Using a tangent plane to approximate

Question

Use a tangent plane at (7,2) to approximate the value of the following function at the point (7.1,1.9):

-4x^28y^2+7z=263

Present your answer accurate to 4 decimal places.

Answer 1

The equation for the tangent plane is

z = f(x₀,y₀) + f_x(x₀,y₀)(x-x₀) + f_y(x₀,y₀)(y-y₀),

where the subscripts represent partial derivatives.

This means we need to solve for z and create the partial derivatives.

z = f(x,y) = (263 + 4x²-8y²)/7

I'm assuming this is the equation you meant.

Now we need to just crank everything out.

f_x(x,y) = 8/7 x

f_x(7,2) = 8

f_y(x,y) = -16/7 y

f_y(7,2) = -32/7

f(7,2) = 61

Tangent plane:

z = 61 + 8(x-7) - 32/7(y-2)

Approximation:

f(7.1,1.9) ≈ 61 + 8(7.1-7) - 32/7(1.9-2) ≈ 62.2571

Checking by finding actual answer.

z = (263 + 4*(7.1)²-8*(1.9)²)/7 = 62.2514

Make sure to double check my algebra.