Video 1 Pre-calculus : Graphs of a Rational Function
Graphs of a Rational Function can have discontinuities has a polynomial in the dominator. For Example : f(x)= x+3/x-4
. If x=4 so f(4)= 4+3/4-4
equal7/0 , 0 be denominator, so x=1 wrong choice. In graph not may be to drawn. The graph that discontinuities
Polynomial
Polynomial has graph smooth unbroken curve, and in rational function denominator not may be valuable zero. So that in the graph is not broken off in certain points. the example 1234 so line will be broken off if 1234. There is a way of so that 123 is not cut. That is with simplified the numerator. in example on 1234 be 12 so 1234 so can be simplified menjadi12345, so that the graph will not be broken off.
Video 2 Determining limits by inspection
limits by inspection have 2 conditions, that is :
1. x goes to positive or negative infinity
2. limit involves a polynomial divided by a polynomial.
Example : lim x to ~ = x^3+4/x^2+x+1
limit may be polynomial over polynomial or x approaches infinity.
the key to determining limits by inspection is in looking at the powers of x in the numerator and the denominator.
to apply these rulers, must the dividing by polynomials, and the first shortcut rules, if the highest power of x is greater in numerator, so the limit is positive or negative infinity
Example : lim x to ~ = x^3+4/x^2+x+1
the highest power of x in numerator is 3 an the highest power of x in denominator is 2, so the limit is for infinity, since all the number are positive and x is going to positive infinity. So the solve of limit is positive infinity. If you can’t tell if the answer is positive or negative, you can substitute a large number for x and see if you end up with a positive or negative number.
whatever sign you get is the sign of infinity for the limit.
the second shortcut rule if the highest power of x is in the denominator, so the limit is zero.
example :lim x to ~ = x^2+3/x^3+1
the highest power of x in numerator = 2 and the highest power of x in denominator = 3 the limit expression is zero.
the last shortcut rule is a little bit trickier this rule used when : the highest power of x in numerator is same as highest power of x in denominator and limit x
the quotient of the coefficient = the number that goes with a variable example 2 is the coefficient of 2x² and 75 is the coefficient of 75.
the example : lim x to ~ =4x^3+x^2+1/3+3+4
look the highest power of the both, numerator and denominator is thee. according to this rule lim= coefficient of x³ is over each other the coefficient x in the numerator is 4 and in the denominator is 3, so the limit is .
Video 3 Inverse Functions
functions f (x,y) =0
function y = f(x) : V L T
1.1 function x = g(y) : H L T : Invertible
2x-1 = y
2x=y+1
x =½(y+1)
x= ½y+½ y= ½x + ½
f(x) = 2x-1
g(x) = ½x + ½
f(g(x)) = 2. (½ x - ½) – 1
f(g(x)) = x+1-1 =x
g(f(x)) = ½ ( 2x-1) + ½
g(f(x)) = x-½ + ½ = x
g = f ˉ¹
f(g(x)) = f(fˉ¹ (x))
f(g(x)) = x
g(f(x)) = fˉ¹ (f(x)) = x
for example :
y = x-1/x+2
y(x+2) = x-1
yx+2Y = x-1
yx-x = -1-2y
(y-1)x = -1-2y
x =-1-2y/y-1
y= -1-2x/x-1
at x=0;y=-1
at y=0; -1-2x=0
-2x=1
x=-1/2
v asm. x=1
HA at y =-1
first function picture y=g(x), then see function h and insert existing value in h(1). value y =g(x), if x=2; y=g(x), 1=g(2)
so that value h(1) 3
how much is f when x=3p in function f(x)=x+1, if 2f(p)=20
Video 4 functions and graph
the figure above shows the graph of y=g(x). if the function h is defined by h(x)=g(2x)+2. what is the value of h(I)?
answer h(1)
h(x)=g(2x)+2
h(1)=g(2)+2
h(1)=3
first function picture y=g(x), then see function h and insert existing value in h(1). use similarity g(x)=y, so that g(2)=1. result from g(x)=y pack into similarity h(x)=g(2x)+2, and obtainable h(1)=3
let the function f be defined by f(x)=x+1 if 2f(p)=20. what is the value of f(3p)?
answer
f(3p) what is f when x-3p
f(x) = x+1
2f(p)=20
f(p)=10
f(p)=p+1=10
p=9
x=3p
x=27
f(x)=x+1
f(27)= 27+1=28
how much is f when x=3p in function f(x)=x+1, if 2f(p)=20. from similarity 2f(p)=20 obtainable similarity f(p)=10 and can be produced f(p)=p+1=10, so that p+9, then value p insert to x, and knowable x=3p 27 and f(27)=27+1, so that the result 28.
in the x-y coordinate plane, the graph of x=y²-4 intersects line f at(0,p) and (5,t). what is the greatest possible value of the slope of f?
answer
greatest slope (m)
x=y²-4
line l : m=y2-y1/x2-x1=t-p/s
similarity slope putted into value x and y so that appear similarity.
Senin, 12 Januari 2009
At end English lecturing II
At end English lecturing II I get task from Sir Marsigit, that is how teach mathematics in English. Difficulties in teach, with what is I teach understandable by my friend. Then in my execution takes cube matter. Then I look for friend to practice it. Then i get partner that is Anas Yoga Nugroho. I begin to explain from cube definition, cube elements, cube formula, until to cube concept applications. After I am finished explain to Anas. I also give it exercise practice. In do, Anas not too afflictions in do it. May be, caused by the practice exercise too easy. So that Anas fluent in finish the practice exercise.
Partner Anas Yoga Nugroho 07305144028
Partner Anas Yoga Nugroho 07305144028
My impression is during 1 semester in English lecturing II
My impression is during 1 semester in English lecturing II. During 1 semester, I am taught to how to discuss mathematics in English. Besides I also am to know to how process teaches mathematics in English. As well as how write mathematics article into blog, so that I detect that blog very be of benefit to student university.
Benefit Blog In Communication
Benefit blog in communication: blog very useful once in communicating, insides blog we can write a certain matter unlimited and article result legible by who ever besides can write, we also can read article result from another person. Insides the education blog also very useful, we can mutual swop science with another person.
Rabu, 17 Desember 2008
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