MTH 251 – Differential Calculus Chapter 3 – DifferentiationComposition of FunctionsAn algebraic example …Slide 4Generalizing these 2 examples …The Chain Rule ExamplesSlide 7Parametric EquationsParametric Equations – Example 1Parametric Equations – Example 2Parametric Equations – Example 3Eliminating the ParameterSlopes of Parametric CurvesSlope ExamplesThe Second DerivativeMTH 251 – Differential CalculusChapter 3 – DifferentiationSection 3.5The Chain Rule andParametric EquationsA B CImage: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-WesleyComposition of Functions•How will x turns of gear A affect gear C?x turns of A will result in 2x turns of B•B(x) = 2x = nn turns of B will result in 1/3 turn of C•C(n) = 1/3 ntherefore …•C(B(x)) = 1/3(2x) = 2/3 xNote that …2dB dndx dx= =13dCdn=23dC dC dndx dn dx= = gAn algebraic example …•Consider …2 24 20 25 (2 5)y x x x= - + = -2( )u=Let u = 2x – 52 2(2 5) 4 10dyu x xdu= = - = -2dudx=8 20dyxdx= -(4 10)2x= -dy dudu dx= gAn algebraic example …•Consider …2 2( ) 4 20 25 (2 5)f x x x x= - + = -[ ]22( ) ( )u g x= =Let u = g(x) = 2x – 5'( ) '( ( )) 2 2 ( ) 2(2 5) 4 10f u f g x u g x x x= = = = - = -'( ) 2g x ='( ) 8 20f x x= -(4 10)2x= -'( ( )) '( )f g x g x= gwith alternate notation …Generalizing these 2 examples …•The Chain Rule•Or the alternate form …If ( ( )), then'( ) '( ( )) '( )y f g xf x f g x g x== gIf ( ) and ( ), theny f u u g xdy dy dudx du dx= == g“Differential Form” (more in section 3.10)The Chain Rule Examples[ ]( ( )) '( ( )) '( )df g x f g x g xdx= g2sin( 3)dxdx� �- =� �3 11( 3 1)dx xdx� �- + =� �2 7xdedx-� �=� �1 of 2The Chain Rule Examples[ ]( ( )) '( ( )) '( )df g x f g x g xdx= g12 5ddx x� �=� �+� �tandxdx� �=� �sec3cos( )xdedx� �=� �2 of 2Parametric Equations•Explicit functions of a single variable:Independent Variable (usually x)Dependent Variable … determined in terms of xGraphs consist of ordered pairs: (x, f(x)) or (x, y)•Parametric functions/curves:Independent Variable (usually t)Two dependent variables each determined in terms of t and independent of each otherGraphs consist of ordered pairs: (f(t), g(t)) or (x, y)( )y f x=( ), ( )x f t y g t= =Where does the ball hit the ground?Parametric Equations – Example 1 •The path of a tennis ball.A tennis ball machine launches balls 40 mph at a 10° angle and 3 feet off the ground. Describe the path of a ball in a vacuum (ignore air resistance).10°40 mphxy40cos10 39.4 mph 57.8 ft/secx = = =o40sin10 6.9 mph 10.1 ft/secy = = =o57.8 x t=feet23 10.1 16 y t t= + -feett = time in secondsGravity: 32 ft/sec2Parametric Equations – Example 2 •Circular MotionA particle moves around a circle of radius r, starting at the point (r, 0) at time t = 0. Describe its path if the particle rotates at t radians per second.cosx r t=siny r t=t(x, y)(r, 0)What values of t are needed to get the entire circle?0 2t p� <Parametric Equations – Example 3 •Linear MotionA particle moves around along the line containing the points (1,2) and (2, 5). Determine its location in terms of a parameter t.Since the relationship is linear, one would expect the x and y values of any point to be determined by linear equations. That is …1 1 2 2, x m t b y m t b= + = +(1, 2)(2, 5)Let (x, y) = (1, 2) when t = 01 21, 2x m t y m t= + = +Let (x, y) = (2, 5) when t = 11, 3 2x t y t= + = +Note that any two values of t could have been used. You will get different equations, but they will generate the same line.Eliminating the Parameter•To convert parametric equations of a curve to and explicit equation …Solve one of the equations for the parameter.Substitute into the other equation.•Example 3 …•Example 1 …1, 3 2x t y t= + = +1t x= -( )3 1 2y x= - +257.8 , 3 10.1 16x t y t t= = + -57.8xt =( ) ( )257.8 57.83 10.1 16x xy = + -23 0.17474 0.0047892y x x= + -3 1y x= -Slopes of Parametric Curves•Since x & y are given independently in terms of t, the slope (i.e. derivative) also needs to be determined in terms of t .•Using the chain rule (differential form) … •… and changing the multiplication to division …dy dy dtdx dt dx= gdydtdxdtdydx=Or ... if ( ) & ( ),'( )then '( )x f t y g tdy g tdx f t= ==Slope Examples•Example 3 …•Example 1 …•Example 2 …1, 3 2x t y t= + = +331dydx= =257.8 , 3 10.1 16x t y t t= = + -10.1 3257.8dy tdx-=cos , sinx r t y r t= =coscotsindy r ttdx r t= =--The Second Derivativedydtdxdtdydx=[ ]2'2''dydtdxdtd y d dyydx dx dx= =
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