AP Calculus AB Comprehensive Review Flashcards

Comprehensive vocabulary flashcards covering every major topic of AP Calculus AB including limits, differentiation, and integration.

Average rates

A measure of change over a specific time interval, such as average speed.

Instantaneous rates of change

A measure of how fast something is changing at a specific, exact moment.

Limits

A value that a function approaches as $x$ approaches a given point from both sides; this concept allows for the definition of a curve's slope at a point.

Squeeze theorem

A theorem used when a function is bounded between two others; if the bounding functions approach the same limit at a point, the squeezed function must also approach that limit.

Removable discontinuity

A type of discontinuity where there is a hole in the graph.

Jump discontinuity

A type of discontinuity where the function leaps to another value.

Infinite discontinuity

A type of discontinuity that occurs near vertical asmmptotes.

Continuity at a point

A condition met if the function is defined at the point, the limit exists at that point, and the value of the function equals the limit.

Intermediate Value Theorem (IVT)

A theorem stating that if a function is continuous on the closed interval from $a$ to $b$ and a number lies between $f(a)$ and $f(b)$, then there is at least one value $c$ in the open interval $(a, b)$ where $f(c)$ equals that number.

Derivative

The slope of the tangent line at a point on a function, often noted as $f'(x)$, $y'$, or $rac{dy}{dx}$.

Differentiability

A condition implying continuity and smoothness; a derivative does not exist at points with sharp corners, cusps, or discontinuities.

Power rule

A basic differentiation rule that applies to polomials.

Product rule

A differentiation rule applied to the product of non-constant functions.

Quotient rule

A differentiation rule applied to the quotient of non-constant functions.

Trig derivatives

Fundamental rules including: the derivative of $an(x)$ is $ext{secant}^2(x)$, the derivative of $ext{seccant}$ is $ext{seccant} imes an(x)$, and the derivative of $ext{coseant}$ is negative $ext{coseant} imes ext{cotangent}$.

Chain rule

A differentiation rule used when one function is nested inside another to calculate the derivative of complex composite functions.

Implicit differentiation

A method used when $y$ cannot be easily isolated, involving differentiating both sides with respect to $x$ while treating $y$ as a function of $x$ to solve for $rac{dy}{dx}$.

Velocity

The first derivative of position with respect to time.

Acceleration

The derivative of velocity or the second derivative of position.

Related rates problems

Problems that involve finding the rate of change of one quantity in terms of another that is changing over time.

Tangent line approximations

A method for estimating a function's values near a specific point using its derivative and the point slope formula.

Indeterminate form

A result such as $rac{0}{0}$ obtained when taking a limit.

Li Tall's rule

A rule stating that for an indeterminate form, one can differentiate the numerator and denominator separately and then take the limit again.

Mean value theorem

A theorem stating that for a continuous and differentiable function on an interval, there must be a point where the instantaneous rate of change equals the average rate of change.

Extreme value theorem

Guarantees that a continuous function on a closed interval from $a$ to $b$ has at least one minimum and one maximum value.

Critical point

Any point where the first derivative equals zero or is undefined.

Point of inflection

A point where a function changes concavity or the second derivative changes signs.

Second derivative test

A method to classify critical points where if the second derivative is less than zero ($f''(x) < 0$), it is a local max, and if it is greater than zero ($f''(x) > 0$), it is a local min.

Remon sum

An approximation of the area under a function's graph using shapes such as rectangles (left, right, or midpoint) or trapezoids.

Definite integrals

Integrals with specific bounds that equal the area under the function between those bounds.

Fundamental theorems of calculus

Theorems stating that derivatives and integrals are inverses, and that the integral of $f$ from $a$ to $b$ is the anti-derivative evaluated at $b$ minus the anti-derivative evaluated at $a$.

Indefinite integrals

Integrals with no bounds that result in an anti-derivative plus a constant $C$.

U substitution

A technique used to simplify integrals by letting $u$ equal an expression and then substituting variables.

Slope fields

A visualization of the first derivative of a function at different points on the coordinate plane.

Average value

The mean value of a function over an interval calculated as a = rac{1}{b-a} imes ext{the integral from } a ext{ to } b ext{ of } f(x)dx.

Washer method

A method to find the volume of a function revolved around an axis when there is space between the area and the axis, calculated by subtracting the inner radius squared from the outer radius squared.