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Fig. 366
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Show that the given lter is stable. Fill in the following table of values for the given values of r, then sketch the curve of jH r j versus r.
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r 0.00 0.05 0.10 0.20 0.25 0.30 0.40 0.45 0.50 H r jH r j
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Note 1. Some Basic Algebra
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This item constitutes a very brief review of some basic rules and operations of algebra. We begin with the notation used to denote multiplication, as follows. Let A and B represent two numbers; then A times B, called the product of A and B, is, in algebra, denoted in any of the following ways A B A B AB In the same way, if A, B, and C represent three numbers, then the product of the three can be denoted in any of the following ways A B C A B C ABC In the above, A, B, and C are referred to as the factors of the product ABC. It should be noted that multiplication is a commutative operation, which simply means that it makes no di erence in what order the factors of a product are written; that is AB BA Multiplication is also associative, which means that the product of three or more numbers is the same in whatever way they may be grouped together; that is ABC A BC AB C Lastly, multiplication is distributive with respect to addition, which is summarized in the statement that A B C AB AC which is read as A, times the quantity B plus C, is equal to A times B, plus A times C. In regard to positive and negative numbers, the rules concerning MULTIPLICATION are the PRODUCT of two numbers having LIKE SIGNS is POSITIVE, the PRODUCT of two numbers having UNLIKE SIGNS is NEGATIVE.
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The absolute or numerical value of a number is its value without regard to sign. The absolute value of A is A, which is shown symbolically by writing j Aj A Thus j 2j 2, which is read the absolute value of minus 2 is 2. If no sign is shown with a number, the number is understood to be positive; thus, 2 2, and so on. In the addition of two numbers the following rules apply. (a) To add two numbers having LIKE SIGNS, add their absolute values and pre x the common sign. Thus, 2 5 7, 2 5 7, and so on. (b) To add two numbers having UNLIKE SIGNS, take the di erence of their absolute values and pre x to it the sign of the number having the larger absolute value. For example, 2 5 3, and 2 5 3.
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To SUBTRACT one number from another, change the sign of the number to be taken away and proceed as in addition. Thus, to subtract 5 (meaning 5) from 2, we have 2 5 3. Or, to subtract 5 from 2 we have 2 5 7, that is, 2 5 7. Next, if one number A is to be DIVIDED BY another number B, this is indicated algebraically by the fractional form, A A=B B A C, this says that B A divided by B is equal to C, in which the SIGN of the quotient C is POSITIVE if A and B have LIKE SIGNS but NEGATIVE if A and B have UNLIKE SIGNS. Thus, 6=3 6= 3 2, but, 6=3 6= 3 2. In the expression A/B, A is called the numerator of the fraction and B is called the denominator of the fraction. The value of a fraction is not changed if the numerator and denominator are both multiplied or divided by the same quantity. In regard to the multiplication of fractions, the PRODUCT of two fractions is equal to the product of the two numerators over the product of the two denominators ; that is which is read as A over B, meaning A divided by B. If we write A C AC B D BD In regard to an EQUATION, the equality of the two sides is preserved if the same operation is applied to both sides of the equation. For instance, multiplying both sides of the equation A=B C by B shows that A BC; thus, A=B C and A BC denote the same relationship among the quantities A, B, and C. Next we have the algebraic form B a , in which B is called the base number and in which the exponent a is the power to which B is to be raised. The exponent a can be any positive or negative integer or fraction. If a is a positive integer (positive whole number), then B a is simply a shorthand notation for the number of times B is to be multiplied by itself; thus, B 2 BB, B 3 BBB, and so on. Or, if a is an integer, then B a denotes the reciprocal of B a ; that is, B a 1=B a . Thus, B 1 1=B, B 2 1=BB, B 3 1=BBB, and so on. If the exponent is a fraction 1/a, then B1=a is the ath ( aye th ), root of B. For instance, if a 2, then B1=a B1=2 , which is called the square root of B, which is also written using the radical sign , thus p B1=2 B C
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