**antiderivative**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.

**chain rule** The chain rule tells how to find the derivative of composite functions. In symbols, the chain rule says

In words, the chain rule says the derivative of a composite function is the derivative of the outside function, done to the inside function, times the derivative of the inside function.

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

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

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

**continuous** A function *f(x*) is continuous at a point *x* =; *c* when *f(c*) exists, [img id:59930] exists, and [img id:59931]. 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.

**critical point** 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.

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

**definite integral** 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.

**derivative** 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 [img id:59928]. 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.

**differentiable** 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.

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

**Extreme Value Theorem** 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*].

**First Derivative Test for Local Extrema** 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.

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

**general form** 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.

**higher order derivatives** The second derivative, third derivative, and so forth for some function.

**implicit differentiation** A procedure for finding the derivative of a function which has not been given explicitly in the form "*f(x*) =;".

**indefinite integral** 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

**instantaneous rate of change** 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.

**instantaneous velocity** 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*).

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

**intercept form** 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*).

**limit** 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*.

**Mean Value Theorem** 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

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

**point of inflection** 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.

**point-slope form** The point-slope form for the equation of a line is *y* – *y*_{1} =; *m(x* – *x*_{1}), where *m* stands for the slope of the line and (*x*_{1},*y*_{1}) is a point on the line.

**Riemann sum** A Riemann sum is a sum of several terms, each of the form *f*(*x _{i}*)Δ

*x*, 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.

**Second Derivative Test for Local Extrema** 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.

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

**slope-intercept form** 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*).

**standard form** 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.

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

**tangent line** 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.

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

**washer method** 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.