14.5 Gradient and Directional Derivatives

What is the gradient of a function f(x, y)?

The gradient is ∇f = (∂f/∂x, ∂f/∂y); for f(x, y, z), it's ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). It points in the direction of steepest increase.

What is the formula for the directional derivative of f in direction u?

Dₙf = ∇f ⋅ u, where u is a unit vector.

What is the maximum rate of change of a function at a point?

The magnitude of the gradient vector, ||∇f||. It occurs in the direction of ∇f.

What direction gives the minimum rate of change?

The direction opposite ∇f, with a rate of -||∇f||.

How do you compute a unit vector from a vector v = (a, b)?

u = (a/√(a² + b²), b/√(a² + b²))

What does it mean if the directional derivative is 0?

The function is not changing in that direction; the direction is tangent to a level curve or surface.

How is the gradient related to level curves or surfaces?

∇f is perpendicular (normal) to level curves in 2D and to level surfaces in 3D.

How can you estimate a small change in f using the gradient?

Δf ≈ ∇f ⋅ Δr, where Δr is a small vector (dx, dy).

What is the equation of the tangent plane to F(x, y, z) = k at point (a, b, c)?

Fₓ(x-a) + Fᵧ(y-b) + F_z(z-c) = 0

Why do we use a unit vector in the directional derivative formula?

To measure the rate of change per unit distance, not scaled by the length of the direction vector.

How does the gradient tell us the direction of fastest increase?

The gradient ∇f points exactly in that direction, and its magnitude gives the steepest slope.

How do you compute the gradient of f(x, y) = cos(x² + y)?

∇f = (–2x sin(x² + y), –sin(x² + y))

What is the Chain Rule for paths (parametrized curves)?

d/dt f(r(t)) = ∇f(r(t)) ⋅ r'(t)