1
Q
The Can Problem
A
- Values that make the derivative of a function equal to 0 or undefined are candidates for maxima and minima of the function.
- The can problem involves maximizing the volume of a cylindrical can constructed from a given quantity of material.
2
Q
note
A
- Here you are asked to find the dimensions of a can made from a certain amount of material in a way that maximizes the volume of the can.
- You know the surface area of the can as well as the formula for the volume of a cylinder.
- By solving the surface area for h, you can relate the surface area and the volume. Substituting gives you an equation for volume in terms of the radius of the can.
- Once you have the formula for the volume, you can
differentiate. - Set the derivative of the volume equal to 0 and solve for r.
- Since the radius cannot be negative, one of the answers can be rejected.
- Checking the derivative to the left and the right of the
maximum candidate will show that the volume increases to the left and decreases to the right. So the candidate does correspond to a maximum. - Remember, the question asked for the dimensions of the can, not just the radius. Make sure that you give both dimensions.
3
Q
The average soda can is made up of 15.625π square inches of material. What dimensions (to the nearest hundredth) would maximize the volume of this cylindrical can?
A
r = 1.61 inches h = 3.23inches
4
Q
What is the maximum volume of an open cylinder (open at both ends)with surface area equal to 16π ft2 and a radius between 1 and 8 feet?
A
64π ft3
5
Q
Your average soda can has a height of 5 inches and a circular top and bottom with a diameter of 2.5 inches. What is the surface area of this soda can?
A
15.625π in2
6
Q
You are an engineer working for a shipping company who is air mailing huge amounts of gasoline across the country. Your supervisor tells you that the guys upstairs in the suits want to package the gasoline in cylindrical containers that hold 10π cubic feet of gasoline. It is your job to minimize the cost of the packaging materials. What radius should the cylinders have to minimize the amount of packaging material?
A
^3√5 feet
7
Q
What is the maximum volume for a closed cylinder with surface area equal to 150 square inches?
A
141.047 cubic inches
8
Q
Your average soda can has a height of 5 inches and a circular top and bottom with a diameter of 2.5 inches. What is the volume of this soda can?
A
7.8125π in3
AP Calculus AB
flashcards
Decks in class (190)
# Cards
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1.1.1 An Introduction to Thinkwell Calculus1
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1.1.2 The Two Questions of Calculus3
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1.1.3 Average Rates of Change14
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1.1.4 How to Do Math1
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1.2.1 Functions16
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1.2.2 Graphing Lines14
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1.2.3 Parabolas15
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1.2.4 Some Non-Euclidean Geometry1
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Chapter 1 Practice Test20
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Chapter 1 Test20
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2.1.1 Finding Rate of Change over an Interval18
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2.1.2 Finding Limits Graphically9
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2.1.3 The Formal Definition of a Limit6
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2.1.4 The Limit Laws, Part I7
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2.1.5 The Limit Laws, Part II17
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2.1.6 One-Sided Limits11
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2.1.7 The Squeeze Theorem13
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2.1.8 Continuity and Discontinuity13
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2.2.1 Evaluating Limits17
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2.2.2 Limits and Indeterminate Forms12
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2.2.3 Two Techniques for Evaluating Limits7
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2.2.4 An Overview of Limits11
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Chapter 2 Practice Test20
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Chapter 2 Test20
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3.1.1 Rates of Change, Secants, and Tangents12
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3.1.2 Finding Instantaneous Velocity14
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3.1.3 The Derivative13
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3.1.4 Differentiability9
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3.2.1 The Slope of a Tangent Line9
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3.2.2 Instantaneous Rate14
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3.2.3 The Equation of a Tangent Line17
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3.2.4 More on Instantaneous Rate8
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3.3.1 The Derivative of the Reciprocal Function19
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3.3.2 The Derivative of the Square Root Function16
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Chapter 3 Practice Test20
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Chapter 3 Test20
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4.1.1 A Shortcut for Finding Derivatives14
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4.1.2 A Quick Proof of the Power Rule12
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4.1.3 Uses of the Power Rule24
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4.2.1 The Product Rule12
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4.2.2 The Quotient Rule9
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4.3.1 An Introduction to the Chain Rule15
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4.3.2 Using the Chain Rule14
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4.3.3 Combining Computational Techniques14
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Chapter 4 Practice Test20
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Chapter 4 Test20
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5.1.1 A Review of Trigonometry13
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5.1.2 Graphing Trigonometric Functions10
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5.1.3 The Derivatives of Trigonometric Functions17
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5.1.4 The Number Pi16
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5.2.1 Graphing Exponential Functions13
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5.2.2 Derivatives of Exponential Functions8
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5.3.1 Evaluating Logarithmic Functions11
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5.3.2 The Derivative of the Natural Log Function10
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5.3.3 Using the Derivative Rules with Transcendental Functions8
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Chapter 5 Practice Test20
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Chapter 5 Test20
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6.1.1 An Introduction to Implicit Differentiation13
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6.1.2 Finding the Derivative Implicitly14
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6.2.1 Using Implicit Differentiation14
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6.2.2 Applying Implicit Differentiation9
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6.3.1 The Exponential and Natural Log Functions12
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6.3.2 Differentiating Logarithmic Functions12
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6.3.3 Logarithmic Differentiation14
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6.3.4 The Basics of Inverse Functions12
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6.3.5 Finding the Inverse of a Function12
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6.4.1 Derivatives of Inverse Function12
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6.5.1 The Inverse Sine, Cosine, and Tangent Functions12
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6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions12
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6.5.3 Evaluating Inverse Trigonometric Functions12
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6.6.1 Derivatives of Inverse Trigonometric Functions12
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6.7.1 Defining the Hyperbolic Functions12
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6.7.2 Hyperbolic Identities12
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6.7.3 Derivatives of Hyperbolic Functions12
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Chapter 6 Practice Test20
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Chapter 6 Test20
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Practice Midterm Exam20
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Midterm Exam20
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7.1.1 Acceleration and the Derivative9
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7.1.2 Solving Word Problems Involving Distance and Velocity15
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7.2.1 Higher-Order Derivatives and Linear Approximation14
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7.2.2 Using the Tangent Line Approximation Formula14
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7.2.3 Newton's Method6
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7.3.1 The Connection Between Slope and Optimization12
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7.3.2 The Fence Problem8
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7.3.3 The Box Problem8
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7.3.4 The Can Problem8
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7.3.5 The Wire-Cutting Problem5
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7.4.1 The Pebble Problem8
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7.4.2 The Ladder Problem6
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7.4.3 The Baseball Problem7
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7.4.4 The Blimp Problem7
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Chapter 7 Practice Test20
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Chapter 7 Test20
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8.1.1 An Introduction to Curve Sketching7
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8.1.2 Three Big Theorems12
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8.2.1 Critical Points5
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8.2.2 Maximum and Minimum13
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8.2.3 Regions Where a Function Increases or Decreases12
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8.2.4 The First Derivative Test5
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8.3.1 Concavity and Inflection Points9
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8.3.2 Using the Second Derivative to Examine Concavity6
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8.4.1 Graphs of Polynomial Functions9
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8.4.2 Cusp Points and the Derivative8
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8.4.3 Domain-Restricted Functions and the Derivative6
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8.4.4 The Second Derivative Test8
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8.5.1 Vertical Asymptotes7
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8.5.2 Horizontal Asymptotes and Infinite Limits9
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8.5.3 Graphing Functions with Asymptotes5
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8.5.4 Functions with Asymptotes and Holes5
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8.5.5 Functions with Asymptotes and Critical Points5
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Chapter 8 Practice Test20
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Chapter 8 Test20
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9.1.1 Antidifferentiation8
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9.1.2 Antiderivatives of Powers of x9
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9.1.3 Antiderivatives of Trigonometric and Exponential Functions11
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9.2.1 Undoing the Chain Rule9
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9.2.2 Integrating Polynomials by Substitution12
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9.3.1 Integrating Composite Trigonometric Functions by Substitution12
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9.3.2 Integrating Composite Exponential and Rational Functions by Substitution10
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9.3.3 More Integrating Tirgonometric Functions by Substitution12
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9.3.4 Choosing Effective Function Decompositions6
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9.4.1 Approximating Areas of Plane Regions5
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9.4.2 Areas, Riemann Sums, and Definite Integrals12
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9.4.3 The Fundamental Theorem of Calculus, Part I12
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9.4.4 The Fundamental Theorem of Calculus, Part II5
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9.4.5 Illustrating the Fundamental Theorem of Calculus10
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9.4.6 Evaluating Definite Integrals13
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9.5.1 An Overview of Trigonometric Substitution Strategy12
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9.5.2 Trigonometric Substitution Involving a Definite Integral: Part One2
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9.5.3 Trigonometric Substitution Involving a Definite Integral: Part Two9
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9.6.1 Deriving the Trapezoidal Rule12
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9.6.2 An Example of the Trapezoidal Rule14
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Chapter 9 Practice Test20
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Chapter 9 Test20
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10.1.1 Antiderivatives and Motion12
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10.1.2 Gravity and Vertical Motion11
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10.1.3 Solving Vertical Motion Problems8
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10.2.1 The Area between Two Curves7
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10.2.2 Limits of Integration and Area7
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10.2.3 Common Mistakes to Avoid When Finding Areas6
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10.2.4 Regions Bound by Several Curves6
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10.3.1 Finding Areas by Integrating with Respect to y: Part One8
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10.3.2 Finding Areas by Integrating with Respect to y: Part Two8
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10.3.3 Area, Integration by Substitution, and Trigonometry8
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10.4.1 Finding the Average Value of a Function8
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10.5.1 Finding Volumes Using Cross-Sectional Slices8
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10.5.2 An Example of Finding Cross-Sectional Volumes8
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10.6.1 Solids of Revolution7
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10.6.2 The Disk Method along the y-Axis9
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10.6.3 A Transcendental Example of the Disk Method8
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10.6.4 The Washer Method across the x-Axis8
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10.6.5 The Washer Method across the y-Axis8
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10.7.1 Introducing the Shell Method8
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10.7.2 Why Shells Can Be Better Than Washers6
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10.7.3 The Shell Method: Integrating with Respect to y6
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10.8.1 An Introduction to Work12
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10.8.2 Calculating Work12
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10.8.3 Hooke's Law12
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10.9.1 Center of Mass9
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10.9.2 The Center of Mass of a Thin Plate11
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10.10.1 An Introduction to Arc Length12
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10.10.2 Finding Arc Lengths of Curves Given by Functions12
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Chapter 10 Practice Test25
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Chapter 10 Test25
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11.1.1 An Introduction to Differential Equations12
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11.1.2 Solving Separable Differential Equations13
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11.1.3 Finding a Particular Solution12
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11.1.4 Direction Fields15
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11.1.5 Euler's Method for Solving Differential Equations Numerically13
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11.2.1 Exponential Growth11
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11.2.2 Logistic Growth13
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11.2.3 Radioactive Decay10
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Chapter 11 Practice Test12
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Chapter 11 Test12
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12.1.1 Indeterminate Forms14
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12.1.2 An Introduction to L'Hôpital's Rule14
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12.1.3 Basic Uses of L'Hôpital's Rule10
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12.1.4 More Exotic Examples of Indeterminate Forms14
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12.2.1 L'Hôpital's Rule and Indeterminate Products10
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12.2.2 L'Hôpital's Rule and Indeterminate Differences14
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12.2.3 L'Hôpital's Rule and One to the Infinite Power7
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12.2.4 Another Example of One to the Infinite Power10
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12.3.1 The First Type of Improper Integral10
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12.3.2 The Second Type of Improper Integral10
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12.3.3 Infinite Limits of Integration, Convergence, and Divergence11
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Chapter 12 Practice Test15
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Chapter 12 Test15
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Practice Final Exam30
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Final Exam30
