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AP Calculus BC Study Guide

1. Introduction to Calculus BC

AP Calculus BC is an advanced version of the AP Calculus AB course, covering all AB topics and expanding to more advanced concepts such as parametric, polar, and vector functions, as well as series. This course prepares students for college-level mathematics.

  • Exam Format:
    • Multiple-choice questions: Focus on a range of topics, including calculus concepts, theorems, and applications.
    • Free-response questions: Require detailed solutions to more complex problems, involving multiple calculus concepts.

2. Limits and Continuity

Review of Limits:
Limits describe the behavior of a function as x approaches a certain value.

Continuity:
A function is continuous at x = a if:

  1. f(a) exists.
  2. lim(x→a) f(x) exists.
  3. lim(x→a) f(x) = f(a)

3. Derivatives

Definition of the Derivative:
The derivative measures the rate of change of a function. For a function f(x), the derivative is defined as the limit:

f'(x) = lim(h→0) [f(x+h) - f(x)] / h

Differentiation Rules:

  • Power Rule: The derivative of xⁿ is n x^(n-1).
  • Product Rule: The derivative of the product of two functions is u'(x)v(x) + u(x)v'(x).
  • Quotient Rule: The derivative of u(x)/v(x) is [v(x)u'(x) - u(x)v'(x)] / (v(x))².
  • Chain Rule: The derivative of f(g(x)) is f'(g(x)) * g'(x).
  • Higher Derivatives:
    • The second derivative is f''(x).
    • The third derivative is f'''(x).

Derivatives of Special Functions:

  • The derivative of sin(x) is cos(x).
  • The derivative of cos(x) is -sin(x).
  • The derivative of eˣ is eˣ.
  • The derivative of ln(x) is 1/x.

Applications of Derivatives:

  • Related Rates: These problems involve finding the rate at which one quantity changes in relation to another.
  • Optimization Problems: Use derivatives to find maximum or minimum values of a function.

4. Integrals

Definite and Indefinite Integrals:

  • Indefinite Integral: Represents a family of functions and includes a constant of integration.
  • Definite Integral: Represents the net area under the curve from x = a to x = b.

Fundamental Theorem of Calculus:

  • Part 1: If F(x) is an antiderivative of f(x), then ∫ₐᵇ f(x) dx = F(b) - F(a).
  • Part 2: If a function is continuous, then the derivative of its integral is the original function.

Techniques of Integration:

  • Integration by Substitution: Used when the integrand is a composite function.
  • Integration by Parts: ∫ u dv = uv - ∫ v du.
  • Partial Fractions: Decompose rational functions into simpler fractions for integration.

Applications of Integrals:

  • Area Between Curves: The area between two curves y = f(x) and y = g(x) is given by ∫ₐᵇ [f(x) - g(x)] dx.
  • Volume of Solids of Revolution: Use the disk or shell method to find volumes of solids formed by rotating a curve around an axis.

5. Series and Approximations

Taylor Series:
A Taylor Series approximates a function around a point x = a. The Taylor Series of f(x) at x = a is:

f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + ...

Maclaurin Series:
A special case of the Taylor Series where a = 0. For example, the Maclaurin Series of eˣ is:

eˣ = 1 + x + x²/2! + x³/3! + ...

Convergence Tests:

  • Ratio Test: Used to determine the convergence of series.
  • Root Test: Another method for testing convergence.
  • Alternating Series Test: Used for series with alternating signs.

6. Parametric, Polar, and Vector Functions

Parametric Equations:
Parametric equations describe curves using parameters. For example, the parametric equations of a circle are:

x(t) = cos(t)
y(t) = sin(t)

Calculating Derivatives of Parametric Equations:
If x = f(t) and y = g(t), the derivative of y with respect to x is:

dy/dx = (dy/dt) / (dx/dt)

Polar Coordinates:
Polar coordinates represent points in a plane by their distance from the origin r and angle θ. The equations are:

x = r cos(θ)
y = r sin(θ)

Area in Polar Coordinates:
The area inside a polar curve r(θ) from θ = a to θ = b is:

A = ∫ₐᵇ ½ r²(θ) dθ

Vector Functions:

  • Position Vector: Describes the location of a point in space, r(t) = <x(t), y(t)>.
  • Velocity and Acceleration: The derivative of the position vector gives velocity, and the derivative of velocity gives acceleration.

7. Differential Equations

Separable Differential Equations:
A differential equation is separable if it can be written as:

dy/dx = g(x) h(y)

Solve by separating the variables and integrating both sides.

Slope Fields:
Slope fields visually represent the solutions of a differential equation without finding an explicit solution.