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

VIDIO 2 FAKTORING POLYNOMIAL
In mathematics, a polynomial is an expression constructed from variables (also known as indeterminates) and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and also because its third term contains an exponent that is not a whole number.
For example,is a term.
The coefficient is –5, the variables are x and y, the degree of x is two, and the degree of y is one.
The degree of the entire term is the sum of the degrees of each variable in it. In the example above, the degree is 2 + 1 = 3.
A polynomial is a sum of terms. For example, the following is a polynomial:

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero. Here "− 5x" stands for "+ (−5)x", so the coefficient of the middle term is −5.
When a polynomial in one variable is arranged in the traditional order, the terms of higher degree come before the terms of lower degree. In the first term above, the coefficient is 3, the variable is x, and the exponent is 2. In the second term, the coefficient is –5. The third term is a constant. The degree of a non-zero polynomial is the largest degree of any one term. In the example, the polynomial has degree two.
A polynomial function is a function defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies

for all arguments x, where n is a nonnegative integer and a0, a1,a2, ..., an are constant coefficients.
Example :
Factoring of (x^3 – 7x – 6)
(x^3 -7x-6)/(x-3)
(x^3-7x-6)/ (x^3) no remainder
(x-3) is factor f (x^3-7x-6)
x^2 + 3x + 2 also a factor of (x^3-7x-6)
x^3-7x-6) = (x^3) (x^2 + 3x + 2)
the roots is x-3 = 0; x = 3 or
x + 1 = 0; x = -1
x + 2 = 0; x = -2
three roots this 3 rd degree equation quadaratic (2nd degree) equations always have at most 2 roots.
Long division for a 3 rd order polynomial:
1. Find a partial quotient of x^2, by dividing x into x^3 to get x^2
2. Multiply x^2 by the division and subtract the product from the product from the dividend
3. Repeat the process until you either “clear it out” or reach a reminder

VIDIO INVERS FUNCTION

The Inverse of a function

Definition: The inverse of the function is when the domain and the range trade places. All elements of the domain become the range, and all elements of the range become a domain.

Example of the inverse of a simple function

Original function f(x)	Inverse of function or f-1(x)
{ (0,3 ) , (1,4) , (2, 5) }	{ (3, 0 ) , (4,1) , (5, 2) }

Function invers to shape the function is “invers function”. Invers function of f can written f^-1 (read invers function f).

Given a function , its inverse is defined by

Therefore, and are reflections about the line . In Mathematica, inverse functions are represented using InverseFunction[f].

Example :

v y = 2x – 1

if y = 0 so

0 = 2x – 1

x = 1/2 so coordinat’s (1/2,0)

if x = 0 so

y = 0- 1

y = -1 coordinat’s (0,-1)

v x = 2x -1

1 + x = 2x

1 = x

v y = 2x -1

2x = y + 1

x = ½ (y + 1)

x = ½ y + ½

· function f and function g can combination with a rule to be certain, and a rule this as some function composition

· function composition f continued function g(given sign g0f)

example :

let f(x) = 2x – 1

g(x) = 1/2x + ½

determine fog and gof

solution :

· (fog) = f (g(x))

= 2(1/2x + ½) – 1

= x + 1 – 1

= x

· (gof) = g(f(x))

= ½(2x-1) + ½

= x – ½ + ½

= x

TUGAS VII
how to express mathematic

a. evidences
information that gives a strong reason for believing a prover
example : there were evidences that teachers perception of their student learning in lesson study
activities were positive
b. awareness
having knowledge of sb/sth, realizing sth interested in and knowing abouth sth
example : she became awareness that something was burning.
c. barisan tak berhingga
impossible to measure, calculate or imagine because of being so great; very great; an infinite
number of possibilities
example : i can't to work infinite row problem
d. deret pangkat
a figure or symbol that shows how many times a quantity must be multiplied by it self