Par df x dx 1 l imit fx1 x fx1 limit f limit f

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Unformatted text preview: hat the Limit f(x) does NOT exist anywhere. x→a However, Value f(x) is well-defined everywhere. x=a Since L imit f(x) ≠ Value f(x) we say that f(x) is not continuous at x = a. not x→a x=a But a can be any point on the Real number line. So f(x) is not continuous anywhere. not 52 DIFFERENTIATION Par t 2 : DIFFERENTIATION 53 (0, 0) y(t) = u . s i n θ . t -- 1- g t 2 2 y(t ) SPEED (2-Dimension) 1 u u.sin θ Height y(t) Over view Ov er vie w Let us continue with our main example. VERTICAL POSITION (1-Dimension) h1 t1 Time T /2 T t θ (0, 0) u.cos θ When an object is projected into the air with a given initial velocity u a nd angle of projection or direction θ , the cur ve that describes its path is a parabola. There are two functions that describe the two components of its position. A VERTICAL function y(t) which describes its HEIGHT at any chosen INSTANT and a HORIZONTAL function x(t) which describes its RANGE at any chosen INSTANT (refer Preface iii). We know from Dynamics that: RANGE function x(t) = u . cos θ . t HEIGHT function y(t) = u . s i n θ . t -- 1 g t 2 -2 Let us concentrate on the VERTICAL function y(t). Let t 1 the time for the object to travel from t = 0 to y(t1). Let h1 be the height or ver tical displacement described in time t 1. y(t1) = u . s i n θ . t1-- 1 g t12 -2 This is a polynomial of the form: f(x) = a 2 x 2 + a 1 x + a 0 which is CONTINUOUS everywhere. Now that we know the TIME AXIS is a CONTINUOUS set of INSTANTS and y(t) is a SINGLE VALUED and CONTINUOUS function over [A , B], we may proceed to describe the DIFFERENTIATION operation and perform the calculation. 54 Before we do any calulation (differentiation ) we will develop the concept and definition of the FIRST DERIVATIVE or the INSTANTANEOUS RATE OF CHANGE. Using our main example we will illustrate the concept to find the INSTANTANEOUS RATE OF CHANGE of the par ticular height function y(t) at some par ticular par par ticular instant t 1. LIMIT δy LIMIT δy L IMIT y(t1 + δ t) − y(t1) = t → t δt = δt→ 0 δt = δt→...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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