Find an equation of the tangent plane to the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ at the point $( 1,2,2 )$ .
A) $2 x + y + z = 4$

B) $x + y + z = 5$
C) $x + 2 y + 2 z = 9$
D) $x + 4 y + 4 z = 17$
E) $2 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 10$
F) $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$

G) $x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } = 17$

H) $x ^ { 2 } + 4 y ^ { 2 } + 4 z ^ { 2 } = 33$

Question

Find an equation of the tangent plane to the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ at the point $( 1,2,2 )$ .
A) $2 x + y + z = 4$ 
 
B) $x + y + z = 5$ 
C) $x + 2 y + 2 z = 9$ 
D) $x + 4 y + 4 z = 17$ 
E) $2 x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 10$
F) $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$

G) $x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } = 17$

H) $x ^ { 2 } + 4 y ^ { 2 } + 4 z ^ { 2 } = 33$

Quizplus · Accepted Answer

The equation of the tangent plane to the surface $x^2 + y^2 + z^2 = 9$ at the point $(1,2,2)$ can be found by first computing the gradient of the given function, which is $\nabla f = (2x, 2y, 2z)$. At the point $(1,2,2)$, the gradient is $(2, 4, 4)$. The equation of the tangent plane is then given by $2(x-1) + 4(y-2) + 4(z-2) = 0$, which simplifies to $x + 2y + 2z = 9$.

Find an Equation of the Tangent Plane to the Surface x2+y2+z2=9x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9x2+y2+z2=9

Find an Equation of the Tangent Plane to the Surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$