Matlab program

Suppose that a rabbit is initially at point (0,100) and a fox is at (0,0). Suppose that the rabbit runs to the right at speed Vr = 5 ft/sec and the fox always runs toward the rabbit at speed Vf = 6 ft/sec. Write a Matlab program that determines to within 1 second, when the fox catches the rabbit. The program should also plot rabbit and fox positions. Use the modified Euler method to estimate Xf(t),Yf(t) and the time to capture. Please do not respond without a written program in Matlab.

The one norm

Prove rigorously that ||x + y||1 <= ||x||1 + ||y||1

Local minimum

Please see the attached file for the fully formatted problems. The profit P(t) for a firm as a function of time (t) is described by the equation P(t) = t — 3/8 t^2 + 1/24t^3 For 0 < t =< 10 show that there is exactly one local minimum, and find the value of t at which it occurs. Is the profit positive at this local minimum?

Real Analysis: Differential Equations (Leibnitz Formula)

Let I be an open interval and n be a natural number. Suppose that both f:I->R and g:I->R have n derivatives. Prove that fg:I->R has n derivatives, and we have the following formula called Leibnitz's formula: (fg)^n(x) = the sum as k=0,1,2,...n of(n choose k)f^k(x)g^(n-k)(x) for all x in I. Write the formula out explicitly for n=2 and n=3.

Real Analysis: Criteria for Integrability

Define f(x) = x^2 for all x in [0,1]. For each natural number n, compute L(f,Pn) and U(f,Pn), where Pn is the regular partition of [0,1] into n subintervals.Then use the Integrability Criterion to show that the function f:[0,1]->R is integrable.