5 - Finding a Derivative In Exercises 75-82, find the...Ch. 5 - Finding a Derivative In Exercises 75-82, find the...Ch. 5 - Finding a Derivative In Exercises 75-82, find the...Ch. 5 - Finding a Derivative In Exercises 75-82, find the...Ch. 5 - Finding a Derivative In Exercises 75-82, find the...Ch. 5 - Finding a Derivative In Exercises 75-82, find the...Ch. 5 - Finding a Derivative In Exercises 75-82, find the...Ch. 5 - Finding an Indefinite Integral In Exercises 83 and...Ch. 5 - Finding an Indefinite Integral In Exercises 83 and...Ch. 5 - Evaluating a Definite Integral In Exercises 85 and...Ch. 5 - Evaluating a Definite Integral In Exercises 85 and...Ch. 5 - Compound Interest (a) A deposit of $550 is made in...Ch. 5 - Climb Rate The lime t (in minutes) for a small...Ch. 5 - Evaluating a Limit In Exercises 8996, use LHpitals...Ch. 5 - Evaluating a Limit In Exercises 8996, use LHpitals...Ch. 5 - Evaluating a Limit In Exercises 8996, use LHpitals...Ch. 5 - Evaluating a Limit In Exercises 8996, use LHpitals...Ch. 5 - Evaluating a Limit In Exercises 8996, use LHpitals...Ch. 5 - Evaluating a Limit In Exercises 8996, use LHpitals...Ch. 5 - Evaluating a Limit In Exercises 8996, use LHpitals...Ch. 5 - Evaluating a Limit In Exercises 8996, use LHpitals...Ch. 5 - Evaluating an Expression In Exercises 97 and 98,...Ch. 5 - Evaluating an Expression In Exercises 97 and 98,...Ch. 5 - Finding a Derivative In Exercises 99-104, find the...Ch. 5 - Finding a Derivative In Exercises 99-104, find the...Ch. 5 - Finding a Derivative In Exercises 99-104, find the...Ch. 5 - Finding a Derivative In Exercises 99-104, find the...Ch. 5 - Finding a Derivative In Exercises 99-104, find the...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Evaluating a Definite Integral In Exercises...Ch. 5 - Evaluating a Definite Integral In Exercises...Ch. 5 - Evaluating a Definite Integral In Exercises...Ch. 5 - Evaluating a Definite Integral In Exercises...Ch. 5 - Area In Exercises 115 and 116, find the area of...Ch. 5 - Area In Exercises 115 and 116, find the area of...Ch. 5 - Verifying an Identity In Exercises 117 and 118,...Ch. 5 - Verifying an Identity In Exercises 117 and 118,...Ch. 5 - Finding a Derivative In Exercises 119-124, find...Ch. 5 - Finding a Derivative In Exercises 119-124, find...Ch. 5 - Finding a Derivative In Exercises 119-124, find...Ch. 5 - Finding a Derivative In Exercises 119-124, find...Ch. 5 - Finding a Derivative In Exercises 119-124, find...Ch. 5 - Finding a Derivative In Exercises 119-124, find...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Finding an Indefinite Integral In Exercises...Ch. 5 - Evaluating a Definite Integral In Exercises...Ch. 5 - Evaluating a Definite Integral In Exercises...Ch. 5 - Evaluating a Definite Integral In Exercises...Ch. 5 - Evaluating a Definite Integral In Exercises...Ch. 5 - Approximation To approximate ex, you can use a...Ch. 5 - Symmetry Recall that the graph of a function...Ch. 5 - Finding a Value Find the value of the positive...Ch. 5 - Finding a Value Find the value of the positive...Ch. 5 - Finding Limits Use a graphing utility to estimate...Ch. 5 - Areas and Angles (a) Let P(cost,sint) be a point...Ch. 5 - Intersection Graph the exponential function y=ax...Ch. 5 - Length The line x=1 is tangent to the unit circle...Ch. 5 - Area Consider the three regions A, B, and C...Ch. 5 - Distance Let L be the tangent line to the graph of...Ch. 5 - Distance Let L be the tangent line to the graph of...Ch. 5 - Gudermannian Function The Gudcrmannian function of...Ch. 5 - Decreasing Function Show that f(x)=lnxnx is a...Ch. 5 - Area Use integration by substitution to find the...Ch. 5 - Area Use integration by substitution to find the...Ch. 5 - Mortgage A $120,000 home mortgage for 35 years at...Ch. 5 - Approximating a Function (a) Use a graphing...
True or False? In Exercises 113-118, determine whether the statement is true or false. If it is false, explain ...
f(x) = (x + 3)3/2
Find the Taylor series for f(x) centered at the given value of a. Assume that f has a power series expansion. D...
Sketch the region enclosed by the given curves and find its area. y = sec2x, y = 8 cos x, /3 x /3
Evaluate the integral, if it exists. 19u2u2udu
In Exercises 2330, factor each expression and simplify as much as possible. (x+1)(x+2)+(x+1)(x+3)
Find the absolute maximum and absolute minimum values of f on the given interval. f(t)=tt3,[1,4]
Evaluate the expression sin Exercises 116. 323
For the limit limx2(x33x+4)=6 illustrate Definition 2 by finding values of that correspond to = 0.2 and = 0....
In Exercises 3-6, simplify the expression. (14)x3/4
Let r(t) = 2-t, (et 1)/t, ln(t + 1). (a) Find the domain of r. (b) Find limt0 r(t). (c) Find r'(t).
Combining Functions Find f + g, f g, fg, and f/g and their domains. 8. f(x) = x, g(x)=x
Here is a stem-and-leaf display of the escape time data introduced in Exercise 36 of this chapter. 32 55 33 49 ...
GER Twenty-five residents of a community for senior citizens were asked to keep track of the number of hours th...
Determine whether the argument is an example of inductive reasoning or deductive reasoning. We had rain each da...
In Exercises 11 to 22, use the Given information to draw a conclusion based on the stated property or definitio...
Statistical Literacy Consider a hypothesis test of difference of proportions for two independent populations. S...
4. If total costs are and total revenues Are ,find the break-even points.
Simplify each power of i. i15
For Problems 1-10, identify each statement as true or false. Objective 1 All whole numbers are integers.
Place the following set of n = 20 scores in a frequency distribution table. 6, 2, 2, 1, 3, 2, 4, 7, 1, 2 5, 3, ...
Evaluate the expression sin Exercises 116. (23)2
Write each number in decimal form: 3.241018
Participants enter a research study with unique characteristics that produce different scores from one person t...
True or False Label each of the following statements as either true or false. 1. Every permutation can be writt...
Sunny land Orange Groves planted fruit trees valued at $375,000 on February 12, 2014. The accountant for the co...
Finding and Evaluating a Derivative In Exercises 1320, And f'(x) and f'(c). FunctionValue of c y=(x23x+2)(x3+1)...
List all the subsets of the set A={IBM, U.S.Steel,UnionCarbide,Boeing}, Which of these are proper subsets of A?
In Exercises 27 to 32, use a calculator and reciprocal relationships to find each ratio correct to four decimal...
Even and Odd Functions In Exercises 73-76, evaluate the integral using the properties of even and odd functions...
True or False: The Mean Value Theorem for Integrals is a restatement of the Mean Value Theorem (of Chapter 4) b...
Finding Partial DerivativesIn Exercises 1926, find all finding partial derivatives. w=x2y2z2
Using Properties of the GradientIn Exercises 6166, find the gradient of the function and the maximum value of t...
The y-coordinate of the center of mass of a lamina D with density ρ(x, y) and mass m is:
A Deloitte employment survey asked a sample of human resource executives how their company planned to change it...
The American Association of Individual Investors (AAII) On-Line Discount Broker Survey polls members on their e...
In Exercises 1-12, find the value of u, v or b. log3u=0
Describe the basic elements of APA-style writing including the use of verb tense and guidelines for citations a...
Express these ratios in lowest fractional form. 10. 2 ft to8 in.
Assume that each sequence is defined recursively. Find the first four terms of a sequence. a1=5 and an+1=an3
Describe the basic characteristics of a pre-post design and explain why these designs are not true experiments.
Let x be the amount of time (in minutes) that a particular San Francisco commuter must wait for a train. Suppos...