CHAPTER 55 DIFFERENTIATION OF PARAMETRIC EQUATIONS

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1 CHAPTER 55 DIFFERENTIATION OF PARAMETRIC EQUATIONS EXERCISE 7 Page. Given an ( ), deermine in erms of. If, hen If ( ), hen ( ). A parabola has parameric equaions:,. Evaluae d d when 0.5 If, hen If, hen When 0.5, 0.5. The parameric equaions for an ellipse are cos θ, sin θ. Deermine (a) (b) d (a) If cos θ, hen If sin θ, hen sinθ cosθ 9 0, John Bird

2 (b) d cosθ θ sinθ co θ d d co ( cosec θ θ ) sinθ sinθ sin θsinθ sin θ d cosec θ. Evaluae a θ radians for he hperbola whose parameric equaions are sec θ, an θ If sec θ, hen secθ anθ If an θ, hen sec θ When θ, θ θ cosθ secθ anθ anθ sin θ sinθ cosθ d sec sec sin The parameric equaions for a recangular hperbola are, 0.0. Evaluae when If, hen If, hen d 9 0, John Bird

3 When 0.0, d (0.).5. The equaion of a angen drawn o a curve a poin (, ) is given b: ( ) Deermine he equaion of he angen drawn o he ellipse cos θ, sin θ a θ ( ) A poin θ, cosθ sinθ sinθ cosθ cosθ coθ sinθ co cos The equaion of a angen is: sin θ θ( θ) A θ, sin co cos i.e. ( )( ) i.e..55(.598) i.e an The equaion of a angen drawn o a curve a poin (, ) is given b: ( ) 5 Deermine he equaion of he angen drawn o he recangular hperbola 5, a ( ) A poin θ, , John Bird

4 The equaion of a angen is: 5 A 5 5 ( 5), 5 ( 0) i.e i.e i.e , John Bird

5 EXERCISE 8 Page. A ccloid has parameric equaions (θ sin θ), ( cos θ). Evaluae, a θ 0. rad, correc o significan figures, (a) (b) d (a) If (θ sin θ), hen cosθ If ( cos θ), hen sinθ sin sin d θ θ θ ( cos θ) cosθ sin 0. When θ 0. rad,., correc o significan figures cos 0. d d sinθ ( cos )(cos ) (sin )(sin ) cos cos sin d cosθ ( cos θ) ( cos θ) (b) ( cos θ) ( cos θ) ( cos θ) θ θ θ θ θ θ θ cos θ (cosθ + sin θ) cosθ ( cos θ) ( cos θ) When θ 0. rad, d (cos 0.) 0.85 ( cos 0.) (0.0078)., correc o significan figures. The equaion of he normal drawn o a curve a poin (, ) ( ) is given b: Deermine he equaion of he normal drawn o he parabola, a If, If, 95 0, John Bird

6 Equaion of a normal is: ( ) i.e. A, equaion of normal is: ( ) i.e. + or +. The equaion of he normal drawn o a curve a poin (, ) ( ) is given b: Finhe equaion of he normal drawn o he ccloi (θ sin θ), ( cos θ) a θ rad. If (θ sin θ) θ sin θ, If ( cos θ) cosθ, sin cos cosθ sinθ θ θ Equaion of a normal is: ( ) i.e. ( cos θ) ( ( θ sin θ) ) A θ rad, equaion of normal is: sinθ cosθ ( cos ) ( sin ) sin cos 9 0, John Bird

7 i.e. ( ) i.e. + or +. Deermine he value of d, correc o significan figures, a θ rad for he cardioid 5(θ cos θ), 5( sin θ sin θ). + + If 5(θ cos θ), hen 0 0sin θ 0( sin θ) If 5( sin θ sin θ), hen 0 cosθ 0 cos θ 0(cosθ cos θ) 0(cosθ cos θ) cosθ cos θ 0( + sin θ) + sin θ d d cosθ cos θ (+ sin θ)( sinθ + sin θ) (cosθ cos θ)(cos θ) + ( sin θ ) d + sin θ 0( + sin θ) 0( + sin θ) When θ rad, d + sin sin + sin cos cos cos + sin 0 + sin (.80)(.05) (0.05)() , correc o significan figures 97 0, John Bird

8 5. The radius of curvaure, ρ, of par of a surface when deermining he surface ension of a liquid is given b: d + d ρ d / Finhe radius of curvaure (correc o significan figures) of he par of he surface having parameric equaions (a), a he poin (b) cos, sin a rad. (a), hence, hence a d, d d d ( ) a, d radius of curvaure, / d + + ( ) 7 ρ. d (b) cos, hence cos ( sin ) cos sin sin, hence sin cos an d sin cos sin d cos sin cos a rad, an , John Bird

9 d d ( an ) sec d cos sin cos sin cos sin a rad, d 0.90 cos sin radius of curvaure, / d + + ( ). ρ 5.9 d , John Bird