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Section 1

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indefinite integral

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Calculus Math

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Mar 1, 2020

Cards (37)

Section 1

(37 cards)

indefinite integral

Front

The indefinite integral of f(x) is another term for the general antiderivative of f(x). The indefinite integral of f (x) is represented in symbols as

Back

Riemann Sum

Front

A Riemann sum is a sum of several terms, each of the form f(xi)Δx, each representing the area below a function f(x) on some interval if f(x) is positive or the negative of that area if f(x) is negative. The definite integral is mathematically defined to be the limit of such a Riemann sum as the number of terms approaches infinity.

Back

washer method

Front

A procedure for finding the volume of a solid of revolution by treating it as a collection of thin slices with cross sections shaped like washers.

Back

point of inflection

Front

A point is called a point of inflection of a function if the function changes from concave upward to concave downward, or vice versa, at that point.

Back

continuous

Front

A function f(x) is continuous at a point x =; c when 1. f(c) exists, 2. the limit as x goes to c exists, 3. and f(c) equals the limit as x goes to . In words, this means the curve could be drawn without lifting the pencil. To say that a function is continuous on some interval means that it is continuous at each point in that interval.

Back

higher order derivatives

Front

The second derivative, third derivative, and so forth for some function.

Back

First Derivative Test for Local Extrema

Front

A method used to determine whether a critical point of a function is a local maximum or local minimum. If a continuous function changes from increasing (first derivative positive) to decreasing (first derivative negative) at a point, then that point is a local maximum. If a function changes from decreasing (first derivative negative) to increasing (first derivative positive) at a point, then that point is a local minimum.

Back

normal line

Front

The normal line to a curve at a point is the line perpendicular to the tangent line at that point.

Back

change of variables

Front

A term sometimes used for the technique of integration by substitution.

Back

Second Derivative Test for Local Extrema

Front

A method used to determine whether a critical point of a function is a local maximum or local minimum. If f'(x) = 0 and the second derivative is positive at this point, then the point is a local minimum. If f'(x) = 0 and the second derivative is negative at this point, then the point is a local maximum.

Back

general antiderivative

Front

If F(x) is an antiderivative of a function f(x), then F(x) + C is called the general antiderivative of f(x).

Back

limit

Front

A function f(x) has the value L for its limit as x approaches c if as the value of x gets closer and closer to c, the value of f(x) gets closer and closer to L.

Back

concave downward

Front

A function is concave downward on an interval if f"(x) is negative for every point on that interval.

Back

implicit differentiation

Front

A procedure for finding the derivative of a function which has not been given explicitly in the form f(x).

Back

definite integral

Front

The definite integral of f(x) between x =; a and x =; b, denoted gives the signed area between f (x) and the x-axis from x =; a to x =; b, with area above the x-axis counting positive and area below the x-axis counting negative.

Back

tangent line

Front

The tangent line to a function is a straight line that just touches the function at a particular point and has the same slope as the function at that point.

Back

derivative

Front

The derivative of a function f (x) is a function that gives the slope of f (x) at each value of x. The derivative is most often denoted f'(x) or d/dx. The mathematical definition of the derivative is or in words the limit of the slopes of the secant lines through the point (x, f(x)) and a second point on the graph of f(x) as that second point approaches the first. The derivative can be interpreted as the slope of a line tangent to the function, the instantaneous velocity of the function, or the instantaneous rate of change of the function.

Back

slope-intercept form

Front

The slope-intercept form for the equation of a line is y =; mx + b, where m stands for the slope of the line and the line has its y-intercept (the place where the line crosses the y-axis) at the point (0,b).

Back

point-slope form

Front

The point-slope form for the equation of a line is y - y1 =; m(x - x1), where m stands for the slope of the line and (x1,y1) is a point on the line.

Back

substitution

Front

Integration by substitution is one of the most common techniques of integration, used to reduce complicated integrals into one of the basic integration forms.

Back

antiderivative

Front

a function F(x) is called an antiderivative of a function f(x) if F'(x)=f(x) for all x in the domain of f. In words, this means that an antiderivative of f is a function which has f for its derivative.

Back

instantaneous rate of change

Front

One way of interpreting the derivative of a function is to understand it as the instantaneous rate of change of that function, the limit of the average rates of change between a fixed point and other points on the curve that get closer and closer to the fixed point.

Back

cylindrical shell method

Front

A procedure for finding the volume of a solid of revolution by treating it as a collection of nested thin rings.

Back

disk method

Front

A procedure for finding the volume of a solid of revolution by treating it as a collection of thin slices with circular cross sections.

Back

standard form

Front

The standard form (sometimes also called general form) for the equation of a line is ax + by =; c, where a and b are not both zero.

Back

chain rule

Front

The chain rule tells how to find the derivative of composite functions.

Back

slope of the tangent line

Front

One way of interpreting the derivative of a function is to understand it as the slope of a line tangent to the function.

Back

Mean Value Theorem

Front

If a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists some c in the interval [a,b] for which

Back

instantaneous velocity

Front

One way of interpreting the derivative of a function s(t) is to understand it as the velocity at a given moment t of an object whose position is given by the function s(t).

Back

intercept form

Front

The intercept form for the equation of a line is x/a + y/b =; 1, where the line has its x-intercept (the place where the line crosses the x-axis) at the point (a,0) and its y-intercept (the place where the line crosses the y-axis) at the point (0,b).

Back

differentiable

Front

A function is said to be differentiable at a point when the function's derivative exists at that point. A function will fail to be differentiable at places where the function is not continuous or where the function has corners.

Back

general form

Front

The general form (sometimes also called standard form) for the equation of a line is ax + by =; c, where a and b are not both zero.

Back

integration by parts

Front

One of the most common techniques of integration, used to reduce complicated integrals into one of the basic integration forms.

Back

trigonometric subsititution

Front

A technique of integration where a substitution involving a trigonometric function is used to integrate a function involving a radical.

Back

concave upward

Front

A function is concave upward on an interval if f"(x) is positive for every point on that interval.

Back

Extreme Value Theorem

Front

A theorem stating that a function which is continuous on a closed interval [a, b] must have a maximum and a minimum value on [a, b].

Back

critical point

Front

A critical point of a function is a point (x, f(x)) with x in the domain of the function and either f'(x) =; 0 or f'(x) undefined. Critical points are among the candidates to be maximum or minimum values of a function.

Back