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prerequisites

# prerequisites - What you need to know to take Calculus 221...

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What you need to know to take Calculus 221 August 16, 2000 1 Arithmetic § 1 . Here are some things which should be easy for you. (If they are not, you may not be ready for calculus.) 1. Factor x 2 - 6 x + 8. 2. Find the values of x which satisfy x 2 - 7 x + 9 = 0. (Quadratic formula.) 3. x 2 - y 2 =? Does x 2 + y 2 factor? 4. True or False: x 2 + 4 = x + 2? 5. True or False: (9 x ) 1 / 2 = 3 x ? 6. True or False: x 2 x 8 x 3 = x 2+8 - 3 = x 7 ? 7. Which x satisfy x - 2 x + 4 < 7? 8. Find x if 3 = log 2 ( x ). 9. What is log 7 (7 x )? 10. True or False: log( x + y ) = log( x ) + log( y )? 11. True or False: sin( x + y ) = sin( x ) + sin( y )? § 2 . There are conventions about the order of operations. For example, ab + c means ( ab ) + c and not a ( b + c ) , a b c means a/ ( b/c ) and not ( a/b ) /c, a b c means ( a/b ) /c and not a/ ( b/c ) , log a + b means (log a ) + b and not log( a + b ) . 1

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If necessary, we use parentheses to indicate the order of doing the operations. § 3 . There is analogy between the laws of addition and the laws of multiplication: a + b = b + a ab = ba ( a + b ) + c = a + ( b + c ) ( ab ) c = a ( bc ) a + 0 = a a · 1 = a a + ( - a ) = 0 a · a - 1 = 1 a - b = a + ( - b ) a/b = a · b - 1 a - b = ( a + c ) - ( b + c ) a/b = ( ac ) / ( bc ) ( a - b ) + ( c - d ) = ( a + c ) - ( b + d ) ( a/b ) · ( c/d ) = ( ac ) / ( bd ) ( a - b ) - ( c - d ) = ( a + d ) - ( b + c ) ( a/b ) / ( c/d ) = ( ad ) / ( bc ) The last line explains why we invert and multiply to divide fractions . The only other law of arithmetic is the distributive law ( a + b ) c = ac + bc, c ( a + b ) = ca + cb. Note that ( a + b ) /c = ( a/c ) + ( b/c ) , but 1 c/ ( a + b ) = ( c/a ) + ( c/b ) . 2 Polynomials and Rational Functions § 4 . A polynomial is a function of the form P ( x ) = a 0 + a 1 x + a 2 x 2 + · · · a n x n where a 0 , a 1 , a 2 , . . . , a n are constants. When a n = 0 we say the polynomial has degree n . Thus P ( x ) = 3 + 7 x + 2 x 5 is a polynomial of degree 5. A rational function is a ratio of two polynomials like F ( x ) = x 3 + x x - 1 . When (as in the example) the degree of the numerator is greater than or equal to the degree of the denominator, you can do long division and write x 3 + x x - 1 = ( x 2 + x + 2) + 2 x - 1 which expresses the rational function as a polynomial plus another rational function where the degree of the numerator is smaller than the degree of the denominator. In other words N ( x ) D ( x ) = Q ( x ) + R ( x ) D ( x ) 1 usually 2
where N ( x ), D ( x ), Q ( x ), R ( x ) are polynomials and the degree of the remainder R ( x ) is smaller than the degree of the denominator D ( x ). We can multiply both sides by D ( x ) to get N ( x ) = Q ( x ) D ( x ) + R ( x ) , deg R ( x ) < deg D ( x ) . ( ) § 5 . When the denominator D ( x ) is of degree one it has the form D ( x ) = x - a .

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prerequisites - What you need to know to take Calculus 221...

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