math

continuity at a point

A function f(x) is continuous at a point where x = c if. exists. f(c) exists (That is, c is in the domain of f.)

average value of a function

Average Value of a Function. The average height of the graph of a function. For y = f(x) over the domain [a, b], the formula for average value is given below.

average velocity

Average Velocity, General. The average speed of an object is defined as the distance traveled divided by the time elapsed. Velocity is a vector quantity, and average velocity can be defined as the displacement divided by the time.

total distance traveled

Distance is a scalar quantity that refers to "how much ground an object has covered" during its motion. Displacement is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position.

exponential growth and decay model

Exponential word problems almost always work off the growth / decay formula, A = Pert, where "A" is the ending amount of whatever you're dealing with (money, bacteria growing in a petri dish, radioactive decay of an element highlighting your X-ray), "P" is the beginning amount of that same "whatever", "r" is the growth

first derivative test

If the derivative changes from positive (increasing function) to negative (decreasing function), the function has a local (relative) maximum at the critical point. ... When this technique is used to determine local maximum or minimum function values, it is called the First Derivative Test for Local Extrema.

trig rules

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.

local linear (tangent line) approximation

Linear approximation is the process of finding the equation of a line that is the closest estimate of a function for a given value of x. Linear approximation is also known as tangent line approximation, and it is used to simplify the formulas associated with trigonometric functions, especially in optics.

position function

Motion, Position, Velocity, And Acceleration. ... In one dimension, position is given as a function of x with respect to time, x(t). An object's change in position with respect to time is known as its displacement. The velocity of an object is found by taking the derivative of the position function:

acceleration function

Motion, Position, Velocity, And Acceleration. Position, velocity, and acceleration all describe the motion of an object; all three are vector quantities. In one dimension, position is given as a function of x with respect to time, x(t). An object's change in position with respect to time is known as its displacement.

quotient rule

Similar to product rule, the quotient rule is a way of differentiating the quotient, or division of functions. The quotient rule is defined as the quantity of the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all over the denominator squared

exponential and log rules

Since taking a logarithm is the opposite of exponentiation (more precisely, the logarithmic function log b ⁡ x is the inverse function of the exponential function ), we can derive the basic rules for logarithms from the basic rules for exponents.

exponential and logarithmic rules

Since taking a logarithm is the opposite of exponentiation (more precisely, the logarithmic function log b ⁡ x is the inverse function of the exponential function ), we can derive the basic rules for logarithms from the basic rules for exponents.

l'hospital's rule

So, L'Hospital's Rule tells us that if we have an indeterminate form 0/0 or all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. Before proceeding with examples let me address the spelling of "L'Hospital". The more modern spelling is "L'Hôpital".

mean value theorem (MVT)

The Mean Value Theorem is one of the most important theoretical tools in Calculus. It states that if f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) (that is a < c < b) such that.

net change theorem

The Net Change Theorem. In other words, the net change in a function is the (definite) integral of its derivative. ... The net change in velocity (final velocity minus initial velocity) is the integral of acceleration.

second derivative test

The Second Derivative Test relates the concepts of critical points, extreme values, and concavity to give a very useful tool for determining whether a critical point on the graph of a function is a relative minimum or maximum.

chain rule

The chain rule is a rule for differentiating compositions of functions. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Most problems are average. A few are somewhat challenging. The chain rule states formally that

test for increasing/decreasing

The derivative of a function may be used to determine whether the function is increasing or decreasing on any intervals in its domain. ... If f′(x) > 0, then f is increasing on the interval, and if f′(x) < 0, then f is decreasing on the interval.

derivative of a function

The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x.

disk method

The disk method, also known as the method of disks or rings, is a way to calculate the volume of a solid of revolution by taking the sum of cross-sectional areas of infinitesimal thickness of the solid.

washer method

The disk method, also known as the method of disks or rings, is a way to calculate the volume of a solid of revolution by taking the sum of cross-sectional areas of infinitesimal thickness of the solid.

trapezoidal rule

The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result: It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value.

fundamental theorem of calculus - part 1

The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration.

removable vs. non removable discontinuity

The graph of a removable discontinuity leaves you feeling empty, whereas a graph of a nonremovable discontinuity leaves you feeling jumpy. If a term doesn't cancel, the discontinuity at this x value corresponding to this term for which the denominator is zero is nonremovable, and the graph has a vertical asymptote.

test for concavity

The important result that relates the concavity of the graph of a function to its derivatives is the following one: Concavity Theorem: If the function f is twice differentiable at x=c, then the graph of f is concave upward at (c f(c)) if f (c) 0 and concave downward if f (c) 0.

product rule

The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and is given by. . Remember the rule in the following way. Each time, differentiate a different function in the product and add the two terms together

instantaneous rate of change

The rate of change at a particular moment.

velocity function

The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of its speed and direction of motion (e.g.60 km/h to the north).

fundamental theorem of calculus - part 2

This right over here is the second fundamental theorem of calculus. It tells us that if f is continuous on the interval, that this is going to be equal to the antiderivative, or an antiderivative, of f. ... And this is the second part of the fundamental theorem of calculus, or the second fundamental theorem of calculus.

finding extrema on a closed interval

Verify that the function is continuous on the interval [a,b]. Find all critical points of f(x) that are in the interval [a,b]. This makes sense if you think about it. ... Evaluate the function at the critical points found in step 1 and the end points. Identify the absolute extrema.

volume using a known cross selection

Volumes of Solids with Known Cross Sections. ... Because the cross sections are semicircles perpendicular to the x‐axis, the area of each cross section should be expressed as a function of x.

eulers method

We'll use Euler's Method to approximate solutions to a couple of first order differential equations. The differential equations that we'll be using are linear first order differential equations that can be easily solved for an exact solution. ... Knowing the accuracy of any approximation method is a good thing.

riemann sum - right, left, midpoint

a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations.

differentiability

a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

IVT

if a continuous function takes on two values y1 and y2 at points a and b, it also takes on every value between y1 and y2 at some point between a and b.

normal line

is defined as the line that is perpendicular to the tangent line at the point of tangency

average rate of change

the definition is the change in y over the change of x. ... When you calculate the average rate of change of a function, you are finding the slope of the secant line between the two points.

extreme value theorem (EVT)

the extreme value theorem states that if a real-valued function f is continuous on the closed interval [a,b], then f must attain a maximum and a minimum, each at least once.

displacement

the moving of something from its place or position.

speed

the rate at which someone or something is able to move or operate.

tangent line

the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

inverse trig function rules

they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.