1
Q
The Blimp Problem
A
- Related rate problems involve using a known rate of change to find an associated rate of change.
- Use implicit differentiation when you cannot write the dependent variable in terms of the independent variable.
2
Q
note
A
- A blimp traveling overhead is tied to a rope. The rope is let out at a rate of 3 feet per second.
- Assuming the blimp remains at a constant altitude of 800 feet, how fast is the blimp moving when 100 feet of rope have been let out?
- This situation can be modeled by a right triangle. Notice that it is not important to keep your diagram to scale. In fact, drawing to scale might mislead you, giving you false information.
- The Pythagorean theorem relates the lengths of the sides of a right triangle to each other.
- Use implicit differentiation to find the derivative of b with respect to time. Then substitute the known values into the equation.
3
Q
Suppose that a certain clock has a minute hand that is 4 inches long and an hour hand that is 3 inches long. What is the rate of change of the distance between the tips of the minute and hour hands at 2:00?
A
−16.6 inches / hour
4
Q
Laura is hitting a golf ball at the driving range. Suppose that the altitude of the ball is given in feet by the equation y = 200t − 40t ^2, where t is the time in seconds after she hits the ball. Suppose also that the horizontal velocity of the ball is constant and equal to 80 feet per second. At what rate is the angle of elevation of the ball changing 2 seconds after Laura hits the ball?
A
−0.15 radians / second
5
Q
Two cars leave an intersection at the same time, one headed west and the other north. The westbound car is moving at 40 mph and the northbound car is moving at 50 mph. Fifteen minutes later, what is the rate of change in the perimeter of the right triangle created using the two cars and the intersection?
A
154.02 mph
6
Q
An object is moving along the graph of the curve x^2y^2/2x−y=1. At a particular moment, the particle is at (1,1) and dx/dt=5. Find the value of dy/dt at this moment.
A
0
7
Q
A spherical balloon is being inflated from a compressor. Suppose the volume of the balloon is increasing at a constant rate of 10 cubic inches per second. At what rate is the surface area of the balloon increasing when its radius is 6 inches?
A
3.3 square inches per second
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
