AP Calculus AB - Ultimate Guide (copy)

Limit

The value that a function approaches as the variable within the function gets nearer to a particular value.

Removable Discontinuity

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

Vertical Asymptote

A line that a function cannot cross because the function is undefined there.

Horizontal Asymptote

The end behavior of a function; a horizontal asymptote can be crossed.

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) and lim g(x) = L and lim h(x) = L, then lim f(x) = L.

Intermediate Value Theorem (IVT)

If f(x) is continuous on [a,b] and C is between f(a) and f(b), then there exists at least one number in [a,b] such that f(x) = C.

Average Rate of Change

The change in the value of a function over an interval of time represented by the difference quotient.

Instantaneous Rate of Change

The rate of change at a specific point in time, found by taking the limit of the difference quotient as h approaches 0.

Product Rule

If f(x) = uv, then f’(x) = (u)(dv/dx) + (v)(du/dx).

Quotient Rule

If f(x) = u/v, then f’(x) = (v)(du/dx) - (u)(dv/dx) / v^2.

Chain Rule

To find the derivative of a composite function, take the derivative of the outer function, leaving the inner function alone, and multiply by the derivative of the inner function.

Implicit Differentiation

Taking the derivative of an equation involving both x and y without isolating y.

Reciprocal of Derivative

In inverse function differentiation, the derivative of an inverse function is the reciprocal of the derivative of the original function at that point.

Riemann Sum

A method for estimating the total area under a curve by dividing it into rectangles.

Fundamental Theorem of Calculus

The integral from a to b of f(x) is equal to the antiderivative evaluated at b minus the antiderivative evaluated at a.

Differential Equation

An equation that relates a function with its derivatives, modeling change with respect to another variable.

Average Value of a Function

The average value over an interval can be found using the formula: (1/(b-a)) * ∫f(x)dx from a to b.

Volume by Cross Sectional Area

The volume of a shape can be determined by integrating the area of its cross sections.