Question

Exercise 1. Tangent plane (15 pts) Let (5) be the surface given by the following equation x+y2 = 1 + z? An equation of the tangent plane to (S) at A(1,2,2) is a 2x + 4y - Az = 1 b. x + y - z=0 c. r + 2y - 2z = 1 d. x + y - z = 2 e. None of the above o b ОО c. d e

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Abdul-Rahim Taysir

Answer 1

= 1+z2 Given: Equation of surface. x² + y 2 x² + y - 1 Point A (1,2,2) Then you have to find tangent plane at plane at Point A (1,2,2) Differentiate equation is w.aut e.dit be. 2. (x² + y2 - 1) ax 2x (assume yas constant and apply chain sule) 2(22) az ƏZ an (ac) 2g) acho axe d doe (30) noch-1) V az az 2x ga c V az doc o/N

(xo. Yo. 2o) - (1,2,2) 22 an (7) (122) (1,2,2) 2 -too-lob Again Differentiate with respect to y chain sule) 2. (32 + y2-1) sy ay dz2 2 (0) 2CD + a(z2 az 22 sy 2 (42) sy (7) 2 ry oy 2: as constant) 22. az ay zy az ay y I Point az loles sy

y (no. yo. 2o) use equation of tangent plane formula 22 (7-70 ) - ( 23 oc-not (yoyo) (no. Yo, ro) Z-2) = ģ ( 3-1) + (y-2) (7-2) -10 ( 0-1) + 219-21) 2( 2-2) (orel) + 24-2) az - 4 = x-1+2y- & x+2y-27 = 1 С. ANS