Calculus RⁿMedium

Gradient and tangent plane

For

f(x,y)=x^2+y^2

, calculate |∇f(3,4)| and write the tangent plane.

Given data

f(x,y) = x²+y²(x₀,y₀) = (3,4)

Steps

0 / 3

Calculate ∂f/∂x and ∂f/∂y.
Write your working — the numerical result must appear in the text (logica)
Calculate |∇f(3,4)|.
Write the tangent plane z = f₀ + f_x(x−3) + f_y(y−4) and calculate f(3,4).

Full worked solution

Calculate ∂f/∂x and ∂f/∂y.
$f_x = 2x = 6, \quad f_y = 2y = 8$
$f_x = 2x$ , $f_y = 2y$ . At $(3,4)$ : $f_x(3,4) = 6$ , $f_y(3,4) = 8$ . The gradient is $\nabla f = (2x, 2y)$ .
Calculate |∇f(3,4)|.
$|\nabla f| = \sqrt{6^2+8^2} = 10$
$|\nabla f(3,4)| = \sqrt{f_x^2 + f_y^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = \mathbf{10}$ . The magnitude of the gradient is the Euclidean norm of its components.
Write the tangent plane z = f₀ + f_x(x−3) + f_y(y−4) and calculate f(3,4).
$f(3,4)=25 \Rightarrow z=25+6(x-3)+8(y-4)$
$f(3,4) = 3^2 + 4^2 = 9 + 16 = 25$ . Tangent plane: $z = 25 + 6(x-3) + 8(y-4) = 6x - 18 + 8y - 32 + 25 = \mathbf{6x + 8y - 25}$ .

Result:|∇f(3,4)| = 10; tangent plane:

z = 6x+8y-25

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