Let (W)to be a Brownian Motion. (a) Let 1 if Wt > 0, Xt=sgn(W)= 0 t if W₁ = 0, - 1 if Wt < 0. Check that (X+) is an Itô-integrable process. (b) Let Y₁ = [X, W. Find E[Y₁] and Var(Y;). 0 (c) Check that (W2) is an Itô-integrable process. (d) Consider the function f : [0,T] → R defined by f(t) = ta, where a € R. For which f(s) dW, well-defined? When it is, find E[Z₁] values of a is the process Z₁ = and Var(Z). Bonus question: In part (c), consider a partition 0 = to < ··· < tn = T. Prove that n-1 the simple process W221 +) (+) converges to W2 in the L²-sense. i=0 1

Let F(x) "X F'(x) = f 1 dt. Then 5 (t + t³)17

the linearization of (fi,f2) = (x²y+ 3xy², 3x³ + 2xy) at (1,0) to approximate the value at (0.9,-0.1).

= xye-(a+y*+z²). Find all points (x, y, z) such that Vf(x,y, z) Question 2. Consider the function f(x, y, z) (0,0,0).

Find and classify all stationary points of the function f(x,y) = x³ + xy + y² − x.

Let \n (xy) + y ² = x6 + 3 find dy

Draw the bifurcation diagram for dy/dt = y(a-y)

Write the system æ' = e"x 4ty + 9 sin(t), y' = 7 tan(t)y + 9x – 4 cos(t) in the form d = P(t) dt Use prime notation for derivatives and write r and x', etc., instead of x(t), x'(t), or 4

1) Find the linearization L(x) of f(x) at x = a.f(x) = 2x2 + 4x - 1, a = 4 2) Use implicit differentiation to find dy/dx and d2y/dx2.xy + 3 = y, at the point (4, -1) 3) Provide an appropriate response.The line that is normal to the curve x2 - xy + y2 = 25 at (5, 5) intersects the curve at what other point? 4) Find the second derivative.y = 4x2 + 10x + 2x-3 5) For the given function, find each of the following:(A) Intercepts(B) Vertex(C) Maximum or minimum(D) Rangef(x) = (x + 1)2 - 4 6) Find the slope of the line tangent to the curve y = 3x2 + 9x - 5 at the point (-2, -11). 7) Use implicit differentiation to find dy/dx.cos xy + x4 = y4 Pleas i want answer this Questions (1 -7) Analytical Geometry & Calculus

Evaluate ∫ ▽f*dr Where f(x,y,z)=cos(πx)+sin(πy)-xyz and C is any path that starts at (1,1/2,2) and ends at (2,1,-1)