Tag: AP Subjects

Here are the main topics covered in AP Calculus (which includes both AP Calculus AB and AP Calculus BC):

1. Limits and Continuity

• Understanding limits (numerically, graphically, and analytically)
• Limit properties and the concept of a limit
• One-sided limits and limits at infinity
• Continuity and discontinuity
• Intermediate Value Theorem

2. Derivatives

• Definition of the derivative (as a limit of average rate of change)
• Differentiation rules (power, product, quotient, and chain rules)
• Implicit differentiation
• Higher-order derivatives
• Applications of derivatives (velocity, acceleration, related rates, optimization)
• Derivatives of trigonometric, exponential, and logarithmic functions
• The Mean Value Theorem

3. Applications of Derivatives

• Analyzing functions using the first and second derivatives (increasing/decreasing, concavity)
• Finding local extrema (maxima and minima)
• Curve sketching based on derivative information
• Optimization problems in applied contexts
• Motion problems and rectilinear motion
• L’Hopital’s Rule for indeterminate forms

4. Integrals

• Antiderivatives and indefinite integrals
• Basic integration rules (power rule, substitution, integration by parts)
• Definite integrals and the Fundamental Theorem of Calculus
• Numerical integration (Trapezoidal Rule, Simpson’s Rule)
• Applications of integration (area under a curve, area between curves)
• Volumes of solids of revolution (disk, washer, and shell methods)

5. Applications of Integrals

• Finding displacement and total distance traveled from velocity
• Solving accumulation problems using integrals
• Work done by a force (as an integral)
• Average value of a function
• Differential equations and slope fields

6. Differential Equations (AP Calculus BC only)

• Solving separable differential equations
• Growth and decay models (exponential growth/decay)
• Slope fields and solutions to differential equations
• Logistic growth models

7. Parametric Equations, Polar Coordinates, and Vector Functions (AP Calculus BC only)

• Parametric equations and their derivatives
• Motion in parametric form (velocity, speed, and acceleration)
• Polar coordinates and graphs of polar functions
• Area and arc length in polar coordinates
• Vector-valued functions and their derivatives

8. Series (AP Calculus BC only)

• Sequences and series: Convergence and divergence
• Geometric series and harmonic series
• Power series and interval of convergence
• Taylor and Maclaurin series
• Approximation of functions using Taylor polynomials

9. Convergence Tests for Series (AP Calculus BC only)

• n-th Term Test for Divergence
• Integral Test
• Comparison Test and Limit Comparison Test
• Ratio and Root Tests
• Alternating Series Test and Absolute Convergence

These topics form the basis of both AP Calculus AB and AP Calculus BC, providing students with the foundational concepts and applications of differential and integral calculus. AP Calculus BC covers all AB topics plus additional advanced topics like sequences, series, and parametric equations.