alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

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Unformatted text preview: nd go directly to Par t 4 without loss of continuity. 97 DERIVA angent 1 9 . FIRST DERIVATIVE = Slope of Tang ent y Q P θ (0 , 0 ) x1 y x) z( δy P δx x1 + δ x x θ1 0,0) (0,0) )= z 1(x t gen tan + c1 ). x y’(x 1 x1 x We obser ve that the line z(x) thru P and Q makes an angle θ w ith the x-axis. δy We also obser ve that : tan θ = ---x δ What is the Limit o f the line z(x) as Q → P ? L imit tangent The tangent t o the cur ve of the function y(x) at the point P or at x = x 1. tang Let us call it : z1(x) = m1x + c1 , where m1 is the SLOPE . Note that at P : z1(x 1) = y(x1) This tangent to the curve of the function y(x) at the point P intersects the x-axis tangent tang at an angle. Let us call this angle θ1 . So, when we take the Limit as Q → P the line z(x) becomes the tangent to the Limit tangent curve of y(x) at the point P. So : z1(x) = tan θ 1 . x + c1 . δy This is equivalent to saying: δL imit0 ---- at the point P = tan θ 1 -----x→ δx = S LOPE of TANGENT z1(x) t o the cur ve of y(x) at x = x1. 98 From the Analysis point of view : Analysis δ-y -- - is the AVERAGE RATE OF CHANGE of the function y(x) over the δx small inter val δ x. , y (x) = L imit δ y is the INSTANTANEOUS RATE OF CHANGE. --------δx → 0 δx , y (x 1) = FIRST DERIVATIVE of the function y(x) at x = x 1 = G RADIENT or SLOPE of the TANGENT z1(x) to the curve of y(x) at x = x 1 . y ,(x 1) = tan θ1 Hence : On closer observation of the diagrams, as δ x → 0 it would appear that δ y also TENDS TO zero. Hence: Limit δ y = L imit δ y should = 0/ . ----------------0 δx → 0 δ x P → Q δx What we can depict in a diagram has its limitations. So let us look at what is happening Analysis from an Analysis point of view. Anal Let y(x) = x3 What is δ y ? (x + δ x)3 − x3 = { x3 + 3x2 δ x + 3x δ x2 + δ x3 } − x3 So : = 3x2 δ x + 3x δ x2 + δ x3 δy 3x2 δ x + 3x δ x2 + δ x3 = δx δx 2 + 3x δ x + δ x2 = 3x We may simplify because δ x the LIMIT as P → Q . Limit δ y = L imit --------P → Q δx δx...
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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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