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Transcript

3.3 Rates of change One of the main applications of calculus is determining how one variable changes in relation to another. For example, a manager would want to know how much profit changes with respect to the amount of money spent on advertising. The average rate of change (just like slope of line)in a function f(x) with respect to x for a function f as x changes from a to b is given by f (b) f (a) ba The percentage of men aged 65 and older in the workforce has been declining over the last century. The percent can be approximated by the x function f ( x) 68.7(.986) where x is the number of years since 1900. Find the average rate of change of this percent from 1960 to 2000. The graph below gives the Annual Numbers of New Nonmedical Users of OxyContin®: 1995-2003. Find the average rate of change from 1995 to 2003. Finding the average rate of change of a function over a large interval can lead to answers that are not very helpful. The results are often more useful if the average is found over a fairly small interval. Finding the exact rate of change at a given x-value requires a continuous function. A function is continuous at x = c if the following conditions are satisfied: 1) f(c) is defined 2) lim f(x) exist x c 3)lim f(x) = f(c) x c If a function is not continuous at c, it is discontinuous there. The exact rate of change of f at x = a, called the instantaneous rate of change of f at x = a is f ( a h) f ( a ) lim h 0 h Page 185 ex 3b Find the average rate of change for the function over the given interval. 5 y between x 2 and x 4 2x 3 y 4 x 6 between x 2 and x 5 2 Find the instantaneous rate of change for the function at the given value. s(t ) 4t 6 2 at t = 2 Problem #26 p.190 Problem #30 p.191