Question

Find the (linear) equation of the tangent plane to the surface z=5y2−3x2+x at the point (2,−1,−5). Your answer should be in the form of an equation, e.g., something like ????x+????y+cz=???? or ????(x−x0)+????(y−y0)+c(z−z0)=0 would work.

Answer 1

Answer:

The equation of the tangent plane to the surface is given by $6x+10y+z=-3$

Step-by-step explanation:

We can find the normal of the surface using the gradient over f(x,y,z), where the function is

$f(x,y,z)=-3x^2+x+5y^2-z=0$

And the gradient is

$\nabla f(x,y,z) =$

Then the normal at the point (2,-1, -5) is

$\vec n =\nabla f(2,-1,-5)\\\vec n = \\\vec n =$

Then the equation of the tangent plane to the surface is given by

$\vec n \cdot (P-P_0)=\vec0$

Replacing the given point and the normal we get

$\cdot =0\\-6(x-2)-10(y+1)-(z+5)=0$

We can simplify a bit to get into standard form

$-6x+12-10y-10-z-5=0\\-6x-10y-z-3=0\\-6x-10y-z=3\\\\6x+10y+z=-3$