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Unformatted text preview: Math Review for ECO220Y: PART 2 of 2 For success in our course, it is important that you have the skills needed to solve the problems in both Part 1 and Part 2 of this Math Review. If you find these questions easy then you do not need to spend much time on this. However, if your skills are rusty or missing it is important that you work to get caught up. Both parts of this Math Review are here to help you. Additionally our Teaching Assistants (TA’s) have office hours should you run into difficulty when working through this Math Review. 1 Functions A function f is a rule that assigns to each element x in a set A exactly one element, called f ( x ), in a set B . It is common to write y = f ( x ). For example, if f ( x ) = 2 x + 1, one may write y = 2 x + 1. In this example we have a simple linear function: the equation of a line. In Problem Set #1 we looked at simple linear functions (including the constant function: a horizontal line). In this problem set we explore commonly used nonlinear functions, including how to find their slopes. 2 Power Functions One simple type of nonlinear function is a power function . A function of the form f ( x ) = x a , where a is a constant, is a power function. One power function you probably recall is the parabola. For example, y = x 2 is a power function (and in this example, it is also a parabola). A quadratic function is function of the form f ( x ) = ax 2 + bx + c , where a , b and c are constants. For example, if a = 1, b = 0 . 5 and c = 10 then f ( x ) = x 2 + 0 . 5 x + 10. Sometimes you can solve a quadratic equation by factoring. Otherwise you can always solve a quadratic equation using the quadratic formula. Quadratic Formula: If f ( x ) = ax 2 + bx + c , then x =- b ± √ b 2- 4 ac 2 a 1. Solve ( x- 2) 2 = 0. 2. Solve ( x- 2) 2 = 4. 1 3. Solve ( x- 10)( x- 20) = 0. 4. Solve x 2- 7 x + 12 = 0. 5. Solve x 2 + 2 x + 1 = 0. 6. Solve 6 x 2- 5 x + 1 = 0. 7. Solve 5 x 2 + 3 x- 3. 8. Solve x 2 + x + 2 = 0. 9. Suppose you are given ∑ 100 k =1 x 2 = 3455 and ∑ 100 k =1 x = 509. (a) Find ∑ 100 k =1 ( x- 10) 2 . (b) Find ∑ 100 k =1 ( x- ¯ x ) 2 . (c) Find ∑ 100 k =1 ( x- 3) 2 . (d) Find ∑ 100 k =1 ( x- 7) 2 . (e) In which case is the sum the smallest? Provide an explanation. A polynomial function is of the form f ( x ) = a + a 1 x + a 2 x 2 + a 3 x 3 + ... + a n- 1 x n- 1 + a n x n , where a i for i = 1 , 2 , 3 ,...,n are n + 1 constants. The numbers a ,a 1 ,...,a n are called coefficients and n is called the degree of the polynomial. For example, if n = 3 and a i = 2 for all i = 1 , 2 ,...,n then we’d have f ( x ) = 2 + 2 x + 2 x 2 + 2 x 3 , which is a third degree polynomial, which is also called a cubic function. If n = 2 then we have a second degree polynomial, which is also called a quadratic function. If n = 1 then we have first degree polynomial, which is also called a linear function (a straight line!)....
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This note was uploaded on 04/10/2010 for the course ECO ECO220 taught by Professor Atamazaheri during the Spring '09 term at University of Toronto.
- Spring '09