Partial Differentiation

Question 1

How do you notate a function with two variables (independent of eachother)

Answer 1

f(x,y) = …

Question 2

Taylor’s Theorem for two variables

Answer 2

f(a+h, b+k) ≡ f(a,b) + hf’(a,b) + kf’(a,b) + (h^2/2!) f’’(a,b)+ (k^2/2!) f’’(a,b) + (2kh/2!) f’’(a,b)

Question 3

How do you work out if a stationary point is a maximum or minimum

Answer 3

• find points at fx = 0 and fy = 0
• D = fxx*fyy - fxy^2

D = 0 (unknown)
D < 0 = saddle point
D > 0:
- fxx < 0 = max
- fxx > 0 = min

Question 4

Work out the total differential of z = ax + by

Answer 4

dz = dx( fx ) + dy( fy )
(dx and dy don’t change)