1819 NEED BOTH PLZ PZLZ PLZ Find an equation of the tangent

18-19

NEED BOTH PLZ PZLZ PLZ

Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. r (u, v) = 2u cos h (v)i + 2 u sin h (v)j + 1/2 u^2 k, (-4, 0, 2) Find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the plane r (u, v) = 4ui - vj + vk, where 0 lessthanorequalto u lessthanorequalto 4 and 0 lessthanorequalto u lessthanorequalto 4

Solution

Given that

r(u,v) = 2ucosh(v) i +2u sinh(v) j +(1/2) u²k =>(1)

let s=x i+y j+z k

let s is the tangent plane to the eq (1)

then by tngent property

r.s = 0

=> (2u cosh(v) i+2u sinh(v) j+(1/2)u²k) . (xi+yj+zk) = 0

=> (2ux cosh(v)+2uy sinh(v)+(1/2)u²z) = 0

then this equation at (-4,0,2) is

=> 2u(-4) cosh(v)+ 2u(0) sinh(v) + (1/2)u²(2) = 0

=> -8u cosh(v) +0 + u² =0

=> u²- 8u cosh(v) = 0