AP Calculus AB Study Guide Notes

Key Exam Details

  • The AP® Calculus AB exam is a 3-hour and 15-minute test.

  • Structure:

    • 45 multiple-choice questions (50% of the exam)
    • 6 free-response questions (50% of the exam)

  • Content Breakdown:

    • Limits and Continuity: 10–12%
    • Differentiation: Definition and Basic Derivative Rules: 10–12%
    • Differentiation: Composite, Implicit, and Inverse Functions: 9–13%
    • Contextual Applications of Differentiation: 10–15%
    • Applying Derivatives to Analyze Functions: 15–18%
    • Integration and Accumulation of Change: 17–20%
    • Differential Equations: 6–12%
    • Applications of Integration: 10–15%

Limits and Continuity

Limits

  • The limit of a function ( f ) as ( x ) approaches ( c ) is ( L ) if ( f ) can be made arbitrarily close to ( L ) as ( x ) approaches ( c ) (not equal to ( c )). Denoted ( \lim_{x \to c} f(x) = L ).
  • If no such value exists, we say the limit does not exist (DNE).

Techniques to Find Limits

  1. Tables / Graphs: Evaluate values as ( x ) approaches ( c ).
  2. Algebraic Techniques:
    • Factoring
    • Rationalizing radical expressions
    • Properties of Limits:
      • ( \lim{x \to c} (f(x) + g(x)) = \lim{x \to c} f(x) + \lim_{x \to c} g(x) )
  3. Evaluating Common Functions:
    • Polynomial, rational, exponential, logarithmic, and trigonometric functions usually result in evaluating ( f(c) ) directly if defined.

One-Sided Limits

  • Define ( \lim{x \to c^+} ) for limits approaching from the right and ( \lim{x \to c^-} ) for limits approaching from the left.
  • A limit exists only if both one-sided limits agree.

Special Limits

  • ( \lim_{x \to 0} \frac{\sin x}{x} = 1 )
  • ( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} )

Continuity

A function ( f ) is continuous at ( c ) if:

  1. ( \lim_{x \to c} f(x) ) exists.
  2. ( f(c) ) is defined.
  3. ( \lim_{x \to c} f(x) = f(c) ).

Types of Discontinuities:

  • Jump Discontinuity: Limits approaching from each side exist but are different.
  • Removable Discontinuity: Limit exists, but does not equal the value of the function.
  • Infinite Discontinuity: Limit approaches infinity.

Properties of Continuous Functions

  • Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on their domains.
  • Piecewise functions need boundary continuity checks.

Differentiation: Definition and Fundamental Properties

Definition of the Derivative

The average rate of change of a function ( f ) over ([a, b]) is:
[ \frac{f(b) - f(a)}{b - a} ]

Instantaneous Rate of Change

If ( h \to 0 ), the instantaneous rate of change (derivative) at ( a ) is:
[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} ]

Graphical Interpretation: The derivative at a point is the slope of the tangent line to the function at that point.

Derivative Rules

  • Constant Rule: ( \frac{d}{dx} c = 0 )
  • Power Rule: ( \frac{d}{dx} x^n = nx^{n-1} )
  • Sum Rule: ( \frac{d}{dx}(f + g) = f' + g' )
  • Difference Rule: ( \frac{d}{dx}(f - g) = f' - g' )
  • Product Rule: ( (fg)' = f'g + fg' )
  • Quotient Rule: ( \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} )

Common Derivatives

  • ( \frac{d}{dx}(e^x) = e^x )
  • ( \frac{d}{dx}(\ln x) = \frac{1}{x} )
  • ( \frac{d}{dx}( an x) = \sec^2 x )

Integration and Accumulation of Change

Riemann Sums and the Definite Integral

  • The definite integral represents the accumulation of change:
    [ \int_a^b f(x) \,dx ]
  • Approximated using:
    • Left Riemann Sum: ( \sum{i=0}^{n-1} f(xi^*) \Delta x )
    • Right Riemann Sum: ( \sum{i=1}^{n} f(xi^*) \Delta x )
    • Midpoint Riemann Sum: ( \sum{i=0}^{n-1} f(x{i+1/2}) \Delta x )
    • Trapezoidal Rule: Approximates area using trapezoids.

Fundamental Theorem of Calculus

  • If ( F ) is an antiderivative of ( f ), then:
    [ \int_a^b f(x) \,dx = F(b) - F(a) ]
  • Relationship between differentiation and integration.

Antiderivatives and Basic Rules

Common techniques involve recognizing forms that resemble rules of differentiation and reversing the process.

Differential Equations

Introduction

Differential equations relate functions to their derivatives. Solutions may include infinitely many functions, and given initial conditions can specify a particular solution.

Types of Differential Equations

  • Separable Equations: Can be rearranged for integration.

Slope Fields

Graphical representation of solutions to differential equations.

Applications of Integration

Average Value of a Function

If ( f ) is continuous on ([a,b]), the average value is:
[ \text{Average Value} = \frac{1}{b-a} \int_a^b f(x) \,dx ]

Motion and Displacement

  • Displacement: ( s(t) = \int v(t) \,dt )
  • Total distance can involve integrating the absolute value of velocity.

Area Between Curves

If ( f(x) \geq g(x) ) on ([a,b]), area is:
[ A = \int_a^b (f(x) - g(x)) \,dx ]

Volume of Solids

Volume of solids with cross sections:

  • For circular cross sections: ( V = \int A(x) \,dx )
  • For solids of revolution, use disk/washer methods where applicable.