alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Ga 1 107 21 21 increasing easing decreasing incr

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Unformatted text preview: y(x)? We may consider all the tangents z i (x) to a par ticular cur ve y(x) as belonging to a single family. We say that the curve y(x) ENVELOPES this family z i (x) of straight lines. The curve y(x) is touched by all the straight lines of this family. Analytical geometr eometry tangents From the Analytical geometr y point of view we ask: if we know the tangents Anal t ang at each point to a curve y(x) can we determine the cur ve ? .x + c 2 If we know the tangents z i (x) = y ’(x i) . x + ci for i = 1, 2, 3, . . ., then we can find y (x i) = z i (x i) for i = 1, 2, 3, . . . . en ng ta y(x 2 ) t 2 z( x) = y’( 2 x) y y(x 1 ) (0, 0) nt tange x1 x2 z 1(x) .x+ y ’ ( x 1) = c1 x If we know the tangents z1(x), z2(x), z3(x), . . . , zn (x), (first derivatives of y(x)) at tangents tang sufficiently many points x1, x2, x3, . . . , xn , we may reconstruct the curve y(x). From the Analysis point of view, this insight tells us that we should be able to calculate Analysis Anal y(x) from y’(x). This inver se oper a tion to find y(x) from its deri v a tive y’ (x) is inverse opera deriv tiv in deri known as integration . i ntegration 103 20. Angle 20 . Ang le of Inter section Let y 1 (x) = m 1 x + c 1 a nd y 2 (x) = m 2 x + c 2 b e two linear functions. y y1(x) ) θ1 )φ θ2 ) y2(x) x1 x The ang le of inter section b etween y 1 (x) a nd y 2 (x) is φ . U sually angle intersection a ng the smaller angle is taken. Hence : φ = minimum (| θ1− θ2|, 180 o − | θ1− θ2|). m inimum With our understanding that SLOPE = tan θ , w e may say that : SLOPE m 1 = t an θ1 a nd m 2 = t an θ2 . per Now if y 1 (x) a nd y 2 (x) a re or tho g onal (per pendicular to each other) thog perpendicular then: m1 . m2 = − 1 104 Proof : tan θ1 = s in θ1 / c os θ 1 If y 1 a nd y 2 a re or tho g onal then : θ2 = 9 0 o + θ1 thog o o o sin (90 + θ1) s in 90 . cos θ1 + c os 90 . sin θ1 = So : tan θ2 = o o o cos (90 + θ1) c os 90 . cos θ1 − s in 90 . sin θ1 c os θ1 = − sin θ1 sin θ1 c os θ1 . =...
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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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