AP Calculus BC
10個單元 · 62個課題
免費的AP Calculus BC學習筆記，專為AP (Advanced Placement)學生準備。下列每個章節都涵蓋一個重點主題，附有例題與練習提示，可在 thinka 應用程式中延伸練習。
Course Content
Unit 1: Limits and Continuity
Limit notation and estimating limits from graphs and tables
Determining limits algebraically and selecting procedures
Squeeze Theorem and multiple representations of limits
Continuity, types of discontinuity and removing discontinuities
Infinite limits, limits at infinity and asymptotes
The Intermediate Value Theorem
Units 2-3: Differentiation - Definitions, Rules and Techniques
Rates of change and the definition of the derivative
Differentiability and continuity; when derivatives fail to exist
Power, constant, sum, difference and constant multiple rules
Derivatives of sin x, cos x, e^x and ln x
Product and quotient rules; derivatives of tan, cot, sec and csc
The chain rule
Implicit differentiation and derivatives of inverse functions
Higher-order derivatives and selecting differentiation procedures
Unit 4: Contextual Applications of Differentiation
Interpreting the meaning of the derivative in context
Straight-line motion: position, velocity and acceleration
Rates of change in applied contexts other than motion
Related rates
Local linearity, linearization and approximation
L'Hospital's Rule and indeterminate forms
Unit 5: Analytical Applications of Differentiation
Mean Value Theorem and Extreme Value Theorem
Increasing and decreasing intervals; the first derivative test
Absolute extrema and the candidates test
Concavity and the second derivative test
Sketching and connecting a function with its first and second derivatives
Optimization problems
Behaviors of implicit relations
Unit 6: Integration and Accumulation of Change
Riemann sums, summation notation and definite integral notation
Fundamental Theorem of Calculus and accumulation functions
Properties of definite integrals and evaluating definite integrals
Antiderivatives and integration by substitution
Integrating using long division and completing the square
Integration by parts (BC only)
Linear partial fractions (BC only)
Improper integrals and selecting antidifferentiation techniques (BC only)
Unit 7: Differential Equations
Modelling with and verifying solutions of differential equations
Slope fields and reasoning from them
Euler's method (BC only)
Separation of variables: general and particular solutions
Exponential models with differential equations
Logistic models with differential equations (BC only)
Unit 8: Applications of Integration
Average value of a function on an interval
Motion and accumulation functions in applied contexts
Area between curves
Volumes with cross sections
Volumes of revolution: disc method
Volumes of revolution: washer method
Arc length of a smooth planar curve and distance travelled (BC only)
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC only)
Defining and differentiating parametric equations
Second derivatives and arc length of parametric curves
Defining, differentiating and integrating vector-valued functions
Motion problems with parametric and vector-valued functions
Polar coordinates and differentiating in polar form
Area of polar regions and regions bounded by two polar curves
Unit 10: Infinite Sequences and Series (BC only)
Convergent and divergent infinite series; geometric series
The nth term test and the integral test
Harmonic series, p-series and comparison tests
Alternating series test and the alternating series error bound
Ratio test; absolute and conditional convergence
Taylor polynomial approximations and the Lagrange error bound
Radius and interval of convergence of power series
Taylor and Maclaurin series; representing functions as power series
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