Assignment 16

 "Learning is the only thing the mind never exhausts, never fears, and never regrets." -- (Leonardo Da Vinci) Mathematical Drought in Western World (500 - 1200): The Islamic world was mathematically active during this period, as was the Hindu world. Perhaps the greatest of the Hindu mathematicians was Bhaskara (1114 - 1185), who, among many other things, produced several approximations for the number pi, including the frequently used 22/7. Bhaskara's major work, the Siddantasiromani , was an astronomy text that contained two chapters on mathematics: the Vija-Ganita , on algebra, and the Lilavati , on arithmetic. "It is remarkable to what extent Indian mathematics enters the science of our time. Both the form and the spirit of the arithmetic and algebra of modern times are essentially Indian and not Grecian." -- (Florian Cajori) What does Herkimer call an individual who has to cough every time he tells a joke? Answer: A practical choker. Herky 's friends: PATTY O'FURNITURE ... young lady who sells chairs that are used outside the house. BILL DING ... a contractor. Reading: Section 3.1, pages 139-142. Written: Pages 142-143/20-31. Solve the systems algebraically, and be able to show the solutions graphically.

 Mathematical word analysis:COEFFICIENT: From the Latin words co and efficiens, meaning "to effect together." [In an expression such as 5x, the numerical coefficient 5 effects the variable x.]

Communication will help you considerable when you are solving systems of linear equations. For example, suppose we are to solve the system:

(a) -3x + y = 4

(b) 7x + 2y = -5

Solution:

(a) ==> y = 3x + 4.

Substituting into (b) yields 7x + 2(3x + 4) = -5 ==> 7x + 6x + 8 = -5 ==> 13x = -13 ==> x = -1.

(a) ==> -3(-1) + y = 4 ==> y = 1.

The lines intersect at the point (-1,1).

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What is below represents a review of the very important concept of a function.

A function is a collection of ordered pairs such that each x-value has a unique y-value.
The Y= menu on your calculator graphs only functions. (Think about this!)

The domain of a function {(x,y)} is the set of all possible x-values.
The range of a function {(x,y)} is the set of all possible y-values.

Sometimes it is handy to think of a function as a rule. For instance, consider the line y = 4x - 6. This collection of ordered pairs (a line in this case) is a function, since each value of x produces exactly one y-value. Basically, you can think of this function as a rule that says "take a number, multiply it be 4, then subtract 6." Apply the rule to the number 3, and you get 4(3)-6 = 6. In other words, the point (3,6) is on the line.

If you think of a function as a rule, then the domain is imply the set of all numbers on which the rule can operate, and the domain is the set of numbers produced by the rule.

For instance, consider the function f defined by f(x) = x2 + 6. [Note that f(x) is read "f of x."]
We can think of f as a rule that says "take a number, square it, then add 6."
This rule, f, can operate on all real numbers, so the domain of f, designated by Df , is the set of real numbers. Note that the rule, f, can produced only numbers that are 6 or greater. Hence the range of f, designated by Rf, is all real numbers that are at least 6.

 Problem: Solve the system 4x - 3y = 14 and y = x - 3. Solution (with communication): Substituting the second equation into the first yields 4x - 3(x - 3) = 14 ==> 4x - 3x + 9 = 14 ==> x + 9 = 14 ==> x = 5. Hence y = 5 - 3 = 2. The two lines intersect at the point (5,2). Problem: Solve the system (a): 3x + 5y = 21(b): x + 4y = 14 Solution (with communication): (a): 3x + 5y = 21 (c) = 3(b): 3x + 12y = 42 (d)=(c)-(a): 7y = 21 ==> y = 3. Substituting into (b) yields x + 12 = 14 ==> x = 2. The two lines intersect at the point (2,3).