Limits and the derivative function. Limits and the derivative function

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2 The Velocity Problem A particle is moving in a straight line. t is the time that has passed from the start of motion (which corresponds to t = 0) s(t) is the distance from the particle to the initial position at the time t (s(t) is called the displacement function) s(t) initial position t = 0 position at time t How to find the velocity of the particle at a given moment of time?

3 The Velocity Problem We can define the average velocity of the particle on a particular time interval. Fi a moment of time t and consider the time interval [t,t + t]. s( t + t) 0 s(t) t t + t distance traveled over the time interval [t, t + t] The distance traveled over the period [t,t + t] is s(t + t) s(t). Hence, the average velocity of the particle on the fied interval is s(t + t) s(t) t

4 The Velocity Problem Now, the (instantaneous) velocity v(t) of the particle at the time t is equal to the average velocity on [t,t + t] as the length of the time interval t decreases to 0. s(t + t) s(t) t v(t) as t 0 In this case we write s(t + t) s(t) v(t) = lim t 0 t

5 The Tangent Line Problem Given the graph of a function f(), we would like to find the tangent line at the point P( 0,f( 0 )).

6 The Tangent Line Problem Given the graph of a function f(), we would like to find the tangent line at the point P( 0,f( 0 )). y tangent line y = f() f( 0 ) P 0 0

7 The Tangent Line Problem Given the graph of a function f(), we would like to find the tangent line at the point P( 0,f( 0 )). y tangent line y = f() f( 0 ) P 0 0

8 The Tangent Line Problem Given the graph of a function f(), we would like to find the tangent line at the point P( 0,f( 0 )). y tangent line y = f() f( 0 ) P 0 0

9 The Tangent Line Problem The tangent line only touches the graph of f() at P but does not intersect it anywhere close to P. It can be obtained as a limiting position of secant lines as follows

10 The Tangent Line Problem The tangent line only touches the graph of f() at P but does not intersect it anywhere close to P. It can be obtained as a limiting position of secant lines as follows y y = f() P 0

11 The Tangent Line Problem The tangent line only touches the graph of f() at P but does not intersect it anywhere close to P. It can be obtained as a limiting position of secant lines as follows

12 The Tangent Line Problem The tangent line only touches the graph of f() at P but does not intersect it anywhere close to P. It can be obtained as a limiting position of secant lines as follows

13 The Tangent Line Problem The tangent line only touches the graph of f() at P but does not intersect it anywhere close to P. It can be obtained as a limiting position of secant lines as follows

14 The Tangent Line Problem The tangent line only touches the graph of f() at P but does not intersect it anywhere close to P. It can be obtained as a limiting position of secant lines as follows

15 The Tangent Line Problem The tangent line only touches the graph of f() at P but does not intersect it anywhere close to P. It can be obtained as a limiting position of secant lines as follows y y = f() P 0

16 The Tangent Line Problem Each secant line passes through P and some point Q( 1,f( 1 )) on the graph of f(). y y = f() f Q f( 1 ) f( 0 ) P Q Q Its slope is a ratio f Q Q As Q approaches P, the ratio f Q Q approaches the slope of the tangent line at P.

17 The Tangent Line Problem In other words f Q Q slope of the tangent line as Q P, recall that f Q = f( 1 ) f( 0 ), Q = 1 0, so or, f( 1 ) f( 0 ) 1 0 slope of the tangent line as 1 0, slope of the tangent line = lim 1 0 f( 1 ) f( 0 ) 1 0

18 The Area Problem Given a circle C of radius r, how to compute its area? The idea is to approimate the circle by figures whose areas are easy to compute. The easiest way is to use regular polygons P n with n sides (n-gons) inscribed in the circle C. The area of each P n is easy to epress in terms of the radius of the circle As n the area of the n-gon P n approaches the area of the circle.

19 The Area Problem Area(P n ) Area(C) as n,

20 The Area Problem C Area(P n ) Area(C) as n,

21 The Area Problem P 4 Area(P n ) Area(C) as n,

22 The Area Problem P 5 Area(P n ) Area(C) as n,

23 The Area Problem P 6 Area(P n ) Area(C) as n,

24 The Area Problem P 8 Area(P n ) Area(C) as n,

25 The Area Problem P 12 Area(P n ) Area(C) as n,

26 Informal definition of limit Definition The limit of a function f(), as approaches a R, is L R, and we write lim a f() = L if the values of f() can be made arbitrarily close to L by choosing the values of close enough to a. In other words, the statement lim f() = L a means that we can make the distance between f() and L arbitrarily small by making the distance between and a small enough but not equal to 0. That is, approaching a makes the corresponding value f() approach L.

27 Formal definition of limit 1 First of all, recall that the distance between f() and L is f() L, and the distance between and a is a. 2 To make f() L arbitrarily small means that f() L can be made smaller than any given positive real number, say ε. 3 Once ε > 0 is given, we have to choose a bound δ on the distance a small enough to force f() L < ε. 4 Hence the precise definition of limit Definition We write lim f() = L a if for any ε > 0 there eists δ > 0 such that a < δ implies f() L < ε.

28 Eamples Eample lim 1 1 = 1 Eample lim = 2 Eample lim = undefined

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