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Unformatted text preview: Midterm 1 Review Rate of change We are interested in finding the rate of change of a function. In particular, given a function y = f ( x ) we are interested in finding how fast y is changing with respect to x at some fixed time x = a . The main way we will do this is to combine two observations. 1. Given a line y = mx + b the slope is m and is found by m = rise run = y x = y 2 y 1 x 2 x 1 . In other words, for a line the rate of change of y with respect to x is found by looking at the slope. 2. For a typical function we will see in our class when we look at the function near x = a it will look like a line, namely the tangent line. (The tangent line is the line which touches without crossing the curve.) So to find the rate of change we will find the slope of the tangent line. But, to calculate the slope we need two points and the tangent line only gives one. So first we will work with the simpler case of the secant line (a secant line crosses the curve at two values of x ). So the secant line which intersects the curve y = f ( x ) at ( a,f ( a ) ) and ( b,f ( b ) ) has slope f ( b ) f ( a ) b a , or if we let b = a + h this can be written as f ( a + h ) f ( a ) h . This slope gives the average rate of change of y = f ( x ) from x = a to x = b , i.e., the constant rate that f would have to change at to go from ( a,f ( a ) ) to ( b,f ( b ) ) . The slope of the secant line will approximate the slope of the tangent line and the approximation will get better and better as b gets closer to a (or equivalently as h goes to 0). The problem is that if a = b or if h = 0 then these slopes are 0 / 0 which are undefined, so we need some way to handle this. Limits The way we handle this is to use limits . Intuitively limits tell us what should happen based on what is happening nearby . So for example lim x c g ( x ) = L, which we read the limit as x goes to c of g ( x ) is L , means that as x gets close to c the function g ( x ) is getting close to L (and staying close!). It is possible that the limit does not exist. For example, lim x sin( 1 x ) = Does not exist. To see this we note that the function sin(1 /x ) will do infinitely many oscillations between 1 and 1 around x = 0 and so it does not approach a single fixed L . One way to guess a limit is to plug in values of x closer and closer to c and see if it is approaching some certain value; we can also try plotting a picture of g ( x ) near x = c and seeing how the function is behaving. Both of these methods have short comings, in partic ular they are hard to do without a calculator and can sometimes be deceiving. So we want to have some methods to deal with these limits. One method is to build up a collection of rules that we can use. For example we have the following two rules lim x c K = K and lim x c x = c....
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 Winter '10
 BUTLER
 Rate Of Change

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