This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 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 ) 6 = ( c/a ) + ( c/b ) . 2 Polynomials and Rational Functions 4 . A polynomial is a function of the form P ( x ) = a + a 1 x + a 2 x 2 + a n x n where a ,a 1 ,a 2 ,...,a n are constants. When a n 6 = 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...
View
Full
Document
 Spring '08
 MarvinSolomon
 Data Structures

Click to edit the document details