Glossary
Absolute Extrema
The highest or lowest points of a function over a given interval, found by comparing relative extrema and endpoint values.
Example:
To find the absolute extrema of an accumulation function on a closed interval, one must evaluate the function at its critical points and the interval endpoints.
Accumulation Function (Integrally-defined function)
A function defined by an integral where one of the limits of integration is a variable, representing the accumulated net change from a fixed point to that variable.
Example:
If F(x) = ∫ from 0 to x of sin(t) dt, then F(x) is an accumulation function that gives the net area under sin(t) from 0 to x.
Accumulation of Change / Net Change
The total amount a quantity has changed over an interval, found by calculating the definite integral of its rate of change.
Example:
If a function represents the rate of water flowing into a tank, the accumulation of change over an hour tells you the total volume of water added.
Antiderivative
A function whose derivative is the original function; the reverse operation of differentiation.
Example:
An antiderivative of 2x is x^2 + C, because the derivative of x^2 + C is 2x.
Area (under a curve)
The geometric region between a function's graph and the x-axis over a specified interval, whose signed value represents the definite integral.
Example:
Finding the area under the curve of a marginal cost function from x=0 to x=100 gives the total cost of producing 100 units.
Completing the Square (for integrals)
An algebraic technique used to rewrite quadratic expressions in the denominator of an integrand into a sum or difference of squares, often to facilitate integration using inverse trigonometric formulas.
Example:
To integrate ∫ 1 / (x^2 + 4x + 5) dx, one would use completing the square on the denominator to get 1 / ((x+2)^2 + 1), which is an inverse tangent form.
Concavity (in context of accumulation function)
The direction in which the graph of an accumulation function opens (up or down), determined by the sign of its second derivative (the derivative of the integrand).
Example:
An accumulation function H(x) = ∫ from a to x of g(t) dt has concavity determined by the sign of g'(x).
Constant of Integration (+C)
An arbitrary constant added to the antiderivative of a function in an indefinite integral, accounting for the fact that the derivative of any constant is zero.
Example:
When finding the indefinite integral of x, we write (1/2)x^2 + C because any constant would differentiate to zero.
Converge (Improper Integral) (BC ONLY)
An improper integral is said to converge if the limit used to evaluate it exists and is a finite real number.
Example:
The improper integral ∫ from 1 to ∞ of 1/x^2 dx converges to 1.
Decreasing Function (in context of accumulation function)
An accumulation function is decreasing when its derivative (the integrand) is negative.
Example:
If G(x) = ∫ from a to x of f(t) dt, then G(x) is decreasing when f(x) is negative.
Definite Integral
An integral with specified upper and lower limits of integration, representing the exact net accumulation of a quantity over a given interval.
Example:
The definite integral from 0 to 5 of a velocity function gives the total displacement of an object over those 5 seconds.
Diverge (Improper Integral) (BC ONLY)
An improper integral is said to diverge if the limit used to evaluate it does not exist or is infinite.
Example:
The improper integral ∫ from 1 to ∞ of 1/x dx diverges because its limit is infinite.
Fundamental Theorem of Calculus (FTC)
A foundational theorem linking differentiation and integration, establishing that these two operations are inverses of each other and providing a method to evaluate definite integrals.
Example:
The Fundamental Theorem of Calculus allows us to find the exact area under a curve by simply evaluating its antiderivative at the upper and lower bounds.
Improper Integrals (BC ONLY)
Integrals that have either infinite limits of integration or an integrand that is undefined at some point within the interval of integration, requiring evaluation using limits.
Example:
The integral ∫ from 1 to ∞ of 1/x^2 dx is an improper integral because of its infinite upper limit.
Increasing Function (in context of accumulation function)
An accumulation function is increasing when its derivative (the integrand) is positive.
Example:
If G(x) = ∫ from a to x of f(t) dt, then G(x) is increasing when f(x) is positive.
Indefinite Integral
The general form of the antiderivative of a function, including an arbitrary constant of integration (+C), representing a family of functions.
Example:
∫ cos(x) dx = sin(x) + C is an indefinite integral, representing all functions whose derivative is cos(x).
Inflection Points (in context of accumulation function)
Points on the graph of an accumulation function where its concavity changes, corresponding to where the derivative of the integrand changes sign.
Example:
If F(x) = ∫ from a to x of f(t) dt, then inflection points of F(x) occur where f'(x) changes sign.
Integral
A major concept in calculus representing the accumulation of change of a function, often visualized as the signed area between a graph and the x-axis.
Example:
Calculating the total distance traveled by a car given its velocity function over time involves finding the integral of the velocity.
Integrand
The function being integrated within an integral expression.
Example:
In the integral ∫(x^2 + 3x) dx, the integrand is (x^2 + 3x).
Integration by Parts (BC ONLY)
A technique used to integrate products of functions, derived from the product rule for differentiation, often summarized by the formula ∫ u dv = uv - ∫ v du.
Example:
To integrate ∫ x * e^x dx, one would use Integration by Parts by choosing u=x and dv=e^x dx.
Left Riemann Sum
A Riemann Sum where the height of each rectangle is determined by the function's value at the left endpoint of its subinterval.
Example:
Approximating the area under f(x) = x^2 from 0 to 4 using 4 subintervals, a Left Riemann Sum would use f(0), f(1), f(2), and f(3) for heights.
Midpoint Riemann Sum
A Riemann Sum where the height of each rectangle is determined by the function's value at the midpoint of its subinterval.
Example:
For f(x) = x^2 from 0 to 4 with 4 subintervals, a Midpoint Riemann Sum would use f(0.5), f(1.5), f(2.5), and f(3.5) for heights.
Overestimate (Riemann Sum)
An approximation of an integral that yields a value greater than the true value of the integral, often occurring with increasing functions using a Right Riemann Sum or decreasing functions using a Left Riemann Sum.
Example:
For an increasing function like f(x) = e^x, a Right Riemann Sum will always result in an overestimate of the area.
Partial Fraction Decomposition (BC ONLY)
An algebraic technique used to rewrite a complex rational function as a sum of simpler rational functions, each with a linear denominator, making them easier to integrate.
Example:
To integrate ∫ 1 / (x^2 - 1) dx, one would use Partial Fraction Decomposition to rewrite the integrand as 1/(2(x-1)) - 1/(2(x+1)).
Polynomial Long Division (for integrals)
A technique used to rewrite rational functions (where the numerator's degree is greater than or equal to the denominator's degree) into a form that is easier to integrate.
Example:
To integrate ∫ (x^2 + 1) / (x - 1) dx, one would first perform polynomial long division to rewrite the integrand as (x + 1) + 2/(x - 1).
Relative Extrema (in context of accumulation function)
Local maximum or minimum points of an accumulation function, occurring where its derivative (the integrand) changes sign.
Example:
For an accumulation function A(x) = ∫ from c to x of h(t) dt, a relative extremum occurs where h(x) changes sign.
Relative Maximum
A point where a function changes from increasing to decreasing, corresponding to where its derivative changes from positive to negative.
Example:
If the integrand of an accumulation function changes from positive to negative at x=c, then the accumulation function has a relative maximum at x=c.
Relative Minimum
A point where a function changes from decreasing to increasing, corresponding to where its derivative changes from negative to positive.
Example:
If the integrand of an accumulation function changes from negative to positive at x=c, then the accumulation function has a relative minimum at x=c.
Riemann Sum
A method for approximating the definite integral of a function by dividing the area under its curve into a series of rectangles or trapezoids and summing their areas.
Example:
Estimating the distance traveled by a runner whose speed varies, by summing the areas of rectangles representing speed over small time intervals, is a Riemann Sum.
Right Riemann Sum
A Riemann Sum where the height of each rectangle is determined by the function's value at the right endpoint of its subinterval.
Example:
To approximate the area under f(x) = x^2 from 0 to 4 using 4 subintervals, a Right Riemann Sum would use f(1), f(2), f(3), and f(4) for heights.
Summation Notation (Sigma Notation)
A mathematical notation using the Greek letter sigma (Σ) to represent the sum of a sequence of terms.
Example:
The expression Σ (i^2) from i=1 to 5 uses Summation Notation to represent 1^2 + 2^2 + 3^2 + 4^2 + 5^2.
Trapezoidal Sum
A method for approximating the definite integral by dividing the area under the curve into trapezoids instead of rectangles.
Example:
When estimating the volume of water in a reservoir from depth measurements at irregular intervals, a Trapezoidal Sum can provide a more accurate approximation than rectangles.
U-substitution
A technique for integrating composite functions by replacing a part of the integrand with a new variable 'u' and its differential 'du', simplifying the integral.
Example:
To integrate ∫ 2x * cos(x^2) dx, one can use u-substitution by letting u = x^2, which simplifies the integral significantly.
Underestimate (Riemann Sum)
An approximation of an integral that yields a value less than the true value of the integral, often occurring with increasing functions using a Left Riemann Sum or decreasing functions using a Right Riemann Sum.
Example:
For a decreasing function like f(x) = 1/x, a Right Riemann Sum will always result in an underestimate of the area.