Download equations of higher order

EQUATIONS OF HIGHER ORDER
Hallo guys. Today we are going to have our math lesson in English.
We will study equations of higher order than the second.
First, however, we review the vocabulary.
1. VOCABULARY :
Can you read this mathematical sentence? 7+3=10
Addition: is the operation
7+ (plus is the sign) 3 = (is equal to) 10 (sum)
Subtraction: is the operation
7 - (minus is the sign) 3 = (is equal to) 4
(difference)
Multiplication: is the operation
3 * 5 (multipl ied by) = (is equal to) 15
(product)
Division: is the operation 15: 3 (divided by)= (is equal to) 5 (quotient)
Power of (number or letter) :
2 2 = two squared or two to the power of two
2 3 = two cubed or two to the power of three
2 4 = two to the power of four
( open bracket
) closed bracket
An equation is a mathematical sentence that states that two algebraic
expressions are equal. The two expressions are separated by an equal sign.
For example
x + 5 = 7
is an equation.
The expression x+5 is called the left side, or left member , and 7 the right side,
or right member, of the equation.
As in this last case, one or both of the expressions may contain variables.
In this case x is th e variable.
A variable is a symbol for a number we don’ t know yet.
Finall y, you certainly remember the Cartesian system axis , that is a graph
with an x-axis and a y-axis
y-axis
O
the point where they intersect is the origin
x-axis
2. REVISION OF THE CONCEPT OF EQUATION :
What does it mean to make a sentence true?
Are you able to give me some exam ple of math sentences true?
For example 7- 4 = 3 (Is this sentence true or false?) It is true
For example 15- 4 = 13 (Is this sentence true or fal se?) It is false
What does it mean when you are asked to solve an equation?
when you solve an equation, you find out the value of the variable x
(unknown number) that makes the sentence true.
For example in this first degree equation:
x+3=0
x is the variable and you solve it by finding out what it is
x=-3
In fact if you put -3 in place of x you have the true sentence -3+3 =0
Now, can you find out the value of the variable x in this second degree
equation?
x2  9  0
There are two possible values that make the sentence true and these are -3
and +3
Last
year,
we
studied
first-degree
equations
and
second-degree
equations.
It might be useful to know how to solve equations of higher order.
Why?
Why is this important?
See you this movie!
3.TO CREATE MOTIVATION :
Often, you ask me, why you have to study Maths and if they are useful in
real life.
And you might ask me what connection there is between the movie we saw
with Mathematics.
We can try to have an answer , analysing this problem :
“The concentration of a drug, in parts per million, in a patient’s
blood, x hours after the drug is administered , is given by the function
(f open bracket x closed bracket , is equal to…..)
f ( x)   x 4  12 x 3  58 x 2  132 x
This function is the result of a research carried out at the Sautheast
Missouri State University .
This is the question: “How many hours after the drug is administered
will it be eliminate d from the bloodstream?”
Before searching a solution, let’s analyse the text of the problem:
do you know the meaning of all the words in it?
Drug: droga/farmaco
Bloodstream: circolo sangui gno
Function: a function is like a machine. It has an input and an output, and
the output is related somehow to the input.
f ( x)   x 4  12 x 3  58 x 2  132 x
is the classic way of writing a function.
The other way to write a function is to use y in the place of f(x).
x and y are two variables and the second value y/f(x) dep ends on the value
of x.
In fact, if we put a number in the place of x we find the value of y/f(x).
We do this exercise every time we apply the Ruffini’s method!!!!!!
In our example y/f(x) represents the concentration of drug in the blood,
over time x. In fact x represents the number of hours . In other words,
the concentration of drug in the blood depends on time.
It will be useful to understand better to look at this graph .
4.DISCUSSION
How long do you have to wait?
Can you describe the graph? ...the graph starts from the origin and it increases up
to the maximum and than it decreases until reaching the point on the x -axis.
Can you deduce how many hours have passed after the administration of the
drug to have the maximum concentration in the blood?
3 hours
Can you deduce how many hours have passed after the administration of the
drug to eliminate it from the bloodstream? 6 hours
In this point, how much is the drug concentration ( y/f(x) ) in the blood ? zero
In fact, if we put in the equati on
f ( x)   x 4  12 x 3  58 x 2  132 x
the value 6 in the place of x , we will have 0
 (6) 4  12(6) 3  58(6) 2  132(6)  .......  1296  12 * 216-58*36+132*6=….= 0!!!!!
So this mean that to find the unknown value of x we must solve the equation of
fourth degree
 x 4  12 x 3  58 x 2  132 x  0
We have find this value(6) looking at the graph, that is very simple in this case,
but not all the graphics are so easy to read. It would be possible to find it with
algebraic methods, like total factorization and Ruffini’s Method .
For tomorrow morning you have the task to find out the solutions of the equation.
So to conclude, suppose that on Saturday night a boy will go to the disco at 1 A.M.
Suppose he will assume a drug at 2 A.M.
So what time do he should get the car to go home safel y?
Not before 8 A.M. because we know that he needs 6 hours before safely driving a
car.
In fact, if he takes the car at 5 A.M or at 6 A.M. he is under the full effect of the
drug, so he and all the people he meets on his way are in danger!!!!!
This is an example of the usefulness of Mathematics in everyday life.
Problem
“The concentration of a drug, in parts per million, in a patient’s blood x
hours after the drug is administered is given by the function
(f open bracket x closed bracket, is equal to…..)
f ( x)   x 4  12 x 3  58 x 2  132 x
How many hours after the drug is administered will it be eliminated from
the bloodstream?”
(USA Southeast Missouri State Univerity: Math Field Day, 2005)
It will be useful to look at this graph , to find the answer.
How long do you have to wait?
 Can you describe the graph?....................................................................
……………………………………………………………………………………….
 Can you deduce how many hours have passed after the administration of the
drug to have the maximum concentration in the blood?. ......................
 Can you deduce how many hours have passed after the administration of the
drug to eliminate it from the bloodstream?...................................
In this point, how much is the drug concentration (y /f(x)) in the blood ? ……...