Practice Questions and Solutions (Not covered by the written assignments ) Parametric curves, Lengths, Surface Area, Curvature, Frenet’s Frame 1.Find a parametric representation which does not involve radicals, for the curve that is the intersection of the following surfaces: (a)The cone 2 2 z x y = + and the plane 1 z y = + (b)The cylinder and the plane 2 2 1 x y + = 2 y z + = (c)(c) The cylinders: 2 2 , 1 z x y x = = − . Solution: (a)We have 2 2 2 2 2 2 1 1 1 2 ( 2 x y y x y y y y x + = + ⇒ + = + + ⇒ = −1 ) . If set x t= we obtain 2 2 1 1 () , ( 1), ( 1) 2 2 r t t t t =< − + > . (b)If set cos x t= , then and sin y t= 2 sin z t= − . (c)First we notice that . If set 2 2 1 z y + = cos z t= we get sin y t= and 2 1 sin x t= − REMARK: The above parameterizations are not unique! 2.(a) Show that the curve ( ) ( sin ) ( cos ) t t tr e i e t j e t k = + + is on a cone. (b) Find an equation for the tangent line at 0 t= . Solution: (a), hence the cure is on the cone 2 2 2 2 2 [( )] [ ( )] ( sin ) ( cos ) [ ( )] t t ty t z t e t e t e x t + = + = = 2 2 2 2 x y z = + . (b)0 0 , (cos sin ), (cos sin ) 1 ,1 ,1 t t tttdru e e t t e t t dt= = = = < + − > =< > ; At 0 t= we get the point (1,0,1) on the curve. An equation for the tangent line at (1,0,1) is 0 r r tu = + , , 1, 0,1 1,1,1 x y z t ⇔< >=< > + < > 1 , , 1 x t y t z t ⇔ = + = = + . 3.(a) At what point do the curves 2 1 ( 1 ) (3 ) r ti t j t k = + − + + , 2 2 (3 ) ( 2) ( ) r s i s j s = − + − + kintersect? (c)Find their angle of intersection at the intersection point (i.e. the angle between the tangent vectors at the point of intersection). Solution: (a)We must find tand s which satisfy the following equations: 2 3, 1 2, 3 t s t s t s 2 − = − = − + = . We obtain and the point of intersection (1,0,4). 1 and 2 t s = =
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