VIDIO 3
PRE-CALCULUS
Graphic Calculus is a suite of programs to help students of different ages with the visualization, exploration and conceptualization of mathematical concepts.
• Graphic Calculus has a powerful graph plotter.
• Students can test their knowledge of graphs in finding the formula.
• The option Line draws a line through two points and shows the changes of the formula as the point moves.
• With Parabola you can draw the graph of a quadratic equation through two points.
• In the module Exponential functions, the effects of changing of the parameters on the graph is shown.
• Trigonometric functions show the construction of the sine, cosine and tangent from the unit circle.
DiagraAdvanced parts of the program
• Gradient shows the drawing of a slope in a specific point as well as the graph of a gradient function.
• Area shows the area under a graph and calculates its value. Area shows the area function of a given graph.
• In first order differential equations the direction field and the solution through any starting point is plotted.
• Functions of two variables plots perspective drawings of surfaces z=f(x,y), either in 3D or as level lines.
• The Cob-Web graph to solve f(x)=x.
• Taylor polynomials to approximate functions with polynomials.
• Financial mathematics to visualize present value of an annuity or a bond.
• Linear Programming will solve lp-problems straightforward.
• Complex functions represents complex functions w=f(z), like sin(z), by drawing figures in the z-plane and their image in the w-plane.
Definition. A rational function f is a quotient:

where g and h are polynomials
The domain of f consists of all real numbers x such that the denominator h(x) is not equal to 0.
It is not easy to give a general description of the range of f. See the first example below.
Graph of rational function which can have discontinued, because have polynomial in the denominator.
Is possible value x devide by 0 (zero)
Example :
F(x) = (x+2)/(x-1)
Let f(1) = (1+2)/(1-1) = 3/0 bad idea
Graph f(1) = (1+2)/0 break in function graph
f(x) = (x+2)/(x-1) insert 0
f(0) = (0+2)/(0-1)= -2
insert 1 f(x) = (1+2)/(1-1)= 3/0 impossible
rational function don’t always work this way!
Take graph f(x) = (1)/(x^2+1) not all rational function will give zero in denominator.
y = (x^2-x-6)/(x-3) the graph lose like these
if x = 3; (3^2-3-6)/(3-3) = 0/0 that is not
possible not to asible not allowed when you see result of 0/0 and also fell you direction be possible. Factor top and bottom of rational function and simplify.
Rational function denominator can be zero.
 Polynomial have smoth and un broken curve and for rational x approach zero in denominator, that impossible situation.
 Example y = (x^2-x-5)/(x-3)
y = (x-3)(x+2)/(x-3)
y = x+2
if x =3, so that y = x+2
=3 + 2
= 5