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10th Edition

Ron Larson

Publisher: Cengage Learning

ISBN: 9781305860919

Chapter 4.3, Problem 53E

Textbook Problem

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**Normal Probability Density Function** Use Example 6 as a model to show that the graph of the normal probability density function with *μ* = 0,

*σ*.

To determine

**To prove:** The point of inflection at

**Given Information:**

The provided normal probability density function is

**Formula used:**

If the function is

Where

**Proof:**

Consider the normal probability density function,

Evaluate the derivative of the function by applying constant multiple rule and chain rule.

Further solve as,

Now, evaluate the second derivative of the function by applying product rule.

Further solve as,

Set the second derivative equal to zero, to determine the value of *x.*

Further solve as,

Evaluate the result of substitution the values from each region in the derivative of the function.

Consider the interval,

Test the region by substituting the value

Calculus: An Applied Approach (MindTap Course List)

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Changing Bases to Evaluate Logarithms In Exercises...Ch. 4.5 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4.5 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4.5 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4.5 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4.5 - Differentiating Functions of Other Bases In...Ch. 4.5 - Differentiating Functions of Other Bases In...Ch. 4.5 - Differentiating Functions of Other Bases In...Ch. 4.5 - Differentiating Functions of Other Bases In...Ch. 4.5 - Differentiating Functions of Other Bases In...Ch. 4.5 - Differentiating Functions of Other Bases In...Ch. 4.5 - Differentiating Functions of Others Bases In...Ch. 4.5 - Differentiating Functions of Others Bases In...Ch. 4.5 - Differentiating Functions of Others Bases In...Ch. 4.5 - Differentiating Functions of Others Bases In...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding an Equation of a Tangent Line In Exercises...Ch. 4.5 - Finding Derivatives Implicitly In Exercises 53-56,...Ch. 4.5 - Finding Derivatives Implicitly In Exercises 53-56,...Ch. 4.5 - Finding Derivatives Implicitly In Exercises 53-56,...Ch. 4.5 - Finding Derivatives Implicitly In Exercises 53-56,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Finding Second Derivatives In Exercises 57-62,...Ch. 4.5 - Sound Intensity The relationship between the...Ch. 4.5 - Chemistry The temperatures T (inF) at which water...Ch. 4.5 - Finding Relative Extrema In Exercises 65-68, find...Ch. 4.5 - Finding Relative Extrema In Exercises 65-68, find...Ch. 4.5 - Finding Relative Extrema In Exercises 65-68, find...Ch. 4.5 - Finding Relative Extrema In Exercises 65-68, find...Ch. 4.5 - Analyzing a Graph In Exercises 6972, analyze and...Ch. 4.5 - Analyzing a Graph In Exercises 6972, analyze and...Ch. 4.5 - Analyzing a Graph In Exercises 69-72, analyze and...Ch. 4.5 - Analyzing a Graph In Exercises 69-72, analyze and...Ch. 4.5 - Demand In Exercises 73 and 74, find dx/dp for the...Ch. 4.5 - Demand In Exercises 73 and 74, find dx/dp for the...Ch. 4.5 - Demand Solve the demand function in Exercise 73...Ch. 4.5 - Demand Solve the demand function in Exercise 74...Ch. 4.5 - Minimum Average Cost The cost of producing x units...Ch. 4.5 - Minimum Average Cost The cost of producing x units...Ch. 4.5 - Consumer Trends The numbers of employees E (in...Ch. 4.5 - HOW DO YOU SEE IT? The graph shows the temperature...Ch. 4.5 - Home Mortgage The term t (in years) of a $200,000...Ch. 4.5 - Earthquake Intensity On the Richter scale, the...Ch. 4.5 - Learning Theory Students in a learning theory...Ch. 4.6 - Checkpoint 1 Worked-out solution available at...Ch. 4.6 - Checkpoint 2 Worked-out solution available at...Ch. 4.6 - Checkpoint 3 Worked-out solution available at...Ch. 4.6 - Checkpoint 4 Worked-out solution available at...Ch. 4.6 - In Exercises 1-4, solve the equation for k....Ch. 4.6 - In Exercises 1-4, solve the equation for k....Ch. 4.6 - In Exercises 1-4, solve the equation for k....Ch. 4.6 - In Exercises 1-4, solve the equation for k....Ch. 4.6 - In Exercises 5-8, find the derivative of the...Ch. 4.6 - In Exercises 5-8, find the derivative of the...Ch. 4.6 - In Exercises 5-8, find the derivative of the...Ch. 4.6 - In Exercises 5-8, find the derivative of the...Ch. 4.6 - In Exercises 9-12, simplify the expression. eln4Ch. 4.6 - In Exercises 9-12, simplify the expression....Ch. 4.6 - In Exercises 9-12, simplify the expression....Ch. 4.6 - In Exercises 9-12, simplify the expression....Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Modeling Exponential Growth and Decay In Exercises...Ch. 4.6 - Determining Exponential Growth and Decay In...Ch. 4.6 - Determining Exponential Growth and Decay In...Ch. 4.6 - Determining Exponential Growth and Decay In...Ch. 4.6 - Determining Exponential Growth and Decay In...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Modeling Radioactive Decay In Exercises 11-16,...Ch. 4.6 - Radioactive Decay What percent of a present amount...Ch. 4.6 - Radioactive Decay Find the half-life of a...Ch. 4.6 - Carbon Dating Carbon-14 (14C) dating assumes that...Ch. 4.6 - Carbon Dating Repeat Exercise 19 for a piece of...Ch. 4.6 - Finding Exponential Models In Exercises 21 and 22,...Ch. 4.6 - Finding Exponential Models In Exercises 21 and 22,...Ch. 4.6 - Population Growth The number of a certain type of...Ch. 4.6 - HOW DO YOU SEE IT? The graph shows the populations...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Modeling Compound Interest In Exercises 25-32,...Ch. 4.6 - Finding Present Value In Exercises 33 and 34,...Ch. 4.6 - Finding Present Value In Exercises 33 and 34,...Ch. 4.6 - Effective Rate The effective rate of interest reff...Ch. 4.6 - Effective Rate Use the results of Exercise 35 to...Ch. 4.6 - Investment: Rule of 70 Verify that the time...Ch. 4.6 - Investment: Rule of 70 Use the Rule of 70 from...Ch. 4.6 - Depreciation A sports utility vehicle that costs...Ch. 4.6 - Population The table shows the populations P (in...Ch. 4.6 - Sales The cumulative sales S (in thousands of...Ch. 4.6 - Sales The cumulative sales S (in thousands of...Ch. 4.6 - Learning Curve The management of a factory finds...Ch. 4.6 - Learning Curve The management in Exercise 43...Ch. 4.6 - Revenue A small business assumes that the demand...Ch. 4.6 - Revenue Repeat Exercise 45 given that when...Ch. 4 - Applying Properties of Exponents In Exercises 1...Ch. 4 - Applying Properties of Exponents In Exercises 1...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Graphing Exponential Functions In Exercises 3-8,...Ch. 4 - Population Growth The resident populations P (in...Ch. 4 - Revenue The revenues R (in millions of dollars)...Ch. 4 - Property Value A piece of property sells for...Ch. 4 - Inflation Rate With an annual rate of inflation of...Ch. 4 - Applying Properties of Exponents In Exercises 13...Ch. 4 - Applying Properties of Exponents In Exercises 13...Ch. 4 - Graphing Natural Exponential Functions In...Ch. 4 - Graphing Natural Exponential Functions In...Ch. 4 - Graphing Natural Exponential Functions In...Ch. 4 - Graphing Natural Exponential Functions In...Ch. 4 - Finding Account Balances In Exercises 19-22,...Ch. 4 - Finding Account Balances In Exercises 19-22,...Ch. 4 - Finding Account Balances In Exercises 19-22,...Ch. 4 - Finding Account Balances In Exercises 19-22,...Ch. 4 - Comparing Account Balances In Exercises 23 and 24,...Ch. 4 - Comparing Account Balances In Exercises 23 and 24,...Ch. 4 - Effective Rate Find the effective rate of interest...Ch. 4 - Effective Rate Find the effective rate of interest...Ch. 4 - Present Value How much should be deposited in an...Ch. 4 - Present Value How much should be deposited in an...Ch. 4 - Demand The demand function for a product is...Ch. 4 - Demand The demand function for a product is...Ch. 4 - Profit The net profits P (in millions of dollars)...Ch. 4 - Population The populations P (in thousands) of...Ch. 4 - Biology A lake is stocked with 500 fish, and the...Ch. 4 - Medicine On a college campus of 5000 students, the...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Differentiating Exponential Functions In Exercises...Ch. 4 - Finding an Equation of a Tangent Line In Exercises...Ch. 4 - Finding an Equation of a Tangent Line In Exercises...Ch. 4 - Finding an Equation of a Tangent Line In Exercises...Ch. 4 - Finding an Equation of a Tangent Line In Exercises...Ch. 4 - Analyzing a Graph In Exercises 4548, analyze and...Ch. 4 - Analyzing a Graph In Exercises 4548, analyze and...Ch. 4 - Analyzing a Graph In Exercises 4548, analyze and...Ch. 4 - Analyzing a Graph In Exercises 4548, analyze and...Ch. 4 - Logarithmic and Exponential Forms of Equations In...Ch. 4 - Logarithmic and Exponential Forms of Equations In...Ch. 4 - Logarithmic and Exponential Forms of Equations In...Ch. 4 - Logarithmic and Exponential Forms of Equations In...Ch. 4 - Graphing Logarithmic Functions In Exercises 5356,...Ch. 4 - Graphing Logarithmic Functions In Exercises 5356,...Ch. 4 - Graphing Logarithmic Functions In Exercises 5356,...Ch. 4 - Graphing Logarithmic Functions In Exercises 5356,...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Expanding Logarithmic Expressions In Exercises...Ch. 4 - Condensing Logarithmic Expressions In Exercises...Ch. 4 - Condensing Logarithmic Expressions In Exercises...Ch. 4 - Condensing Logarithmic Expressions In Exercises...Ch. 4 - Condensing Logarithmic Expressions In Exercises...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Solving Exponential and Logarithmic Equations In...Ch. 4 - Compound Interest A deposit of $400 is made in an...Ch. 4 - Hourly Earnings The average hourly wages w (in...Ch. 4 - Learning Theory Students in a psychology...Ch. 4 - Demand The demand function for a product is given...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Differentiating Logarithmic Functions In Exercises...Ch. 4 - Evaluating Logarithms In Exercises 99-102,...Ch. 4 - Evaluating Logarithms In Exercises 99-102,...Ch. 4 - Evaluating Logarithms In Exercises 99-102,...Ch. 4 - Evaluating Logarithms In Exercises 99-102,...Ch. 4 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4 - Changing Bases to Evaluate Logarithms In Exercises...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Differentiating Functions of Other Bases In...Ch. 4 - Finding Relative Extrema In Exercises 113-116,...Ch. 4 - Finding Relative Extrema In Exercises 113-116,...Ch. 4 - Finding Relative Extrema In Exercises 113-116,...Ch. 4 - Finding Relative Extrema In Exercises 113-116,...Ch. 4 - Music The numbers of music albums downloaded D (in...Ch. 4 - Minimum Average Cost The cost of producing x units...Ch. 4 - Modeling Exponential Growth and Decay In Exercises...Ch. 4 - Modeling Exponential Growth and Decay In Exercises...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Radioactive Decay In Exercises 121-126,...Ch. 4 - Modeling Compound Interest In Exercises 127-130,...Ch. 4 - Modeling Compound Interest In Exercises 127-130,...Ch. 4 - Modeling Compound Interest In Exercises 127-130,...Ch. 4 - Modeling Compound Interest In Exercises 127-130,...Ch. 4 - Medical Science Soon after an injection, the...Ch. 4 - Population Growth The number of a certain type of...Ch. 4 - In Exercises 14, use the properties of exponents...Ch. 4 - In Exercises 14, use the properties of exponents...Ch. 4 - In Exercises 14, use the properties of exponents...Ch. 4 - In Exercises 14, use the properties of exponents...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 5-10, sketch the graph of the...Ch. 4 - In Exercises 1113, use the properties of...Ch. 4 - In Exercises 1113, use the properties of...Ch. 4 - In Exercises 1113, use the properties of...Ch. 4 - In Exercises 1416, use the properties of...Ch. 4 - In Exercises 1416, use the properties of...Ch. 4 - In Exercises 1416, use the properties of...Ch. 4 - In Exercises 17-19, solve for x. ln3x=3.5Ch. 4 - In Exercises 17-19, solve for x. 10e2x+1=900Ch. 4 - In Exercises 17-19, solve for x. 50(1.06)x=1500Ch. 4 - A deposit of $500 is made in an account that earns...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - In Exercises 21-26, find the derivative of the...Ch. 4 - The sales S (in billions of dollars) of organic...Ch. 4 - What percent of a present amount of radioactive...Ch. 4 - A population is growing continuously at the rate...

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