absolute minimum
See Also: global minimum
acceleration
The instantaneous rate of change of velocity with respect to time. The first derivative of the velocity function with respect to time. The second derivative of the distance function with respect to time.
algebraic function
A function that can be expressed as a finite number of sums, differences, multiples, quotients, and radicals involving xn.
algebraically
A form of expressing a mathematical relationship with variables and formulas.
asymptote
A straight line whose perpendicular distance from a curve decreases to zero as the distance from the origin increases without limit. average rate of change
average rate of change
f(x) has an average rate of change over an interval [a, b] given by
average velocity
The distance traveled divided by the time of travel.
Chain Rule, The
The derivative of a composition is the derivative of the "outside" times the derivative of what's "inside."If y is a differentiable function of u, and u is a differentiable function of x, then is the composition of two differentiable functions y = g(u) and u = f(x), then h' (x) = g' (f (x)) • f ' (x).
Chain Rule, The
The derivative of a composition is the derivative of the "outside" times the derivative of what's "inside." If y is a differentiable function of u, and u is a differentiable function of x, then is the composition of two differentiable functions y = g(u) and u = f(x), then h' (x) = g' (f (x)) • f '(x).
chord
A line segment between two points on a curve.
concave down
A curve is concave down if it is bowed or cupped in such a way so that it would spill water.
concave up
A curve is concave up if it is bowed or cupped in such a way that it would hold water.
concavity
Which way a curve is bowed or cupped.
continuous
is continuous at a if f(a) is defined, exists, and . Continuous functions have no discontinuities; no jumps, vertical asymptotes, or holes.
critical point
Points where a function f(x) satisfies either f ' (x) = 0 or f ' (x) does not exist.
dependent variable
The variable that represents the output, or result of the function. (usually a "y" value, on the vertical axis of a graph)
derivative
The derivative is the slope of a curve. The derivative of a function y = f(x) at a point (a,f(a)) is the slope of the tangent line to the curve y = f(x) at that point.The derivative of a function y = f (x) at a point (a,f(a)) is given by: The derivative for all values of x" is a function in its own right and is given by:
derivative with respect to x
, or the slope expressed as "rise over run.
derivatives of sums and differences
The derivative of a sum is the sum of the derivatives. Also, the derivative of a difference is the difference of the derivatives.
derived graph
The graph of the derivative of a function, often obtained by graphical differentiation.
difference quotient
A quotient of differences, particularly the slope of a secant line:
differentiable
A function is differentiable at a point if the derivative of the function exists at that point. At that point, the function must be continuous, and it must not have a "corner" ("teetering tangent") or a vertical tangent.
differentiable
A function is differentiable at a point if the derivative of the function exists at that point. At that point, the function must be continuous, and it must not have a "corner" ("teetering tangent") or a vertical tangent.
differentiable function
A function that is differentiable at every point.
differentiable function
A function that is differentiable at every point.
differential, or fractional, notation
means "the derivative of y with respect to x.
differentiating
The process of finding a derivative.
domain
The input of a function; the set of values that a function will act upon. On a graph, the domain values are usually shown on the horizontal axis, which is usually labeled x though it may be labeled with other letters.
dot notation
x with dot on top" means "the derivative of x with respect to t." This notation always means the derivative with respect to time. Dot notation is traditional in physics settings.
even Functions
A function y = f(x) is called an even function if f(- x) = f(x). Even functions are symmetrical about the y-axis.
explicit equations
Equations where y is given in terms of x: y = f(x). (e.g. y = 3x - 8)
exponential function (precalculus definition)
A function of the form where "a" is a positive constant not equal to 1.
extrema
An extremum is either a maximum or a minimum. Extrema is the plural of extremum.
Extreme Value Theorem
If "f " is a function continuous for all points in the closed interval [a, b] then "f " takes on both a maximum value M and a minimum value m over that interval.
first derivative test
If f '(x) changes from negative to positive at a "critical point", then the original function has a local minimum at that point. If f '(x) changes from positive to negative at a "critical point", the original function has a local maximum at that point.
function
A correspondence that relates one set of values (the domain) to a second set (the range) such that there is one and only one value of the range for each value of the domain.
global maximum
The largest value that the function has over its domain. (Also called an "absolute maximum.")
global minimum
The smallest value that the function has over its domain. (Also called an "absolute minimum.")
graphical differentiation
Using the graph of a function to sketch the graph of its derivative.
graphically
A form of expressing a mathematical relationship with a plot line that shows the relationship between the input (domain) of a function and the output (range) of a function.
horizontal asymptote
When the y value approaches a number as the "x" value increases or decreases without bound. An asymptote parallel to the x-axis.
implicit differentiation
Finding the derivative of implicit equations by differentiating each term of the equation with respect to the independent variable.
implicit equations
Equations where y is imbedded in the equation and not explicitly stated.
independent variable
The variable that represents the input of the function; the variable is acted upon by the function. (usually an "x" value, represented on the horizontal axis on a graph.)
infinite discontinuity
f(x) has an infinite discontinuity at "a" if:
inflection point
A point where the concavity changes. Inflection points can only occur where f "(x)= 0 or f "(x) does not exist.
instantaneous rate of change
f(x) has an instantaneous rate of change at "a" given by
instantaneous rate of change
f(x) has an instantaneous rate of change at "a" given by
instantaneous velocity
The limit of the average velocity as time of travel approaches zero.
Intermediate Value Theorem
If f is a function, continuous for all points in the closed interval [a, b] then, for every value C between f(a) and f(b) there is at least one c between a and b for which f(c) = C.
inverse function
The inverse of f is written f - 1. The domain and range of f - 1 are respectively the range and domain of f. f : f - 1(x) = y if and only if f(y) = x.f must be one to one for f - to exist.
irrational function
A function that cannot be written as a quotient of polynomial functions.
jump discontinuity
f(x) has a jump discontinuity at "a" if the one sided limits exist but:
left-hand limit
The number that a function is approaching as x approaches a particular value from the left.
left-hand limit
The number that a function is approaching as x approaches a particular value from the left.
limit
The number that a function is approaching as x approaches a particular value from both the right and the left.
limit
The number that a function is approaching as x approaches a particular value from both the right and the left.
local linearity
The tangent line to a curve lies close to the curve near the point of tangency-so close, in fact, that the curve and the line are almost the same.
local linearity
Very small parts of curves are nearly straight lines; the tangent line to a curve is nearly the same as the curve within a small neighborhood of the point of tangency.
local maximum
A value of a function that is larger than or equal to the function values to its left and right. (Also called a "relative maximum.")
local minimum
A value of a function that is smaller than or equal to the function values to its left and right. (Also called a "relative minimum.")
marginal cost
The derivative of the cost function. The rate of change of the cost with respect to the number of units, or (more simply) the cost of making one more unit.
numerically
A form of expressing a mathematical relationship by presenting raw data (numbers).
odd functions
A function y = f (x) is called an odd function if f (- x) = - f (x). Odd functions are symmetrical about the origin.
one-to-one function
When a function has only one "x" value for each "y" value.
optimization
The process of using the derivative to find local (relative) or global (absolute) maxima or minima on a function, thereby maximizing or minimizing a quantity.
oscillating limits
Oscillating limit: or do not exist because f(x) continues to oscillate between two (unequal) numbers, we say it is an oscillating limit.
parameter
A variable in an equation that is not the domain or the range variable. For instance, in slope-intercept form of the equation for a line (y = mx + b), m and b are the parameters for slope and y-intercept, while x and y are the domain and range variables. In the function y = 2x + 4, 2 and 4 are the values for the slope and y intercept parameters.
piecewise function
A function that has different equations that describe the value of the function over different parts of the domain.
piecewise function
A function that has different equations that describe the value of the function over different parts of the domain.
power functions
A function of the form where "a" is a non-zero constant.
prime notation
f'(x) or f' means "the derivative of f(x).
product rule
(fg)' = f ' g + g' f
properties of limits
NOTE: all of where it says "function" the "with respect to X" is implied except when need of the explicit statement is needed. Also these statements are conditional with both the right hand limit and left hand limit existing.The limit of two functions being added is equal to the limit of the first function plus the limit of the second function,The limit of two functions being multiplied together is the same as the limit of the first function multiplied to the limit of the second function.The limit of two functions where one is divided by the other: is equal to the limit of the first function divided by the limit of the second function, ( only if the limit of the divisor function does not equal zero.)The limit of a function multiplied by a constant K is the same as the limit of the function multiplied by K.And the limit of a function with respect to X is equal to the function of X when the general function is a polynomial function.
range
The output of a function; the set of values that results from a function. On a graph, the range values are usually shown on the vertical axis, which is usually labeled "y," though it may be labeled with other letters.
rates of change
See Also: average rate of change instantaneous rate of change
rationalizing
Multiplying by an appropriate term in order to remove a radical.
rectilinear motion
The motion of an object moving back and forth in a straight line.
related rates
A kind of calculus problem in which you find a desired rate of change from information about other rates of change.
relative growth of functions
Exponential growth functions grow faster than polynomials.Power functions generally grow faster than logarithms. Exponential growth functions will always outgrow polynomials and power functions, while all three of these will generally outgrow logarithms. (The only exceptions here are the constant polynomials, p(x) = constant, which don't outgrow logarithms.) polynomial p(x).
relative maximum
See Also: local maximum
relative minimum
See Also: local minimum
removable discontinuity
f(x) has a removable discontinuity at a if: the left hand limit exists and the right hand limit exists. The function has to not be continous at f(a). This includes the case where f(a) is not defined.
right-hand limit
The number that a function is approaching as x approaches a particular value from the right.
Rolle's Theorem
Suppose f is a function, differentiable for all points in the open interval (a, b) and continuous for all points in the closed interval [a, b]. If f(a) = f(b) = 0, then there is at least one number c between a and b for which f ' (c) = 0.
rules for comparing functions
When using to compare magnitudes of polynomials, consider only the terms with the largest exponent. As x goes to infinity, a polynomial will usually outgrow a logarithm. Exponential functions with bases greater than 1 will always outgrow polynomials and for any positive constants a, b > 1. Exponential functions with bases greater than 1 will outgrow power functions, which will usually outgrow logarithmic functions.
rules for shifting or distorting graphs
First: Shift, Second: Squash or stretch, Third: Flip y + a = f(x) or (y = f(x) - a) is shifted down a units. y = f(x + a) is shifted to the left a units. is times as tall. y = f(ax)+ is times as wide. - y = f(x) (or y = - f (x)) is upside down. y = f(- x) is mirror image, left to right.
secant
A line segment between 2 points on a curve.
second derivative number line test
Concavity changes can occur only at points where either f"(x)= 0 or f"(x) does not exist. If f"(x) changes sign at one of these points, then the original function has an inflection point there.
second derivative test
At a critical point where f' (x) = 0, if f'' (x) < 0, there is a relative max; if f'' (x) > 0, there is a relative min; if f'' (x) = 0, then the test fails. These cartoon faces will help you remember.
slope of a line
The amount of increase or decrease in the y (range) value for each increase or decrease in the x (domain) value;
speed
The absolute value of the velocity.
symmetry
An object has symmetry if it looks exactly the same when seen from two (or more) different vantage points. A function has symmetry if it is identical with its own reflection in an axis or point of symmetry.
tangent line
A line that touches a curve at one point and has the same slope as the curve at that point.
tangent line
A line that touches a curve at one point and has the same slope as the curve at that point.
theorem for limits of rational functions
If "f " is a rational function and "a" is in the domain of "f " then: .
velocity
The rate of change of position.
vertical asymptote
When the y value increases or decreases without bound as the x value approaches a number. An asymptote parallel to the y-axis.
Mean Value Theorem
Suppose f is a function, differentiable for all points in the open interval (a, b), and continuous for all points in the closed interval [a, b]. Then there is at least one point c between a and b for which the derivative of f(c) equals the average rate of change of the function.