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AP Calculus BC

Math Academy official course import: AP Calculus BC

305 节课

课程大纲

  1. 1.Limits
    1. 1.1.1. Estimating Limits from Graphs
      1. 1
        The Finite Limit of a Function
      2. 2
        The Left and Right-Sided Limits of a Function
      3. 3
        Finding the Existence of a Limit Using One-Sided Limits
      4. 4
        Limits at Infinity from Graphs
      5. 5
        Infinite Limits from Graphs
    2. 2.1.2. The Algebra of Limits
      1. 1
        Limits of Power Functions, and the Constant Rule for Limits
      2. 2
        The Sum Rule for Limits
      3. 3
        The Product and Quotient Rules for Limits
      4. 4
        The Power and Root Rules for Limits
    3. 3.1.3. Limits of Functions
      1. 1
        Limits at Infinity of Polynomials
      2. 2
        Limits of Reciprocal Functions
      3. 3
        Limits of Exponential Functions
      4. 4
        Limits of Logarithmic Functions
      5. 5
        Limits of Radical Functions
      6. 6
        Limits of Trigonometric Functions
      7. 7
        Limits of Reciprocal Trigonometric Functions
      8. 8
        Limits of Inverse Trigonometric Functions
      9. 9
        Limits of Piecewise Functions
    4. 4.1.4. Determining Limits Using Algebraic Manipulation
      1. 1
        Calculating Limits of Rational Functions by Factoring
      2. 2
        Limits of Absolute Value Functions
      3. 3
        Calculating Limits of Radical Functions Using Conjugate Multiplication
      4. 4
        Calculating Limits Using Trigonometric Identities
      5. 5
        Limits at Infinity and Horizontal Asymptotes of Rational Functions
      6. 6
        Evaluating Limits at Infinity by Comparing Relative Magnitudes of Functions
      7. 7
        Evaluating Limits at Infinity of Radical Functions
      8. 8
        Vertical Asymptotes of Rational Functions
      9. 9
        Connecting Infinite Limits and Vertical Asymptotes of Rational Functions
    5. 5.1.5. Special Limits
      1. 1
        The Squeeze Theorem
      2. 2
        Special Limits Involving Sine
      3. 3
        Evaluating Special Limits Involving Sine Using a Substitution
      4. 4
        Special Limits Involving Cosine
      5. 5
        Limits Involving the Exponential Function
  2. 2.Continuity
    1. 1.2.1. Continuity
      1. 1
        Determining Continuity from Graphs
      2. 2
        Defining Continuity at a Point
      3. 3
        Left and Right Continuity
      4. 4
        Further Continuity of Piecewise Functions
      5. 5
        Point Discontinuities
      6. 6
        Jump Discontinuities
      7. 7
        Discontinuities Due to Vertical Asymptotes
      8. 8
        Continuity Over an Interval
      9. 9
        Continuity of Functions
      10. 10
        The Intermediate Value Theorem
    2. 2.2.2. Removing Discontinuities
      1. 1
        Removing Point Discontinuities
      2. 2
        Removing Jump Discontinuities
      3. 3
        Removing Discontinuities From Rational Functions
  3. 3.Introduction to Differentiation
    1. 1.3.1. Introduction to Differentiation
      1. 1
        The Average Rate of Change of a Function over a Varying Interval
      2. 2
        The Instantaneous Rate of Change of a Function at a Point
      3. 3
        Defining the Derivative Using Derivative Notation
      4. 4
        Connecting Differentiability and Continuity
      5. 5
        The Power Rule for Differentiation
      6. 6
        The Sum and Constant Multiple Rules for Differentiation
      7. 7
        Calculating the Slope of a Tangent Line Using Differentiation
      8. 8
        Calculating the Equation of a Tangent Line Using Differentiation
      9. 9
        Calculating the Equation of a Normal Line Using Differentiation
    2. 2.3.2. Derivatives of Functions and the Rules of Differentiation
      1. 1
        Differentiating Exponential Functions
      2. 2
        Differentiating Logarithmic Functions
      3. 3
        Differentiating Trigonometric Functions
      4. 4
        Second and Higher Order Derivatives
      5. 5
        The Product Rule for Differentiation
      6. 6
        The Quotient Rule for Differentiation
      7. 7
        Differentiating Reciprocal Trigonometric Functions
      8. 8
        Calculating Derivatives From Data and Tables
      9. 9
        Calculating Derivatives From Graphs
      10. 10
        Recognizing Derivatives in Limits
  4. 4.Advanced Differentiation
    1. 1.4.1. Differentiating Composite Functions
      1. 1
        The Chain Rule for Differentiation
      2. 2
        The Chain Rule With Exponential Functions
      3. 3
        The Chain Rule With Logarithmic Functions
      4. 4
        The Chain Rule With Trigonometric Functions
      5. 5
        Calculating Derivatives From Data Using the Chain Rule
      6. 6
        Calculating Derivatives From Graphs Using the Chain Rule
      7. 7
        Selecting Procedures for Calculating Derivatives
    2. 2.4.2. Differentiating Implicit and Inverse Functions
      1. 1
        Implicit Differentiation
      2. 2
        Calculating Slopes of Circles, Ellipses, and Parabolas
      3. 3
        Calculating dy/dx Using dx/dy
      4. 4
        Differentiating Inverse Functions
      5. 5
        Differentiating an Inverse Function at a Point
      6. 6
        Differentiating Inverse Trigonometric Functions
      7. 7
        Differentiating Inverse Reciprocal Trigonometric Functions
  5. 5.Contextual Applications of Differentiation
    1. 1.5.1. Contextual Applications of Differentiation
      1. 1
        Interpreting the Meaning of the Derivative in Context
      2. 2
        Rates of Change in Applied Contexts
    2. 2.5.2. Estimating Derivatives
      1. 1
        Estimating Derivatives Using a Forward Difference Quotient
      2. 2
        Estimating Derivatives Using a Backward Difference Quotient
      3. 3
        Estimating Derivatives Using a Central Difference Quotient
    3. 3.5.3. Related Rates of Change
      1. 1
        Introduction to Related Rates
      2. 2
        Related Rates With Implicit Functions
      3. 3
        Calculating Related Rates With Circles and Spheres
      4. 4
        Calculating Related Rates With Squares
      5. 5
        Calculating Related Rates With Rectangular Solids
      6. 6
        Calculating Related Rates Using the Pythagorean Theorem
      7. 7
        Calculating Related Rates Using Similar Triangles
      8. 8
        Calculating Related Rates Using Trigonometry
      9. 9
        Calculating Related Rates With Cones
    4. 4.5.4. L'Hopital's Rule
      1. 1
        L'Hopital's Rule
      2. 2
        L'Hopital's Rule Applied to Tables
  6. 6.Analytical Applications of Differentiation
    1. 1.6.1. Analytical Applications of Differentiation
      1. 1
        The Mean Value Theorem
      2. 2
        Global vs. Local Extrema and Critical Points
      3. 3
        The Extreme Value Theorem
      4. 4
        Using Differentiation to Calculate Critical Points
      5. 5
        Determining Intervals on Which a Function Is Increasing or Decreasing
      6. 6
        Using the First Derivative Test to Classify Local Extrema
      7. 7
        The Candidates Test
      8. 8
        Intervals of Concavity
      9. 9
        Relating Concavity to the Second Derivative
      10. 10
        Points of Inflection
      11. 11
        The Second Derivative Test
    2. 2.6.2. Analysis of Curves
      1. 1
        Sketching the Derivative of a Function From the Function's Graph
      2. 2
        Interpreting the Graph of a Function's Derivative
      3. 3
        Interpreting the Graph of a Function's Derivative: Concavity and Points of Inflection
      4. 4
        Sketching a Function From the Graph of its Derivative
      5. 5
        Sketching a Function Given Some Derivative Properties
    3. 3.6.3. Approximating Values of a Function
      1. 1
        Approximating Functions Using Local Linearity and Linearization
      2. 2
        Approximating the Roots of a Number Using Local Linearity
      3. 3
        Approximating Trigonometric Functions Using Local Linearity
    4. 4.6.4. Optimization
      1. 1
        Solving Optimization Problems Using Derivatives
      2. 2
        Optimization Problems Involving Sectors of Circles
      3. 3
        Optimization Problems Involving Boxes and Trays
      4. 4
        Optimization Problems Involving Cylinders
      5. 5
        Optimizing Distances
      6. 6
        Optimizing Distances to Curves
      7. 7
        Optimization Problems With Inscribed Shapes
      8. 8
        Optimization Problems in Economics
  7. 7.Integration
    1. 1.7.1. Indefinite Integrals
      1. 1
        The Antiderivative
      2. 2
        The Constant Multiple Rule for Indefinite Integrals
      3. 3
        The Sum Rule for Indefinite Integrals
      4. 4
        Integrating the Reciprocal Function
      5. 5
        Integrating Exponential Functions
      6. 6
        Integrating Trigonometric Functions
      7. 7
        Integration Using Inverse Trigonometric Functions
    2. 2.7.2. Approximating Areas with Riemann Sums
      1. 1
        Approximating Areas With the Left Riemann Sum
      2. 2
        Approximating Areas With the Right Riemann Sum
      3. 3
        Approximating Areas With the Midpoint Riemann Sum
      4. 4
        Approximating Areas With the Trapezoidal Rule
      5. 5
        Left and Right Riemann Sums in Sigma Notation
      6. 6
        Midpoint and Trapezoidal Rules in Sigma Notation
      7. 7
        Approximating Areas Under Graphs of Composite Functions
    3. 3.7.3. Definite Integrals
      1. 1
        Defining Definite Integrals Using Left and Right Riemann Sums
      2. 2
        The Fundamental Theorem of Calculus
      3. 3
        Applying the Fundamental Theorem of Calculus to Exponential and Trigonometric Functions
      4. 4
        The Sum and Constant Multiple Rules for Definite Integrals
      5. 5
        Properties of Definite Integrals Involving the Limits of Integration
    4. 4.7.4. The Area Under a Curve
      1. 1
        The Area Bounded by a Curve and the X-Axis
      2. 2
        Evaluating Definite Integrals Using Symmetry
      3. 3
        Finding the Area Between a Curve and the X-Axis When They Intersect
      4. 4
        The Area Bounded by a Curve and the Y-Axis
      5. 5
        Calculating the Definite Integral of a Function Given Its Graph
      6. 6
        Calculating the Definite Integral of a Function's Derivative Given its Graph
      7. 7
        Definite Integrals of Piecewise Functions
    5. 5.7.5. Accumulation Functions
      1. 1
        The Integral as an Accumulation Function
      2. 2
        The Second Fundamental Theorem of Calculus
      3. 3
        Maximizing a Function Using the Graph of Its Derivative
      4. 4
        Minimizing a Function Using the Graph of its Derivative
      5. 5
        Further Optimizing Functions Using Graphs of Derivatives
      6. 6
        Integrating Rates of Change
      7. 7
        Integrating Density Functions
  8. 8.Techniques of Integration
    1. 1.8.1. Integration Using Substitution
      1. 1
        Integrating Algebraic Functions Using Substitution
      2. 2
        Integrating Linear Rational Functions Using Substitution
      3. 3
        Integration Using Substitution
      4. 4
        Calculating Definite Integrals Using Substitution
      5. 5
        Further Integration of Algebraic Functions Using Substitution
      6. 6
        Integrating Exponential Functions Using Linear Substitution
      7. 7
        Integrating Exponential Functions Using Substitution
      8. 8
        Integrating Trigonometric Functions Using Substitution
      9. 9
        Integrating Logarithmic Functions Using Substitution
      10. 10
        Integration by Substitution With Inverse Trigonometric Functions
    2. 2.8.2. Integration Using Trigonometric Identities
      1. 1
        Integration Using Basic Trigonometric Identities
      2. 2
        Integration Using the Pythagorean Identities
      3. 3
        Integration Using the Double-Angle Formulas
    3. 3.8.3. Special Techniques for Integration
      1. 1
        Integrating Functions Using Polynomial Division
      2. 2
        Integrating Functions by Completing the Square
    4. 4.8.4. Integration by Parts
      1. 1
        Introduction to Integration by Parts
      2. 2
        Using Integration by Parts to Calculate Integrals With Logarithms
      3. 3
        Applying the Integration By Parts Twice
      4. 4
        The Tabular Method of Integration by Parts
      5. 5
        Integration by Parts in Cyclic Cases
    5. 5.8.5. Integration Using Partial Fractions
      1. 1
        Expressing Rational Functions as Sums of Partial Fractions
      2. 2
        Expressing Rational Functions with Repeated Factors as Sums of Partial Fractions
      3. 3
        Expressing Rational Functions with Irreducible Quadratic Factors as Sums of Partial Fractions
      4. 4
        Integrating Rational Functions Using Partial Fractions
      5. 5
        Integrating Rational Functions with Repeated Factors
      6. 6
        Integrating Rational Functions with Irreducible Quadratic Factors
    6. 6.8.6. Improper Integrals
      1. 1
        Improper Integrals
      2. 2
        Improper Integrals Involving Exponential Functions
      3. 3
        Improper Integrals Involving Arctangent
      4. 4
        Improper Integrals Over the Real Line
      5. 5
        Improper Integrals of the Second Kind
      6. 6
        Improper Integrals of the Second Kind: Discontinuities at Interior Points
  9. 9.Differential Equations
    1. 1.9.1. Introduction to Differential Equations
      1. 1
        Introduction to Differential Equations
      2. 2
        Verifying Solutions of Differential Equations
      3. 3
        Solving Differential Equations Using Direct Integration
      4. 4
        Solving First-Order ODEs Using Separation of Variables
      5. 5
        Solving Initial Value Problems Using Separation of Variables
      6. 6
        Modeling With Differential Equations
      7. 7
        Further Modeling With Differential Equations
    2. 2.9.2. Qualitative Techniques for Differential Equations
      1. 1
        Qualitative Analysis of Differential Equations
      2. 2
        Equilibrium Solutions of Differential Equations
    3. 3.9.3. Modeling Exponential Growth and Decay With Differential Equations
      1. 1
        Exponential Growth and Decay Models With Differential Equations
      2. 2
        Exponential Growth and Decay Models With Differential Equations: Calculating Unknown Times and Initial Values
      3. 3
        Exponential Growth and Decay Models With Differential Equations: Half-Life Problems
    4. 4.9.4. Modeling Logistic Growth With Differential Equations
      1. 1
        Logistic Growth Models With Differential Equations
      2. 2
        Qualitative Analysis of the Logistic Growth Equation
      3. 3
        Solving the Logistic Growth Equation
    5. 5.9.5. Slope Fields
      1. 1
        Slope Fields for Directly Integrable Differential Equations
      2. 2
        Slope Fields for Autonomous Differential Equations
      3. 3
        Slope Fields for Nonautonomous Differential Equations
      4. 4
        Analyzing Slope Fields for Directly Integrable Differential Equations
      5. 5
        Analyzing Slope Fields for Autonomous Differential Equations
      6. 6
        Analyzing Slope Fields for Nonautonomous Differential Equations
    6. 6.9.6. Numerical Solutions of Differential Equations
      1. 1
        Euler's Method: Calculating One Step
      2. 2
        Euler's Method: Calculating Multiple Steps
  10. 10.Applications of Integration
    1. 1.10.1. Applications of Integration
      1. 1
        The Average Value of a Function
      2. 2
        The Area Between Curves Expressed as Functions of X
      3. 3
        The Area Between Curves Expressed as Functions of Y
      4. 4
        Finding Areas Between Curves that Intersect at More Than Two Points
      5. 5
        The Arc Length of a Planar Curve
    2. 2.10.2. Volumes of Solids With Known Cross Sections
      1. 1
        Volumes of Solids with Square Cross Sections
      2. 2
        Volumes of Solids with Rectangular Cross Sections
      3. 3
        Volumes of Solids with Triangular Cross Sections
      4. 4
        Volumes of Solids with Circular Cross Sections
    3. 3.10.3. Volumes of Revolution
      1. 1
        Volumes of Revolution Using the Disc Method: Rotation About the Coordinate Axes
      2. 2
        Volumes of Revolution Using the Disc Method: Rotation About Other Axes
      3. 3
        Volumes of Revolution Using the Washer Method: Rotation About the Coordinate Axes
      4. 4
        Volumes of Revolution Using the Washer Method: Rotation About Other Axes
  11. 11.Parametric Equations
    1. 1.11.1. Parametric Equations
      1. 1
        Differentiating Parametric Curves
      2. 2
        Calculating Tangent and Normal Lines with Parametric Equations
      3. 3
        Second Derivatives of Parametric Equations
      4. 4
        The Arc Length of a Parametric Curve
    2. 2.11.2. Vector-Valued Functions
      1. 1
        Defining Vector-Valued Functions
      2. 2
        Differentiating Vector-Valued Functions
      3. 3
        Integrating Vector-Valued Functions
  12. 12.Polar Equations
    1. 1.12.1. Polar Coordinates
      1. 1
        Differentiating Curves Given in Polar Form
      2. 2
        Further Differentiation of Curves Given in Polar Form
      3. 3
        Horizontal and Vertical Tangents to Polar Curves
      4. 4
        Horizontal and Vertical Tangents to Polar Curves in Non-Differentiable Cases
      5. 5
        Tangent and Normal Lines to Polar Curves
      6. 6
        Finding the Area of a Polar Region
      7. 7
        Finding the Limits of Integration For a Given Polar Region
      8. 8
        The Total Area Bounded by a Single Polar Curve
      9. 9
        The Area Bounded by Two Polar Curves
      10. 10
        The Arc Length of a Polar Curve
  13. 13.Particle Dynamics
    1. 1.13.1. Displacement, Velocity, and Acceleration
      1. 1
        Calculating Velocity for Straight-Line Motion Using Differentiation
      2. 2
        Calculating Acceleration for Straight-Line Motion Using Differentiation
      3. 3
        Determining Characteristics of Moving Objects Using Differentiation
    2. 2.13.2. Connecting Position, Velocity and Acceleration Using Integrals
      1. 1
        Calculating Velocity Using Integration
      2. 2
        Determining Characteristics of Moving Objects Using Integration
      3. 3
        Calculating the Position Function of a Particle Using Integration
      4. 4
        Calculating the Displacement of a Particle Using Integration
      5. 5
        Calculating the Total Distance Traveled by a Particle
      6. 6
        Average Position, Velocity, and Acceleration
    3. 3.13.3. The Planar Motion of a Particle
      1. 1
        Calculating Velocity for Plane Motion Using Differentiation
      2. 2
        Calculating Acceleration for Plane Motion Using Differentiation
      3. 3
        Finding Velocity Vectors in Two Dimensions Using Integration
      4. 4
        Finding Displacement Vectors in Two Dimensions Using Integration
  14. 14.Sequences & Series
    1. 1.14.1. Sequences
      1. 1
        Limits of Sequences
      2. 2
        Convergence of Geometric Sequences
      3. 3
        Further Convergence of Geometric Sequences
      4. 4
        Limits of Sequences With Factorials
      5. 5
        Determining Limits of Sequences Using Relative Magnitudes
      6. 6
        Further Determining Limits of Sequences Using Relative Magnitudes
    2. 2.14.2. Monotonic Sequences
      1. 1
        Monotonic Sequences
      2. 2
        Identifying Monotonic Sequences Using Differentiation
      3. 3
        Identifying Monotonic Sequences Using Ratios
    3. 3.14.3. Infinite Series
      1. 1
        Infinite Series and Partial Sums
      2. 2
        Convergent and Divergent Infinite Series
      3. 3
        Properties of Infinite Series
      4. 4
        Further Properties of Infinite Series
      5. 5
        Telescoping Series
    4. 4.14.4. Geometric Series
      1. 1
        Finding the Sum of an Infinite Geometric Series
      2. 2
        Writing an Infinite Geometric Series in Sigma Notation
      3. 3
        Sums of Infinite Geometric Series Given in Sigma Notation
      4. 4
        Convergence of Geometric Series
      5. 5
        Repeating Decimals as Infinite Geometric Series
    5. 5.14.5. Infinite Series Convergence Tests
      1. 1
        The Nth Term Test for Divergence
      2. 2
        The Integral Test
      3. 3
        Harmonic Series and p-Series
      4. 4
        The Comparison Test
      5. 5
        The Limit Comparison Test
      6. 6
        The Alternating Series Test
      7. 7
        The Ratio Test
      8. 8
        Absolute and Conditional Convergence
      9. 9
        The Alternating Series Error Bound
      10. 10
        Determining Convergence Parameters for Infinite Series
      11. 11
        Selecting Procedures for Analyzing Infinite Series
  15. 15.Power Series
    1. 1.15.1. Taylor Polynomials
      1. 1
        Second-Degree Taylor Polynomials
      2. 2
        Analyzing Second-Degree Taylor Polynomials
      3. 3
        Third-Degree Taylor Polynomials
      4. 4
        Higher-Degree Taylor Polynomials
      5. 5
        The Lagrange Error Bound
    2. 2.15.2. Taylor Series
      1. 1
        Radius of Convergence of Power Series Centered at the Origin
      2. 2
        Radius of Convergence of Power Series
      3. 3
        Maclaurin Series
      4. 4
        Taylor Series
      5. 5
        Representing Functions as Power Series
      6. 6
        Recognizing Standard Maclaurin Series
      7. 7
        Recognizing Standard Maclaurin Series for Trigonometric Functions
      8. 8
        Differentiating Taylor Series
      9. 9
        Approximating Integrals Using Taylor Series
  16. 16.Applications of Technology
    1. 1.16.1. Using Graphing Calculators
      1. 1
        Evaluating Expressions Using a Graphing Calculator
      2. 2
        Finding Roots of Functions Using a Graphing Calculator
      3. 3
        Finding Intersections of Functions Using a Graphing Calculator
      4. 4
        Finding Extrema of Functions Using a Graphing Calculator
      5. 5
        Finding Derivatives Using a Graphing Calculator
      6. 6
        Finding Definite Integrals Using a Graphing Calculator
      7. 7
        Finding Improper Integrals Using a Graphing Calculator
      8. 8
        Exploring Functions Using Technology
      9. 9
        Plotting Parametric and Polar Curves Using Technology