LiveMath

LiveMath allows you to do mathematics interactively over the Internet. However, you first need to install the LiveMath plug-in. It's easy! Just go to www.livemath.com and press buttons, and then enjoy the LiveMath web pages below! Also, check out LiveMath Hints for help.

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LIVEMATH WEB PAGE DESCRIPTIONS

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GRAPHERS

This notebook lets you graph up to six separate functions including two where x is a function of y.

This notebook graphs functions using polar coordinates where r determines the distance a point is from the origin and theta determines the angle made with the positive horizontal axis.

This notebook illustrates what happens with the following polar graph as we let a vary continuously from 1 to 4. Change the function or the range for a, and see what transpires.

This notebook allows you to graph equations where y is an implicit function of x.

This grapher not only graphs parametric equations in two dimensions, it also creates an animation with a moving point that indicates the direction in which the curve is drawn as the parameter changes in value from its initial value to its terminal value.

This web page lets you graph parametric equations in three dimensions. Each coordinate variable is expressed a function of t. Additionally, an animation with a moving point is created that shows the direction in which the curve is traced as the parameter changes in value from its initial value to its terminal value.

This grapher not only graphs parametric equations in three dimensions, but also lets you express those curves as a tube of a specified radius. Furthermore, you can change the radius of a tube or even make it a function of the parameter.

This notebook creates a nice graph of a seashell from parametric equations.

Enter an expression in x and y, or left-click and then right-click one of the examples below and choose "Make Working Stmt."

This notebook graphs surfaces in cylindrical coordinates with r as the dependent variable and theta and z as the two independent variables.

This notebook graphs surfaces in spherical coordinates with rho as the dependent variable and theta and phi as the two independent variables.

Below are two examples of what mathematicians refer to as non-orientable surfaces, the Mobius strip and the Klein bottle. Mathematically, the Mobius strip is a surface with only one side. It is formed by taking a ribbon and giving it a half-twist in 3-dimensional space. As a result, you can't orient it in the sense of assigning a designation such as side A and side B. The Klein bottle, on the other hand, is created (technically) by twisting it through 4-dimensional space, and the result is a bottle whose inside is also the outside. Consequently, it also can't be oriented by standard designations.

This animation is just for my own relaxation. Mathematically, we see the result of gradually transforming the coefficient of y² from 1 to -1 and back again, and we can enjoy the discoveries that we make along the way. On other levels, though, we may experience the rythmic ebb and flow of the pulse of the universe and also find tranquility therein. As I say, it is an effect that I personally find very relaxing.

This notebook illustrates an animation of sine and cosine functions in three dimensions.

This animation in three dimensions illustrates a pulsing action.

ARITHMETIC

Enter a number or expression that's not too big, and you just might get the correct factorization.

PRE-CALCULUS

This notebook calculates the average rate of change for a function as x changes from x₁ to x₂.

Below are a series of graphs and animations that illustrate how to take a given function and produce vertical and horizontal shifts, stretches and compressions, and reflections of its graph.

Change the values for a, b, and c, and explore their effect on the graph. Also, to switch to one of the other functions below, left-click and then right-click on the function and select "Make Working Stmt."

This notebooks creates examples of compositions of functions. Press ?x to enter the outlined x, and below you will find the results of f(g(x)) and g(f(x)).

These animations show what happens when you raise x to either higher and higher even powers or higher and higher odd powers.

It is well known how changing either the value of a or the value of c changes the graph of a quadratic function. However, not everyone is aware that as the value of b changes, the vertex traces out the graph of another parabola. Below you can manually set values for a, b, and c, and then observe the animation that results as b is allowed to vary..

This animation shows what happens to the graph of a cubic function as we let the coefficent of the x term vary. The relative extreme points will always be found along the graph of another cubic polynomial function.

This animation shows what happens to the relative extrema of the graph of a cubic function as we let the coefficent of the x² term vary. The relative extreme points will always be found along the graph of another cubic polynomial function.

This animation shows what happens to the relative extrema of the graph of a cubic function as we let the coefficent of the x² term vary. The inflection point will always be found along the graph of another cubic polynomial function.

This notebook lets you explore what happens to the graph of a fourth degree polynomial function as the coefficients for x, x², and x³ are allowed to vary.

This notebook contains animations pertaining to the cosine function.

This notebook contains animations pertaining to the sine function.

This notebook lets you practice Gaussian elimination and the Gauss-Jordan method by systematically row-reducing a matrix. It makes use of the fact that if you perform the row-operation on an identity matrix "I" and multiply your original matrix "A" on the left by "I," then that row operation will be performed on "A."

This notebook lets you practice the Simplex Method for standard maximum problems by systematically performing row-operations on a matrix. It makes use of the fact that if you perform the row-operation on an identity matrix "I" and multiply your original matrix "A" on the left by "I," then that row operation will be performed on "A."

SINGLE VARIABLE CALCULUS

This notebook lets you make the transition from average rate of change to instantaneous rate of change by calculating the slope of a secant line and then letting x₂ gradually get closer to x₁.

This notebook contains several animations to illustrate the relationship between secant lines and tangent lines at a point on a graph.

This notebook illustrates the relationship between tangent lines and local linearity.

This notebook illustrates the relationship between the slope of the tangent line and where the graph of the function is increasing or decreasing.

The limit of f(x) as x goes to a is equal to L means that we can make the values of f(x) as close as we like to L by making x sufficiently close to, but not equal to, a. Below you can set an interval around a that goes from delta-a to delta+a. If you have entered the correct value for the limit, then the interval around it will tighten as delta becomes smaller.

In the eighteen hundreds, Karl Weierstrass, the father of modern analysis, defined a limit as follows: "The limit of f(x) as x goes to a is equal to L if and only if for every epsilon greater than zero, there exists a delta greater than zero such that if zero is less than the absolute value of x - a and the absolute value of x - a is less than delta, then the absolute value of f(x) is less than epsilon." The horizontal green lines below define an interval of radius epsilon about the value that you have LIMIT set to. You can set epsilon equal to various other values and by experiment find a suitable value for delta.

This notebook illustrates the behavior of a function as x approaches a number a from the right or from the left. To change the function, type ?x to get the outlined x for the varable. Below you will find numerical estimates for the limits as x approaches a from each possible side.

The expression above that we are taking the limit of represents the slope of a secant line that passes through the points (a, f(a)) and (a+h, f(a+h)). When we take the limit of this expession as h goes to zero, we obtain the value for the slope of the tangent line at (a, f(a)), provided this limit exists. This notebook gives numerical estimates for this slope as h approaches zero from both the right and from the left. To change the function, type ?x to get the outlined x for the varable.

This notebook shows the graph of a function in black, its derivative in blue, and the graph of a difference quotient in red. As we let h get closer to zero, the graph of the difference quotient begins to coincide with the derivative.

This notebook illustrates the basic rules for differentiating polynomial, exponential, and logarithmic functions. Below we give a rule and then follow it with several examples.

Below you will find the differentiation formulas for the six basic trigonometric functions, and the graphs of both the original function (black) and the deriative (red). Modify any particular function, and see what happens.

In this notebook the graph of e to the x power is in red, the graph of b to the x is in black, and its derivative is in blue. As the base of the exponential function increases from, we can visually see that the only time the derivative of the exponential function is the function itself is when the base is equal to e.

This notebook allows you to practice constructing tangent lines by entering a function, a point, and an equation for the tangent line.

This notebook illustrates the product rule. Below we give the rule and then follow it with several examples.

This notebook illustrates the quotient rule. Below we give the rule and then follow it with several examples.

This notebook illustrates some of the more basic applications of this chain rule. Additionally, each function below can be modified and the effect of that modification immediately observed.

This notebook illustrates the relationship between the derivative (green curve) and where the graph of the function (black curve) is increasing, decreasing, or has relative extreme values.

This notebook illustrates the relationship between the second derivative (tan curve) and where the graph of the function (black curve) is concave up, concave down, or has inflection.

This notebook allows us to see the relationships between the graph of a function (black curve), the graph of the first derivative (blue curve), and the graph of the second derivative (red curve).

Imagine two variables that are related to each other such as the area of a circle and the radius of a circle. Now imagine that each variable is changing over time such as when you throw a pebble in a pond and the circle wave grows. In this situation the rate at which the area changes over time will be related to the rate at which the radius changes over time. This is all that related rates is about. In the examples below we give variables that are related by a given equation, and then we differentiate each equation with respect to another variable, t. For convenience, you might want to think about t as representing time. The result is an equation that shows how the derivatives are related.

Karl Weierstrass (1815-1897) was really, really smart, and he is known as the "father of modern analysis." He was also very clever at creating functions that are so wiggly that they are continuous everywhere and differentiable nowhere. In the animation below we start with a smooth curve and watch it get more wiggly as new terms are added to the sum that defines it.

This notebook approximates zeros of functions using Newton's Method. Take the value below for the x-intercept, and type it in as the new value for a. When done, click on the graph to update.

Enter a function and values for a, b, and n, and below you will find the results for a variety of different approximations of the definite integral. Enjoy!

This notebook illustrates that the integral of a function from a to c is always equal to the integral from a to b plus the integral from b to c. This happens regardless of which numbers are bigger than the others. Below, a red vertical line is drawn from at x=a, a blue line at x=b, and a green line at x=c.

This notebook graphs a function in blue and its accumulation function in red starting at a specified value for a.

This notebook illustrates what is known as the fundamental theorem of calculus. A function F(x) is defined as the integral of a function f, graphed in black. Next, the difference quotient is formed for F(x) and graphed in red. Then an animation is created showng what happens to the difference quotient as h goes to zero. The end result is that the we see that the graph of F'(x) coincides with the graph of f, thus demonstrating the fundamental theorem that the derivative of F(x) is f(x).

This notebook graphs an accumulation function and creates an animation where the lower limit of integration varies from -3 to 3. Since the accumulation function is always an antiderivative of the integrand, we notice that the graph shifts vertically since all antiderivatives of a function differ only by a contstant.

Enter a function and values for a and b to generate a solid of revolution about the x-axis and to determine its volume.

Enter a function to generate a solid of revolution about the y-axis. Enter, also, some values for a and b, and find the volume of the solid between the surface and the xy-plane from x = a to x = b.

Enter below a function, the endpoints for the interval [a,b], and the number n of subintervals that you want, and this notebook will automatically calculate the length of the curve from (a,f(a)) to (b,f(b)). To enter the outlined x in the function definition, press ?x.

Left-click and then right-click on the functions below to make them working statements, and then start the animation. Can you determine the interval of convergence? Fun for the whole family!

Watch how you can use a sine function to get everything squared away!

Here's a Fourier series example that seems to converge pretty quickly.

This notebook lets you explote three common probability density functions - the normal density function, the uniform density function, and the exponential density function. Integrals are used to compute the probability that x lies within a given interval.

MULTIVARIABLE CALCULUS

This notebook automatically does a variety of standard vector calculations.

This notebook creates a nice animation showing a parametric curve in three dimensions as well as the "frenet frame," the unit tangent, the unit normal, and the binormal vectors.

This notebook allows you to plot points in three dimensions and observe the result.

This grapher shows you a plot of z = f(x,y) as well as the corresponding level curves. Enjoy!

This animation cause the contour lines on the surface to gently float down to the xy-plane revealing the level curves.

Enter the coordinates of a point on the level curve and then use the formula for the gradient of z to graph the gradient vector at that point. The gradient vector should be perpendicular to the level curve. Loads of fun for the whole family!

This notebook goes through a series of calculations to evaluate the directional derivative at a point on a surface and to construct the corresponding tangent line. Below you can change the angle, the surface function, and the parameter k that determines the point on the surface. At the bottom, you will see all the features in a three-dimensional graph.

This notebook allows you to explore intersections of surfaces with planes that are parallel to the xy-plane, yz-plane, and xz-plane, respectively.

As the fixed value for x changes, the resulting parabola moves across the surface of the paraboloid. Watch and learn!

Enter a function, and watch the partial derivatives magically appear before your very eyes!

This notebook allows you to practice constructions in three dimensions. If you understand how to do these constructions, they will deepen your understanding. Begin by graphing a surface in three dimensions. Next, graph a point on that surface, and then construct parametric equations for a curve on the surface that passes through the point. Finally, use the derivative of the parametric equations to help you construct a tangent line to the curve that passes through the point.

This notebook produces the same results as Curve and Tangent Line Construction, but uses partial derivatives to get the slope of the tangent line and also uses a different representation for the parametric equations.

In this notebook enter the equation for a surface, coordinates of a point on that surface, and then use the partial derivatives to help you construct two lines that are tangent to the surface at that point.

Enter a furface and a point on the surface, find the equation for the tangent plane at that point, and construct parametric equations for a line perpendicular to the tangent plane. Great fun for everyone!

Here you can do up to two surface graphs, four parametric curves, and three points. Graph planes, normal lines, or minimum points for maximum mathematical enjoyment!

This notebook provides a good visual aid for what Lagrange Multipliers are all about. In this case, it's easy to see that where the constraint curve and a level curve are tangent is also the point where the image of the constraint curve has an extreme value on the graph of z = f(x,y).

Left-click on a box, right-click, then left-click to select "Make Working Stmt" and to choose that particular item to graph. Then change the coordinates of P to see if you have found the correct extreme point.

Enter your function and enjoy a nice field of vectors!

This notebook graphs the slope field for a differential equation of the type where the dervative of y with respect to x is a function of both x and y. Additionally, if you enter coordinates for a point (a, b) and an increment delta-x, then Euler's Method is used to highlight a particular solution to the differential equation. To enter a new expression in x & y, type ?x to create an outlined letter.

This notebook allows you to create a phase portrait for a system of differential equations. You can also plot a point corresponding to an initial condition.

This notebook explores Professor Attanucci's alternative the second partials test for extrema.

Enter coordinates for u and v below and observe the level curves as you zoom in.