Question

Consider the paraboloid and the point on the level curve. Compute the slope of the line tangent to the level curve at p and verify that the tangent line is orthogonal to the gradient at that point.

Answer 1

The slope of the tangent line to the level curve at point
`\( p \)` is equal to the negative reciprocal of the gradient of the paraboloid at that point, confirming that the tangent line is orthogonal to the gradient at
`\( p \).`

Step-by-step explanation:

Consider a paraboloid defined by the equation
`\( z = f(x, y) \)`, and let
`\( p = (x_0, y_0, z_0) \)` be a point on the level curve of this paraboloid. The gradient of the paraboloid at point
`\( p \)` is given by the vector
`\( \\abla f = ((\partial f)/(\partial x), (\partial f)/(\partial y)) \).` The level curve at point
`\( p \)` is represented by the equation
`\( f(x, y) = z_0 \).`

To find the slope of the tangent line to the level curve at
`\( p \),` we can use the fact that the gradient is orthogonal to the level curve. The tangent vector
`\( \mathbf{v} \)` to the level curve is parallel to the gradient, so
`\( \mathbf{v} = \lambda \\abla f \) for some scalar \( \lambda \)`. The slope of the tangent line is then given by
`\( (\Delta y)/(\Delta x) = (\lambda (\partial f)/(\partial y))/(\lambda (\partial f)/(\partial x)) \).` Simplifying, we find that the slope is
`\( -(\partial f)/(\partial x) / (\partial f)/(\partial y) \).`

Now, the negative reciprocal of the gradient vector is \( -1/\lambda \\abla f \), which is equivalent to
`\( -(\partial f)/(\partial y) / (\partial f)/(\partial x) \).` Comparing this with the slope of the tangent line, we observe that they are equal, confirming that the tangent line is orthogonal to the gradient at point
`\( p \).` This relationship holds for any point on the level curve of the given paraboloid.