Directional Derivatives & Gradients (14.5)

• Keyword: GRADIENT VECTOR

• Problem: "Find ∇f∇f for f(x,y)f(x,y)."

• Steps to Solve:

  1. Compute fxfx​ and fyfy​

  2. Write as vector: ∇f=⟨fx,fy⟩∇f=⟨fx​,fy​⟩

  3. For 3D: ∇f=⟨fx,fy,fz⟩∇f=⟨fx​,fy​,fz​⟩

• Keyword: DIRECTIONAL DERIVATIVE

• Problem: "Find directional derivative of f at P in direction of vector v."

• Steps to Solve:

  1. Compute ∇f(P)∇f(P)

  2. Convert v to unit vector: u⃗=v⃗∣∣v⃗∣∣u=∣∣v∣∣v​

  3. Compute Du⃗f(P)=∇f(P)⋅u⃗Du​f(P)=∇f(P)⋅u

• Keyword: MAX RATE OF CHANGE

• Problem: "Find maximum rate of change of f at P and direction in which it occurs."

• Steps to Solve:

  1. Compute ∇f(P)∇f(P)

  2. Maximum rate = ∣∣∇f(P)∣∣∣∣∇f(P)∣∣

  3. Direction = ∇f(P)∣∣∇f(P)∣∣∣∣∇f(P)∣∣∇f(P)​ (unit vector)

• Keyword: GRADIENT & LEVEL CURVES

• Problem: "Relationship between gradient and level curves/surfaces."

• Steps to Solve:

  1. ∇f∇f is perpendicular to level curves/surfaces

  2. Points in direction of greatest increase

  3. Zero gradient at critical points