Problem 11 Express the function f defined below using step functions and find its Laplace trans- form. f(t) -{ 0

Compute the gradients of each of the following functions: o = xyz & = ln [(x² - y²)-¹/2] r = || = √√√x² + y² + x² 1/r Note: Remember that your tutorial has Hints in it, at the end of each chapter. If you're stuck, they are often quite useful! • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, shift and - for subscripts, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx= x², Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {A₂, Ay, A₂}. Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. Vo= V = Vr = V= (2A, 2A, 2A₂).

Question 6. Consider the function f: R2 R determined by f(x)=x | 12 24 2 3 +2. -> (a) Find the gradient and Hessian of f at the point (1, 1). (b) Find the directional derivative of f at (1, 1) in the direction of the maximal rate of increase. (c) Find a point that satisfies the first order necessary condition. Does the point also satisfy the second order necessary condition for a minimum? [15 marks]