1
Q
The Pebble Problem
A
- Related rate problems involve using a known rate of change to find an associated rate of change.
- The three steps to problem-solving are understanding what you want, determining what you know, and finding a connection between the two.
2
Q
note
A
- If you drop a stone into a body of water, ripples form across the surface of the water.
- Suppose you are told that the radius of a ripple is increasing at a rate of 6 inches per second.
- What is the rate of change in the area enclosed by the ripple?
- This is an example of a related rate. Related rate questions ask you to find information about one rate given information about another.
- You want to know the rate of change in the area enclosed by the ripple.
- You are given the rate of change of the radius of the ripple.
- You also know the formula for the area of a circle.
- Since area is expressed in terms of the radius, you can take the derivative of the area equation with respect to time. Notice that the derivative of the area equation includes two variables: the radius, and the rate of change of the radius.
- Substitute the values for the radius and the rate of change into the equation to find the rate of change in the area.
3
Q
Two identical twelve-inch rulers on a desk form an isosceles triangle with included angle θ. If Mickie moves the rulers so that θ increases at the rate of 2.5 radians per second, what is the rate of change of the area of the triangle when θ = π / 4 radians?
Note: the area of an isosceles triangle with legs of length l and included angle θ is given by the formula
A = 1/2l^2sin θ
A
127 square inches per second
4
Q
Water is draining out of a conical tank at a constant rate of 120 cubic inches per minute. How fast is the level of water in the tank decreasing when there are 7 inches of water in the tank?
A
The level of water in the tank is decreasing at a rate of 3.1 inches per minute.
5
Q
Farmer Bob’s square plot of land is slowly eroding away. Worried about the future of his farm, Farmer Bob measures the rate of erosion and finds that the length of each side of his square plot is decreasing at the constant rate of 2 feet / year. If he currently owns 250,000 square feet of land, what is the current rate of change of the area of Farmer Bob’s land?
A
Farmer Bob is losing 2,000 square feet of land per year.
6
Q
A balloon is being inflated from a helium tank at a constant rate of 50 cubic inches per minute. How fast is the radius of the balloon increasing when the radius is 5 inches? Assume that the balloon is a perfect sphere.
A
1/2π inches per minute
7
Q
Suppose that x and y are differentiable with respect to t. If x^3+y^3=9 and dy/dt=2, what is the value of dx/dt when y=1?
A
dx/dt=−1/2
8
Q
A pebble is dropped into a pool of water, generating circular ripples. The radius of the largest ripple is increasing at a constant rate of 3 inches per second. By how many inches did the circumference of the ripple increase between 0 and 2 seconds?
A
12π in.
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
