Unit 1

Question 1

What is a gradient vector?

Answer 1

For a function of n variables, the GV is the derivative function

Question 2

How is the gradient vector NOT denoted?

Answer 2

f’(x)

Question 3

What does the i’th element or df/dx(i) tell us?

Answer 3

The rate of change of f if x(i) is increased slightly and all other x(j) are fixed

Question 4

What is the direction of vector df/dx?

Answer 4

The direction in which f is increasing most rapidly (the steepest ascent direction) - always orthogonal to the contour at x

Question 5

What is the magnitude of the derivative of f wrt the vector x?

Answer 5

It is the slope/rate or change of f in the steepest direction

Question 6

What are directional derivatives?

Answer 6

The rate of change of f along line from vector(a) to vector(a+b)

Question 7

What is a critical point, x*?

Answer 7

A solution of the first order condition

Question 8

See bottom of unit 1 page 1 my notes on vector differentiation and top of next side on hessian matrix differentiation

Answer 8

Now

Question 9

What does Young’s theorem state?

Answer 9

That the order of partial differentiation doesn’t matter:

d^2f/dxdy = d^2f/dydx

Question 10

If we don’t assume A is symmetric, what is the hessian?

Answer 10

A+A^T