Calculus AB Main Points

This set of flashcards covers key concepts, definitions, and theorems from Calculus AB as outlined in the lecture notes.

Two-Sided Limit

A limit where the left-hand and right-hand limits must be equal for the limit to exist.

Limit of a Function

The limit $ext{lim}_{x o a} f(x) = L$ implies that as x approaches a, the value of f(x) approaches L.

Direct Substitution

A method used for finding limits when a function is continuous at the point being approached.

Indeterminate Form

An expression that does not have a clear limit, such as $0/0$ or $\frac{ ext{∞}}{ ext{∞}}$.

Vertical Asymptote

A line $x = a$ where the limit of the function as x approaches a results in either $ext{±∞}$.

Removable Discontinuity

A point on a graph where a hole exists; the function can be made continuous by defining the missing value.

Jump Discontinuity

A type of discontinuity where the left- and right-hand limits exist but are not equal.

Infinite Discontinuity

Occurs when the function approaches infinity as x approaches a point.

Mean Value Theorem

If a function is continuous on a closed interval and differentiable on the open interval, then there's at least one point c at which the derivative equals the average rate of change over that interval.

Chain Rule

A formula for computing the derivative of the composition of two functions.

Fundamental Theorem of Calculus

Links the concept of differentiation with that of integration, stating that if f is continuous on [a, b], then $ext{∫}_{a}^{b} f(x)dx = F(b) - F(a)$, where F is an antiderivative of f.

Average value of a function

For a function f on the interval [a, b], the average value is given by $f_{avg} = \frac{1}{b-a} ext{∫}_{a}^{b} f(x)dx$.

Rate of change

The slope of the secant line, which is the average rate of change, compared to the instantaneous rate of change defined as the slope of the tangent line.

Higher Order Derivatives

The derivatives of a function beyond the first derivative; for example, the second derivative is denoted as f''(x) or $\frac{d^2y}{dx^2}$.

Product Rule

A formula for finding the derivative of a product of two functions: $[f(x)g(x)]' = f'(x)g(x) + f(x)g'(x)$.

Quotient Rule

A formula for finding the derivative of a quotient: $\frac{d}{dx}\bigg[\frac{f(x)}{g(x)}\bigg] = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.

Volume of Solids: Washer Method

Used to find the volume of a solid of revolution, integrating the area of the washer-shaped cross-section.

U-Substitution

A technique used in integration, where a substitution is made to simplify the integral.

Differential Equations

Equations that describe the relationship between functions and their derivatives, often used in modeling rates of change.