Untitled Flashcards Set

Absolute minimum

The lowest point of a function over a given interval.

Acceleration

The rate of change of velocity of an object with respect to time.

Accumulation

The process of summing up values over intervals.

Antiderivative

A function whose derivative gives the original function.

Area between two curves

The integral of the upper curve minus the integral of the lower curve over a specified interval.

Area under a curve

The definite integral of a function between two points.

Average rate of change

The change in function values divided by the change in input values over an interval.

Average value of a function

The sum of function values over an interval divided by the interval's length.

Average velocity

Total displacement divided by total time.

Axis of revolution

The line about which a two-dimensional area is rotated to create a three-dimensional solid.

Chain Rule

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

Circumscribed Approximation

An approximation of area or volume by using outer boundaries.

Concavity

The direction of the curve; if it's concave up, it opens upwards; concave down opens downwards.

Continuous

A property of a function if it is unbroken and has no gaps.

Critical point

A point on a function where the derivative is zero or undefined.

Cubic Function

A function of the form f(x) = ax^3 + bx^2 + cx + d.

Definite integral

Represents the signed area under a curve on a closed interval.

Derivative

The instantaneous rate of change of a function.

Differentiability

A property of a function that allows it to have a derivative at that point.

Differential equation

An equation that relates a function to its derivatives.

Domain

The set of all possible input values (x-values) for a function.

Even Functions

Functions that satisfy f(-x) = f(x) for all x.

Explicit equation

An equation where the dependent variable is expressed explicitly in terms of the independent variable.

Exponential function

A function of the form f(x) = a * b^x, where b is a positive constant.

Extrema

The maximum and minimum values of a function.

Extreme value theorem

A theorem stating that a continuous function on a closed interval has at least one maximum and one minimum.

First derivative test

A method to determine the local maxima and minima of a function.

Fundamental theorem of Calculus

Connects differentiation and integration; states that integration can be reversed by differentiation.

Greatest Integer Function

A function that maps a real number to the largest previous integer.

Horizontal asymptote

A horizontal line that the graph of a function approaches as x tends to infinity.

Identity Function

A function that returns the same value that was used as the input.

Implicit differentiation

A method of differentiating equations that define y implicitly as a function of x.

Implicit equation

An equation in which the dependent variable is not isolated.

Inflection Point

A point where the function changes concavity.

Inscribed approximation

An approximation of area or volume using the inner boundaries.

Instantaneous rate of change

The derivative at a specific point.

Integration

The process of finding the integral of a function.

Intermediate Value Theorem

A theorem stating that if a function is continuous on a closed interval, then it takes every value between the function values at the endpoints.

Inverse Trigonometric Function

Functions that provide the angle corresponding to a given trigonometric value.

Left endpoint approximation – Riemann sum

An approximation of the area under a curve using rectangles whose heights are determined by the left endpoints of sub-intervals.

Limit

The value that a function approaches as the input approaches some value.

Local Maximum

A point where the function value is higher than all nearby values.

Local Minimum

A point where the function value is lower than all nearby values.

Logarithmic Function

A function of the form f(x) = log_b(x), the inverse of the exponential function.

Mean Value Theorem

A theorem that states if a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Midpoint Approximation – Riemann sum

An approximation of the area under a curve using rectangles whose heights are determined by the midpoints of the sub-intervals.

Natural Logarithm

Logarithm to the base e, where e is an irrational constant.

Normal line

A line that is perpendicular to the tangent line of a curve at a given point.

Odd function

Functions that satisfy f(-x) = -f(x) for all x.

Optimization

The process of finding the maximum or minimum values of a function.

Oscillation

The repeated variation of a function above and below a central value.

Piece-wise Function

A function defined by different expressions for different parts of its domain.

Product Rule

A formula used to find the derivative of the product of two functions.

Quadratic Function

A function of the form f(x) = ax^2 + bx + c.

Quotient Rule

A formula used to find the derivative of a quotient of two functions.

Range

The set of possible output values (y-values) for a function.

Reciprocal/Rational Function

A function of the form f(x) = 1/x or a ratio of polynomials.

Related Rates

A method for finding a rate at which one quantity changes in relation to another quantity.

Riemann sum

A method for approximating the integral of a function using sums of areas of rectangles.

Right endpoint approximation – Riemann sum

An approximation of the area under a curve using rectangles whose heights are determined by the right endpoints of sub-intervals.

Rolle’s Theorem

A theorem that states if a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

Second derivative Test

A method for determining the concavity at a critical point to classify it as a maximum or minimum.

Separable differential equation

A differential equation in which variables can be separated on opposite sides of the equation.

Slope

The ratio of the change in y to the change in x between two points.

Slope field

A graphical representation of the slopes of a differential equation at various points.

Solid of revolution

A three-dimensional object created by rotating a two-dimensional area around an axis.

Speed

The magnitude of velocity; how fast an object is moving.

Symmetry

A property that describes balance; a function is symmetric if it is unchanged when transformed.

Tangent line

A straight line that touches a curve at a given point without crossing it.

Total distance traveled

The overall distance an object covers in its motion, regardless of direction.

Trapezoidal rule – Riemann sum

An approximation method that uses trapezoids to estimate the area under a curve.

Trigonometric Function

Functions related to the angles of triangles, commonly sine, cosine, and tangent.

Velocity

The speed of an object in a given direction.

Vertical asymptote

A vertical line that a graph approaches but never touches or crosses.

Limit at Infinity

The value that a function approaches as the input grows indefinitely large.

L'Hôpital's Rule

A method for finding limits of indeterminate forms by differentiating the numerator and denominator.

Taylor Series

A representation of a function as an infinite sum of terms calculated from the values of its derivatives.

Series Convergence

A condition under which the sum of an infinite series approaches a specific number.

Power Series

A series of the form Σa_n(x-c)^n used to express functions as infinite sums.

Parametric Equations

Equations that express a set of related quantities as explicit functions of an independent parameter.

Polar Coordinates

A two-dimensional coordinate system in which each point is determined by a distance from a reference point and an angle from a reference direction.

Unit Circle

A circle of radius one centered at the origin in the Cartesian coordinate system.

Trigonometric Identities

Equations that involve trigonometric functions and are true for all values of the variables.

Continuous Probability Distribution

A probability distribution with a continuous cumulative distribution function.

Expected Value

The mean of all possible values of a random variable, weighted by their probabilities.

Variance

A measure of how much values in a dataset differ from the mean value.

Standard Deviation

The square root of the variance, indicating the dispersion of a set of values.

Binomial Distribution

A discrete probability distribution of the number of successes in a fixed number of trials.

Normal Distribution

A continuous probability distribution that is symmetrical around the mean.

Central Limit Theorem

A theorem that states, given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the original distribution.