alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

With a little thinking you can infer that when the

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Unformatted text preview: 1 Therefore m 1 . m 2 = t an θ1 . tan θ2 = cos θ1 − sin θ1 Hence : m 2 = − 1/ m 1 From the Analysis point of view : Analysis y 1 ’ (x 1 ) = SLOPE m 1 = t an θ1 y 2 ’ (x 1 ) = SLOPE m 2 = t an θ2 Hence, without drawing any graphs, we may directly analyse a nd say : a nalyse if y 1 ’ (x 1 ) . y 2 ’ (x 1 ) = − 1 then y 1 (x) a nd y 2 (x) m ust be or tho g onal (per pendicular to each other) thog perpendicular per at x 1 . Also y 2 ’ (x 1 ) = − 1 / y ’ (x ) 11 105 Equa tions of the tang ent and nor mal Equations tangent normal Let y(x) be a general function with tangent z1(x) and nor mal n1(x) at (x1, y(x1)) tangent normal tang nor as depicted below. n1(x) y(x) y z1(x) ) θ1 x x1 From the Analysis point of view the equations of the tangent and nor mal are : Analysis tangent normal Anal tang nor tangent z1(x) − y(x1) = y’(x1) (x − x1) nor mal n1(x) − y(x1) = −1/ y’(x ) (x − x1) normal 1 There are two special cases for the tangent . tangent 1. If y ’(x 1) = 0 then the tangent at (x1, y(x1 )) is PARALLEL to the x-axis. tangent t ang So z1(x) = y(x1). Try to imagine what the picture of y(x) will look like around x = x1 . Can we say that y(x1) is either a maximum or a mimimum ? maximum mimimum 2. If y’(x1) = ∞ then the tangent at (x1, y(x1)) is PARALLEL to the y-axis. tangent tang So z1(x) = x1 . normal Likewise, there are two special cases for the nor mal . nor 1. normal If y’(x1) = 0 then the nor mal at (x1 , y(x1)) is PARALLEL to the y-axis. n or So n1(x) = x1 . 2. normal If y’(x1) = ∞ then the nor mal at (x1, y(x1)) is PARALLEL to the x-axis. nor So n1(x) = y(x1) . 106 We may extend this thinking to find the angle of intersection between any two functions f(x) and g(x). zg(x) f(x) z f (x) g(x) o 90 + θ ) )θ a x We may draw the tangents to f(x) and g(x) at the point of intersection a . Let : tangents tang z f (x) = m f . x + c f and z g(x) = m g . x + c g and be the equations of these tangents . Then : tangents tan and tan m f = tan θ f and mg = tan θ g So : mi...
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