Linear FunctionsThis document briefly describes various Mathscribe examples in the following areas:
To open the example files, start Mathscribe, choose Open, open the "reference" folder, open the "more_mm_files" folder inside it, and then open the example file(s) you are interested in. For simpler examples designed for students, see the step-by-step computer lab assignments available at www.mathscribe.com.
Linear Functionseq slope-intercept1.mm, eq slope-intercept2.mm - These examples illustrate the slope-intercept equation for a line: y=m*x+b. Start by opening one of these example files. Your goal is to find values for m and b that will produce the x and y values in the Table window. It is most educational to do this graphically. To do this, click on various m and b values (under Parameters in the main window) to try to get the line in the graph to go through the plotted data values. (Which value should you try to guess first, m or b?) Dragging, shift-clicking and using arrow keys (inside the sliders) can be most instructive. If your values in the Table window are close but not exact, it is often best to refine them analytically rather than graphically.
eq point-slope.mm - This illustrates the point-slope form of an equation for a line: y=m*(x-a)+b. It is similar to the slope-intercept examples above.
line1.mm, line2.mm, line3.mm, line4.mm, line5.mm - Here are several more examples for you to practice on. Notice that in each case, the plotted data values still lie on a line. Your task is again to find an equation for that line. However, in these examples we have not yet introduced any parameters. You are free to use either the slope-intercept or point-slope form for your equation, and to work either analytically or graphically. Click on Graphs after you have entered an equation under Constraints.
linear2args.mm - This problem is much harder, since z depends on two variables (s and t), not just one. Nonetheless, the observed data values can actually be modeled exactly using an equation of the form z=a*s+b*t+c. Try finding the parameters' approximate values through a combination of analytic and graphical reasoning. Don't be discouraged if you can't find the exact answer - this problem is really best solved by more advanced algebraic techniques, for instance by considering it as a system of equations in the three variables a, b, and c.
quadratic.mm - The triangular numbers are given by 1=1, 1+2=3, 1+2+3=6, 1+2+3+4=10, etc. These numbers were very popular among the ancient greeks, and especially the Pythagoreans, who contributed heavily to the early development of mathematics. The triangular numbers grow faster than linearly, but this example shows that they can be computed using polynomials of degree 2. Proceed as in the slope-intercept examples above. Notice that the formulas y=a*x^2+b*x+c and z=a*(x-h)^2+k can both produce any vertical parabola. Which parameters have more geometric meaning, say in terms of the vertex of the parabola: b and c, or h and k?
polynomial1.mm, polynomial2.mm - These are analogous to the line1.mm through line5.mm examples above, except that they are best modeled by polynomials of degree 2 or higher.
polynomial difference.mm
- Here y = x^2, z = (x-d)^2 is y shifted by d, and w = y - z.
What shape is the graph of w as a function of x? Can you explain
this algebraically, by simplifying the formula for w?
polynomial2args.mm - This example is again harder, since z is a function of 2 variables (x and y). As a hint, try to find a relationship between x+y, x-y, and z. If you succeed, notice that your final answer may be expressed either as a factored polynomial, or multiplied out so that it is a simple sum of monomials. Note that since the two forms are equivalent, they each give the correct values.
fractional linear.mm - Fractional linear transformations can be expressed as the quotient of two linear polynomials, as in this example where z=a*(x-b)/(x-c). Such transformations are especially important in the study of complex numbers (which include numbers like square roots of -1, if you can imagine that). What are the geometric meanings of a, b, and c, in terms of the graph of z as a function of x?
partial fractions.mm - Rational functions are defined as quotients of polynomials. They are often particularly interesting (and unpleasant) near roots of their denominator. Partly for this reason, it is often better to split a rational function into a sum of "partial fractions" with simpler (lower degree) denominators, as in this example where z=a/(x-b)+c/(x-d). The simpler partial fractions can then be analyzed one at a time. This function z(x) is said to have singularities or "poles" at b and d.
rational function 2args.mm - You don't have to match any data values in this example. Simply play with the graphs, and notice how z varies as a function of x and y. If x and y are the coordinates of a particle, then z=1/(x^2+y^2) (= 1/r^2 in polar coordinates) is proportional to the gravitational or electrical force exerted on the particle by an object at (0, 0). (Gravitational and electromagnetic forces are 2 of the 4 known forces in the universe.)
circle Lite.mm - This example's graphs form a circle centered at (a, b) with a radius of r. Note the graphs are blurred out near the square root of zero, where the slope is very large and hence the program isn't sure of its accuracy. Change the limits for x to "zoom in" on these regions, if you wish to view them more accurately.
circle non-Lite.mm - A simpler version of the previous file which the full Mathscribe program can graph and calculate with, but not Mathscribe Lite.
ellipse Lite.mm - This example's graphs form an ellipse with semi-major and semi-minor axes given by a and b.
ellipse non-Lite.mm - A simpler version of the previous file which the full Mathscribe program can graph and calculate with, but not Mathscribe Lite.
elliptic curve Lite.mm - This example yields so-called "elliptic curves" (which arise in questions such as arc length along an ellipse), as long as 4*a^3+27*b^2 is nonzero.
elliptic curve non-Lite.mm - A simpler version of the previous file which the full Mathscribe program can graph and calculate with, but not Mathscribe Lite.
xyz distance.mm - This example computes the distance of a point from the origin, given its x, y and z coordinates.
periodic.mm - This gives a sin wave parameterized by its amplitude, angular velocity (inversely proportional to its period), and phase shift.
superposition.mm
- This example illustrates the superposition (sum) s of two sine
waves y and z with the same frequency. Try setting A to 1, so
that y and z have the same amplitude, and varying the phase shift
(delta). Notice how the result transforms between constructive
and destructive interference, i.e. s is always a simple sin wave
with the same frequency as y and z, but its amplitude may be either
larger or smaller than its components' amplitudes.
sin squared.mm - What shape is the graph of y=(sin(x))^2-1/2? Can you match this graph exactly using a simple sin wave with appropriate amplitude, period, and phase shift?
sin Taylor.mm - These polynomials are the first few "Taylor series" approximations to sin(x). They are the polynomials of low degree that best approximate sin(x) near x=0.
xy from polar.mm - This example converts from polar to rectangular coordinates.
xy to polar.mm - This example converts from rectangular to polar coordinates, assuming x > 0.
compound interest.mm - This example computes the future value of an initial amount P, compounded t times at an interest rate of r. Notice the eventual rapid growth!
half life.mm - This example computes the amount at time t of a substance with half-life tau, given an amount P at t=0.
exp shift.mm - How do the shapes of the graphs of a^(x+d) and c*a^x (as functions of x) compare? Given values for a and d, can you choose c so that the graphs are identical?
log10.mm - This example computes the log base 10 of x. The sample values in the Table window show that the log base 10 of x is roughly equal to the number of digits in x. (The natural logarithm of x is thus roughly proportional to the number of digits of x.)
exp Taylor.mm - These polynomials are the first few "Taylor series" approximations to e^x. They are the polynomials of low degree that best approximate e^x near x=0. Notice their similarity to the Taylor series for sin(x) in the "sin Taylor.mm" example above.
log Taylor.mm
- These polynomials are the first few "Taylor series"
approximations to ln(1+x). They are the polynomials of low degree
that best approximate ln(1+x) near x=0.
mystery data.mm - Try to guess a function that matches these data values. You'll need to combine more than one kind of function. Can you think of a physical situation that might produce a graph like this?
Fibonacci.mm - The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ..., with each number being the sum of the two preceding numbers. These numbers occur often in nature, for instance on pine cones. The formulas for F1 and F2 in this example give the k'th Fibonacci number for k odd or even respectively. When k is large, what is an approximation for the k'th Fibonacci number? The number (-1+sqrt(5))/2 is called the golden ratio, and is the number hardest to approximate using fractions with small integer numerators and denominators. In classical art and architecture it is considered the most harmonious ratio, while in classical music it is the least!
Probability
Examples3d6.mm - The sum r of three six-sided dice is a random variable whose probability distribution should be close to the normal (bell-shaped curve) distribution. To check this, we recorded the expected frequency F of each total for 216 trials. We then compared this to the ideal normal distribution, as shown in the graph to the right. We see that the curve is a very good approximation to the plotted data. In fact, the root-mean-square error between the two is approximately 0.835.
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