AP Calculus AB - Ultimate Guide (copy)

Limit

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

Removable Discontinuity

An otherwise continuous curve has a hole in it that can be filled.

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) for values of x near a, and both g and h approach L as x approaches a, then f also approaches L.

Instantaneous Rate of Change

Rate of change at a specific point in time, found using the limit as h approaches 0.

Derivative Notation

The representation of the derivative of a function, typically denoted as f'(x) or dy/dx.

Power Rule

If f(x) = x^n, then f’(x) = nx^(n-1), used for finding derivatives of polynomial functions.

Product Rule

If f(x) = uv, then f’(x) = u(dv/dx) + v*(du/dx), used for finding derivatives of products of functions.

Quotient Rule

If f(x) = u/v, then f’(x) = (v(du/dx) - u(dv/dx))/v^2, used for finding derivatives of quotients of functions.

Mean Value Theorem (MVT)

Guarantees that if a function is continuous and differentiable on an interval, there exists at least one point where the slope of the tangent equals the average slope over that interval.

Indeterminate Form

Occurs in limits when both the numerator and denominator approach 0 or both approach infinity, allowing for further evaluation techniques like L'Hôpital's Rule.

U-Substitution

A method used to evaluate integrals by letting u be a function of x to simplify the integration process.

Average Value of a Function

Calculated by integrating the function over an interval and dividing by the length of that interval.

Critical Points

Points on a graph where the derivative is zero or undefined, potential candidates for local maxima or minima.

Concavity

Describes the direction of the curvature of a graph; if f''(x) > 0, the graph is concave up, and if f''(x) < 0, the graph is concave down.

Intermediate Value Theorem (IVT)

States that if a function is continuous on a closed interval, it takes on every value between f(a) and f(b).

Vertical Asymptote

A vertical line where a function approaches infinity, indicating that the function is undefined at that line.

Horizontal Asymptote

A horizontal line that a function approaches as x approaches infinity or negative infinity, indicating the end behavior of the function.