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 rab ...continues

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? ...continues

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 ...continues

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.

Real Analysis: Criteria for Integrability

Suppose that the function f:[a,b]->R is integrable and there is a postive number m such that f(x) >= m for all x in [a,b]. Show that the reciprocal function 1/f:[a,b]->R is integrable by proving that for each partition P of the interval [a,b], U(1/f,P) - L(1/f,P) <= 1/m^2[U(f,P) - L(f,P)]

Real Analysis: Criteria for Integrability

Suppose the continuous function f:[a,b]->R has the property that: The integral from c to d f<=0 whenever a<=c

Integration: Cauchy-Schwarz Inequality

Suppose that the functions g:[a,b]-> R are continuous. Prove that: The integral from a to b of gf <= (the square root of the integral from a to b of g^2) multiplied by (the square root from a to b of f^2)

Integration

Note: pi = 3.14...... Prove that (2/pi)x <= sinx <= x if 0 <= x <= pi/2, and use this to prove that: 1 <= the integral from 0 to pi/2 of sinx/x dx <= pi/2

Real Analysis: Geometric Interpretation in Terms of Areas

Note: * = infinite Suppose that the function f:[0,*) -> R is continuous and strictly increasing, and that f:(0,*) -> R is differentiable. Moreover, assume f(0) = 0. Consider the formula: the integral from 0 to x of f + the integral from 0 to f(x) of f^-1 =xf(x) for all x>= 0. How can I provide a geometric interpretation ...continues