chapter three

Question 1

Definition of gradient for f(x, y)

Answer 1

∇f(x, y) = (∂f/∂x) i + (∂f/∂y) j

Question 2

Definition of gradient for f(x, y, z)

Answer 2

∇f(x, y, z) = (∂f/∂x) i + (∂f/∂y) j + (∂f/∂z) k

Question 3

Directional derivative

Answer 3

Dû g(x0, y0, z0) = (∇g · û) evaluated at (x0, y0, z0).
It measures the rate of change of g in direction û.

Question 4

properties of directional derivatives

Answer 4

(a) If ∇g = 0 → all directional derivatives = 0. (b) If ∇g ≠ 0 →

Direction of maximum increase = ∇g, value = |∇g|

Direction of maximum decrease = −∇g, value = −|∇g|

Question 5

Tangent plane and normal vector

Answer 5

Tangent plane T: plane that just touches surface f(x, y, z) = c at point P.

Normal vector nP: perpendicular to every vector in tangent plane at P.

Question 6

gradient and normal vector

Answer 6

If f(x0, y0, z0) = c and (∇f)P ≠ 0, then (∇f)P is normal to the surface f(x, y, z) = c at P.

Question 7

equation of tangent plane

Answer 7

At point P = (x0, y0, z0): (x − x0)(∂f/∂x)P + (y − y0)(∂f/∂y)P + (z − z0)(∂f/∂z)P = 0 Form: ax + by + cz = d, with coefficients from partial derivatives at P.