College Algebra MTH 103
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This 7 page Class Notes was uploaded by Donny Graham on Saturday September 19, 2015. The Class Notes belongs to MTH 103 at Michigan State University taught by K. Park in Fall. Since its upload, it has received 13 views. For similar materials see /class/207305/mth-103-michigan-state-university in Mathematics (M) at Michigan State University.
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Date Created: 09/19/15
Section 22 More on Functions and Their Graphs Functions and Difference Quotient The expression f x h f x h for h at 0 is called the difference quotient Example 1 If fx 2x2 x 5 nd and simplify each expression a fx h b fxhhfx h 0 Piecewise Functions A function that is de ned by two or more equations over a speci ed domain is called a piecewise function Example 2 Use the function that describes the cellular phone plan 40 if 0 S t S 450 C0 40 045t 450 if t gt 450 to nd and interpret each of the following a C300 b CSOO Increasing and Decreasing Functions 1 A function is increasing on an open interval 1 if for any x1 and x2 in the interval where x1 lt x2 then fx1 lt fx2 2 A function is decreasing on an open interval 1 if for any x1 and x2 in the interval where x1 lt x2 then fx1 gt fx2 3 A function is constant on an open interval I if for any x1 and x2 in the interval where x1 ltx2then fx1fx2 L x I 39x Increasing Decreasing Constant Example 3 State Intervals on which the given function is increasing decreasing or constant Section 45 Exponential Growth and Decay Modeling Data Exponential Growth and Decay The mathematical model for exponential growth or decay is given by ft 1406 or A 1406 where 140 is the original amount or size at time t 0 A is the amount at time t and k is a constant representing the growth or decay rate If kgt 0 the function models the amount or size of a growing entity If klt 0 the function models the amount or size of a decaying entity Example 1 In 1990 the population of Africa was 643 million and by 2000 it had grown to 819 million a Use the exponential growth model A 1406 in which t is the number of years after 1990 to nd the exponential growth function that models the data b By which year will Africa s population reach 2000 million or two billion Section 34 Zeros of Polynomial Functions Rational Zero Theorem If a x aHxH a1x a0 has integer coefficient and E where g is reduced q q to lowest terms is rational zero of f then p is a factor of the constant term a0 q is a factor of the leading coef cient an Factors of a0 Possible rational zeros Factors of an Example 1 List all possible rational zeros of x3 2x2 5x 6 Example 2 List all possible rational zeros of 4x5 12x4 x 3 Finding Zeros of a Polynomial Function Step 1 List all possible rational zeros Step 2 Use synthetic division to nd a rational zero among the possibilities Step 3 Factor the polynomial using synthetic division Step 4 Repeat Step 1 through Step 3 until you get a quadratic factor Example 3 Find all zeros of x3 8x2 1 1x 20 Example 4 Find all zeros of x3 x2 5x 2 Example 5 Solve x4 6x3 22x2 30x 13 0 Section 13 Models and Application Problem Solving with Linear Equations Example 1 Basketball bicycle riding and football are the three sports and recreational activities in the United States In 2004 the number of injuries from basketball exceeded those from football by 06 million The number of injuries from bicycling exceeded those from football by 03 million Combined basketball bicycling and football accounted for 39 million injuries Determine the number of medically treated injuries from each of these recreational actives in 2004 Step 1 Read the problem carefully and state the problem in your own words Let x represent one of quantities in the problem Step 2 Represent other quantities in terms of x Step 3 Write an equation in x that models the conditions Step 4 Solve the equation and answer the question Step 5 Check your answer in the original wording of the problem Example 2 You are choosing between two longdistance telephone plans Plan A has a monthly fee of 15 with a charge of 008 per minute for all long distance calls Plan B has a monthly fee of 3 with a charge of 012 per minute for all longdistance calls For how many minutes of long distance calls will the costs for the two plans be the same Example 3 You inherited 5000 with the stipulation that for the rst year the money had to be invested in two funds paying 9 and 11 annual interest How much did you invest at each rate if the total interest earned for the year was 487
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