AP Calculus BC Ultimate Guide (copy)

Limits

The value a function approaches as the variable within the function gets closer to a specific value.

Algebraic Manipulation

Techniques like factoring the numerator and denominator to remove removable discontinuities in limits.

Squeeze Theorem

States conditions for functions g(x), f(x), and h(x) where g(x) ≤ f(x) ≤ h(x) and their limits.

Continuity

Conditions for a function to be continuous at a point, including the existence of the limit and the function value.

Removing Discontinuities

Process of redefining a function to eliminate a point causing a discontinuity.

Asymptotes

Vertical and horizontal lines that a function approaches but does not cross.

Horizontal Asymptote Rules

Guidelines for determining horizontal asymptotes based on the highest power of x in a rational expression.

Intermediate Value Theorem

Ensures the existence of a number in an interval for a continuous function.

Rates of Change

Methods for finding average and instantaneous rates of change using the difference quotient.

Slopes of Lines & Definition of Derivative

Techniques for finding slopes of lines and defining derivatives for non-linear functions.

Derivative Notation

Notations for first and second derivatives of functions.

Derivative Rules

Rules like Constant, Constant Multiple, Power, Product, and Quotient Rules for finding derivatives efficiently.

Chain Rule

Method for finding the derivative of composite functions.

Implicit Differentiation

Technique for finding derivatives when isolating one variable is not possible.

Inverse Function Differentiation

Formula for finding the derivative of an inverse function at a specific point.

Inverse Trigonometry

Derivatives of trigonometric functions and their inverses.

Interpreting the Derivative

Understanding the derivative as the slope of the tangent line and its applications.

Straight Line Motion

Relating position, velocity, and acceleration in straight line motion scenarios.

Non-Motion Changes

Using derivatives to analyze changes beyond motion, like volume increase in a pool.

Related Rates

Problems where the change of one thing is related to another, requiring differentiation and relation of rates.

Linearization

Using differentials to approximate the value of a function, involving the derivative and small changes.

L’Hospital’s Rule

A method to evaluate indeterminate limits (0/0 or ∞/∞) by taking the derivative of the numerator and denominator.

Mean Value Theorem (MVT)

Links average rate of change and instantaneous rate of change, ensuring a tangent line equals the secant line slope.

Extreme Value Theorem

States a continuous function on a closed interval has both maximum and minimum values.

Intervals of Increase and Decrease

Using the first derivative to identify where a function is increasing or decreasing.

Relative Extrema

Determining relative maxima and minima using the first derivative.

Function Concavity

Using the second derivative to identify if a function is concave up or down.

Integral & Area Under A Curve

The antiderivative showing total change, with the definite integral giving the area under a curve.

Riemann & Trapezoidal Sums

Methods to estimate area under a curve using rectangles or trapezoids in Riemann sums.

Trapezoids

(1/2)(1.5+2)(1) + (1/2)(2+6)(2) + (1/2)(6+11)(2) + (1/2)(11+15)(3)

Left Sum

(1)(1.5) + (2)(2) + (2)(6) + (3)(11)

Right Sum

(1)(2) + (2)(6) + (2)(11) + (3)(15)

Fundamental Theorem of Calculus & Antiderivatives

Rules for finding antiderivatives, opposite of derivatives, using power rule.

+C

Constant of integration added when finding antiderivatives to account for unknown constant.

Power Rule

Derivative rule to multiply down and decrease power, antiderivative is to divide and increase power.

Definite Integral

Integral with upper and lower limits, finding area under a curve.

First Fundamental Theorem of Calculus

Relates definite integrals to antiderivatives, helps find areas under curves.

U-Substitution

Technique for integration, substituting a term and its derivative to simplify the integral.

Slope Fields

Show slopes at points on a graph, related to differential equations modeling change.

Differential Equations

Equations representing change in one variable with respect to another.

Average Value of Functions

Using integrals to find average value of a function over an interval.

Area Between Two Curves

Finding area between two functions by integrating the difference.

Volume by Cross Sectional Area

Using integrals to find volume of 3D shapes from 2D areas.

Parametric Equations

Equations showing relationship between variables and time.

Arc Length of Curves

Distance along a curve, calculated using derivatives and integrals.

Polar Coordinates

Coordinate system using distance from origin and angle from x-axis.

Sequences & Series

Infinite succession of numbers following a pattern, terms added up in series.

Geometric Series

Series with a common ratio between terms, converges or diverges based on ratio.

nth Term Test For Divergence

Test to determine convergence or divergence of a series based on the nth term limit.

Harmonic Series

Series with the pattern (1/n) that diverges despite appearing to converge

p-Series

Series of the form (1/n^p) converges for p>1 and diverges for 0

Comparison Tests for Convergence

If 0 ≤ aₙ ≤ bₙ for all n, and Σbₙ converges, then Σaₙ converges; if Σaₙ diverges, then Σbₙ diverges

Alternating Series Test for Convergence

Series with terms alternating in sign converges if bₙ > 0, bₙ > bₙ₊₁, and bₙ approaches 0 as n approaches infinity

Absolute Convergence Theorem

Series Σaₙ converges absolutely if Σ|aₙ| converges

Alternating Series Error Bound

Error in summing an alternating series with finite terms is less than the absolute value of the next term after the last term summed

Finding Taylor Polynomial Approximations of Functions

Approximating a function using a Taylor series with terms derived from the function's derivatives

Lagrange Error Bound

Error bound of an nth degree Taylor polynomial, approximated by the next nonzero term in a decreasing series

Radius and Interval of Convergence of Power Series

Radius (R) indicates where a power series converges; interval of convergence is the set of all x values within (-R, R)

Representing Functions as Power Series

Using differentiation and integration of power series to find power series of other functions, approximating integrals and solutions.