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AP Calculus AB Study Guide

1. Introduction to Calculus AB

AP Calculus AB is a foundational college-level mathematics course covering differential and integral calculus concepts. The course focuses on understanding limits, derivatives, integrals, and their applications, along with various methods of solving problems involving these concepts.

  • Exam Format:
    • Multiple-choice questions: Covering a broad range of topics including calculus concepts and applications.
    • Free-response questions: These require detailed solutions and might involve multiple calculus concepts applied to real-world situations.

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)

One-sided Limits:
A one-sided limit is the behavior of a function as x approaches a value from either the left (lim x→a-) or right (lim x→a+).

Limit Laws:

  • Sum Law: lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x)
  • Product Law: lim(x→a) [f(x) * g(x)] = lim(x→a) f(x) * lim(x→a) g(x)
  • Quotient Law: lim(x→a) [f(x)/g(x)] = lim(x→a) f(x) / lim(x→a) g(x) (provided g(x) ≠ 0).

3. Derivatives

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

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).

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 the maximum or minimum values of a function.
  • Mean Value Theorem: If a function is continuous on [a, b] and differentiable on (a, b), then there exists a point c in (a, b) such that f'(c) = [f(b) - f(a)] / (b - a).

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. 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.

6. Applications of Derivatives and Integrals

Motion and Rate Problems:
These problems involve applying derivatives and integrals to real-world situations, such as calculating distance, velocity, and acceleration.

Average Value of a Function:
The average value of a function f(x) on the interval [a, b] is:

1 / (b - a) ∫ₐᵇ f(x) dx

L’Hopital’s Rule:
Used for evaluating limits of indeterminate forms (0/0 or ∞/∞), it states that if lim(x→a) f(x) / g(x) is indeterminate, then:

lim(x→a) f(x) / g(x) = lim(x→a) f'(x) / g'(x), provided the limit exists.