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90 Terms
1
Absolute minimum
The lowest point of a function over a given interval.
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Acceleration
The rate of change of velocity of an object with respect to time.
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Accumulation
The process of summing up values over intervals.
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Antiderivative
A function whose derivative gives the original function.
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5
Area between two curves
The integral of the upper curve minus the integral of the lower curve over a specified interval.
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6
Area under a curve
The definite integral of a function between two points.
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Average rate of change
The change in function values divided by the change in input values over an interval.
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Average value of a function
The sum of function values over an interval divided by the interval's length.
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9
Average velocity
Total displacement divided by total time.
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10
Axis of revolution
The line about which a two-dimensional area is rotated to create a three-dimensional solid.
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11
Chain Rule
A formula for computing the derivative of the composition of two or more functions.
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12
Circumscribed Approximation
An approximation of area or volume by using outer boundaries.
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13
Concavity
The direction of the curve; if it's concave up, it opens upwards; concave down opens downwards.
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Continuous
A property of a function if it is unbroken and has no gaps.
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Critical point
A point on a function where the derivative is zero or undefined.
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16
Cubic Function
A function of the form f(x) = ax^3 + bx^2 + cx + d.
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17
Definite integral
Represents the signed area under a curve on a closed interval.
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18
Derivative
The instantaneous rate of change of a function.
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19
Differentiability
A property of a function that allows it to have a derivative at that point.
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20
Differential equation
An equation that relates a function to its derivatives.
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Domain
The set of all possible input values (x-values) for a function.
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22
Even Functions
Functions that satisfy f(-x) = f(x) for all x.
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23
Explicit equation
An equation where the dependent variable is expressed explicitly in terms of the independent variable.
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Exponential function
A function of the form f(x) = a * b^x, where b is a positive constant.
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25
Extrema
The maximum and minimum values of a function.
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Extreme value theorem
A theorem stating that a continuous function on a closed interval has at least one maximum and one minimum.
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27
First derivative test
A method to determine the local maxima and minima of a function.
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28
Fundamental theorem of Calculus
Connects differentiation and integration; states that integration can be reversed by differentiation.
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29
Greatest Integer Function
A function that maps a real number to the largest previous integer.
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30
Horizontal asymptote
A horizontal line that the graph of a function approaches as x tends to infinity.
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31
Identity Function
A function that returns the same value that was used as the input.
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Implicit differentiation
A method of differentiating equations that define y implicitly as a function of x.
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Implicit equation
An equation in which the dependent variable is not isolated.
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Inflection Point
A point where the function changes concavity.
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Inscribed approximation
An approximation of area or volume using the inner boundaries.
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Instantaneous rate of change
The derivative at a specific point.
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37
Integration
The process of finding the integral of a function.
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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.
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39
Inverse Trigonometric Function
Functions that provide the angle corresponding to a given trigonometric value.
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40
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.
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Limit
The value that a function approaches as the input approaches some value.
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Local Maximum
A point where the function value is higher than all nearby values.
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Local Minimum
A point where the function value is lower than all nearby values.
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Logarithmic Function
A function of the form f(x) = log_b(x), the inverse of the exponential function.
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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).
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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.
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Natural Logarithm
Logarithm to the base e, where e is an irrational constant.
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Normal line
A line that is perpendicular to the tangent line of a curve at a given point.
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Odd function
Functions that satisfy f(-x) = -f(x) for all x.
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Optimization
The process of finding the maximum or minimum values of a function.
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51
Oscillation
The repeated variation of a function above and below a central value.
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52
Piece-wise Function
A function defined by different expressions for different parts of its domain.
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53
Product Rule
A formula used to find the derivative of the product of two functions.
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54
Quadratic Function
A function of the form f(x) = ax^2 + bx + c.
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Quotient Rule
A formula used to find the derivative of a quotient of two functions.
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Range
The set of possible output values (y-values) for a function.
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Reciprocal/Rational Function
A function of the form f(x) = 1/x or a ratio of polynomials.
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Related Rates
A method for finding a rate at which one quantity changes in relation to another quantity.
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Riemann sum
A method for approximating the integral of a function using sums of areas of rectangles.
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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.
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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.
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Second derivative Test
A method for determining the concavity at a critical point to classify it as a maximum or minimum.
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Separable differential equation
A differential equation in which variables can be separated on opposite sides of the equation.
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Slope
The ratio of the change in y to the change in x between two points.
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Slope field
A graphical representation of the slopes of a differential equation at various points.
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Solid of revolution
A three-dimensional object created by rotating a two-dimensional area around an axis.
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Speed
The magnitude of velocity; how fast an object is moving.
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Symmetry
A property that describes balance; a function is symmetric if it is unchanged when transformed.
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Tangent line
A straight line that touches a curve at a given point without crossing it.
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70
Total distance traveled
The overall distance an object covers in its motion, regardless of direction.
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Trapezoidal rule – Riemann sum
An approximation method that uses trapezoids to estimate the area under a curve.
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Trigonometric Function
Functions related to the angles of triangles, commonly sine, cosine, and tangent.
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Velocity
The speed of an object in a given direction.
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Vertical asymptote
A vertical line that a graph approaches but never touches or crosses.
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Limit at Infinity
The value that a function approaches as the input grows indefinitely large.
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L'Hôpital's Rule
A method for finding limits of indeterminate forms by differentiating the numerator and denominator.
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Taylor Series
A representation of a function as an infinite sum of terms calculated from the values of its derivatives.
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Series Convergence
A condition under which the sum of an infinite series approaches a specific number.
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Power Series
A series of the form Σa_n(x-c)^n used to express functions as infinite sums.
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Parametric Equations
Equations that express a set of related quantities as explicit functions of an independent parameter.
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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.
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Unit Circle
A circle of radius one centered at the origin in the Cartesian coordinate system.
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Trigonometric Identities
Equations that involve trigonometric functions and are true for all values of the variables.
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84
Continuous Probability Distribution
A probability distribution with a continuous cumulative distribution function.
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85
Expected Value
The mean of all possible values of a random variable, weighted by their probabilities.
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Variance
A measure of how much values in a dataset differ from the mean value.
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Standard Deviation
The square root of the variance, indicating the dispersion of a set of values.
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Binomial Distribution
A discrete probability distribution of the number of successes in a fixed number of trials.
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Normal Distribution
A continuous probability distribution that is symmetrical around the mean.
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90
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.
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