AP Calculus BC Ultimate Guide Notes

Unit 1: Limits and Continuity

Limits

• Definition: The value a function approaches as the variable nears a specific value.
• Calculation: For simple polynomials, substitute the approaching value.

Ways to Find Limits

• Graph: Observe the graph's behavior.
• Table Estimation
• Algebraic Properties and Manipulation: Factor and cancel removable discontinuities.
• Squeeze Theorem: Use when bounding functions have limits.

Continuity

• Types of Discontinuity:
  • Jump: Curve breaks and restarts elsewhere.
  • Essential/Infinite: Vertical asymptote present.
  • Removable: Curve has a hole.
• Conditions for Continuity: Function value exists, limit exists, and they match.

Removing Discontinuities

• Redefine function without the problematic point.

Limits and Asymptotes

• Vertical: Line function cannot cross.
• Horizontal: End behavior of a function. Rules depend on the degree of polynomials.

Intermediate Value Theorem

• Guarantees a value exists if a function is continuous over an interval.

Unit 2: Differentiation: Definition and Fundamental Properties

Rates of Change

• Average: Difference quotient over an interval.
• Instantaneous: Difference quotient with a limit as h→0.

Slopes and Derivatives

• Secant Line: Used to approximate slope.
• Tangent Line: More accurate for slopes at a specific point.
• Definition of Derivative: Slope as x becomes infinitesimally small.

Derivative Notation

• Depicts first and second derivatives for functions.

Derivative Rules

• Constant Rule: Derivative of a constant is zero.
• Power Rule: nx^(n-1) for x^n.
• Product Rule: (u)(dv/dx) + (v)(du/dx).
• Quotient Rule: (v)(du/dx) - u(dv/dx) / v^2.

Memory Derivatives

• Know derivatives of sinx, cosx, e^x, and lnx.

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Chain Rule

• Differentiate outer function, leave inner, multiply by derivative of inner.

Implicit Differentiation

• Used when y cannot be isolated in terms of x.

Inverse Function Differentiation

• Find reciprocal of derivative at the corresponding y-value.

Inverse Trigonometry

• Derivatives found using implicit differentiation and trig rules.

Unit 4: Contextual Applications of Differentiation

Interpreting the Derivative

• Shows slope of tangent lines at points.

Straight Line Motion

• Derivatives of position give velocity; second derivative gives acceleration.
• Particles speed up when velocity and acceleration have matching signs.

Non-Motion Changes

• Derivatives can indicate change rates for non-motion situations.
• Related rates problems involve finding the rate of change of one variable in relation to another.

Linearization

• Approximates function values using derivatives.

L'Hôpital's Rule

• Used for indeterminate forms 0/0 or ∞/∞.

Unit 5: Analytical Applications of Differentiation

Mean Value Theorem

• Connects average and instantaneous rates of change.

Extreme Value Theorem

• Continuous functions over an interval have both maximum and minimum values.

Intervals of Increase and Decrease

• Use first derivative to determine where functions are increasing or decreasing.

Relative Extrema

• Determined by changes in the sign of the first derivative.

Candidates Test & Absolute Extrema

• Examine endpoints and critical numbers for absolute extrema.

Function Concavity

• Use second derivative to determine concavity (concave up/down).

Unit 6: Integration and Accumulation of Change

Integral & Area Under a Curve

• Integrals represent accumulated change.
• Definite Integrals: Area under curve.
• Riemann Sums: Estimation using rectangles.

Riemann & Trapezoidal Sums

• Approximating area under curves using rectangles or trapezoids.

Fundamental Theorem of Calculus & Antiderivatives

• Power rule for integration involves dividing and increasing power.

Advanced Integration

• Techniques for complex integrals include U-substitution.

Unit 7: Differential Equations

Introduction & Slope Fields

• Slope fields visualize slopes at various points for differential equations.

Differential Equations

• Solve by integrating both sides.
• SIPPY: Separate, Integrate, Plus C, Plug in initial condition, Y equals.

Unit 8: Applications of Integration

Average Value of Functions

• Integrate function over interval and divide by interval length.

Position, Velocity, and Acceleration

• Integrals can reverse derivatives to find position from velocity or acceleration.

Area Between Two Curves

• Subtract one integral from another to find the area between curves.

Volume by Cross-Sectional Area

• Use integration to find volume, often involves disc method.

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Parametric Equations

• Relations between position and time.

Arc Length of Curves

• Distance along a curve calculated using derivatives.

Vector-Valued Functions

• Describe position, velocity, and acceleration; differentiate/integrate components individually.

Polar Coordinates

• Represent positions using radius and angle; area between curves involves integrating differences squared.

Unit 10: Infinite Sequences and Series

Sequences & Series

• Sequences: Patterns in numbers, convergence defined by limits.
• Infinite Series: Sequence of terms added indefinitely, convergence defined by partial sums.

Geometric Series

• Converges if |r|<1, diverges if |r|>1; use formulas to find sums.

Tests for Convergence

• Comparison Tests: Compare terms to identify convergence/divergence.
• Limit Comparison Test and Alternating Series Test: Determine convergence of series.

Harmonic Series & P-series

• Harmonic Series: Diverges.
• P-series: Converges if p>1.

Representing Functions as Power Series

• Use derivatives and integrals to form power series.

Finding Taylor Polynomial Approximations

• Use Taylor and Maclurin series for approximation.

Radius and Interval of Convergence

• Determine where power series converge.

This guide provides a comprehensive summary of key concepts in AP Calculus BC, focusing on limits, derivatives, integrals, differential equations, and infinite series, among others.