Objective: To define and use the concepts of Rates of Change and Limits Average Speed; During an interval is found by dividing the distance covered by the time elapsed ∆y = slope think… mile marker 120 to mile marker 10 and it took 1.5 hours. ∆xfree fall equation y = 16t 2 find average speed for first 2 seconds. = ∆y = 16(2) 2 – 16(0) 2 = 32 ft ∆x 2 – 0 sec

Instantaneous Speed; The speed at any given “instant”. Slope at a point How would we get speed at a point? (ponder) Average Rate of Change and Secant Lines; ∆y = f(x 2 ) – f(x 1 ) = slope of the tangent line ∆x x 2 -x 1 ( x 2, f(x 2 )) ( x 1, f(x 1 ))

In terms of some change in ∆x or h ( x 1 +h, f(x 1 +h)) ∆y = f(x 1 +h) – f(x 1 ) = f(x 1 +∆x) – f(x 1 ) ∆x h ∆x ( x 1, f(x 1 )) h ∆x

Limit of a Function; Informal Definition; Let f(x) be defined on an open interval about x 0, except possibly at x 0 itself. If(x) gets arbitrarily close to L (as close to L as we like) for all x sufficiently close to x 0, we say that f approaches the limit L as x approaches x 0, and we write… Behavior of a Function Near a Point. How does a function behave near a point? What happens to this function f(x) near x=1? Graph this….Notice what happens on both sides of 1, even though the function is not defined at 1 Algebraically?

Calculus Limits Directly; Always try this first Example; Algebraically; just put the number into the equation. Answer = -1

Calculus Limits Removable Discontinuity Use this when you can factor Example; Algebraically; just factor, reduce and then put the number into the equation. Answer = 3 O

Calculus Limits Graphically Use for multipart functions. Example; Answer = does not exist because a limit must approach the same value from the left as from the right at that point. O

Calculus Limits Graphically Use for multipart functions. Example; Answer = The Limit is 5 since it does approach the the same value from the left as from the right at that point. O