Real Analysis : Young's Inequality

Note: * = infinite Suppose that the function f:[0,*)->R is continuous and strictly increasing, with f(0) = 0 and f([0,*)) = [0,*). Then define F(x) = the integral from 0 to x of f and G(x) = the integral from 0 to x of f^-1 for all x>=0 (a) Prove Young's Inequality: ab <= F(a) + G(b) for all a >= 0 and b >= 0 (b) N ...continues

Differential Equations

Define: f(x) = (x^2)sin(1/x)+x if x doesn't equal 0 f(x) = 0 if x=0 Prove that the function f:R-> R is differentiable and that f'(0)=1. Also prove that there is no neighbourhood I of 0 such that the function f:I->R is increasing.

Taylor polynomials

Prove that: 1 + x/2 - (x^2)/8 < squareroot(1+x) < 1 + x/2 if x>0 In particular, show that 1.375 < squareroot(2) < 1.5

Approximation with Taylor polynomials

note:% just a symbol Let I be an open interval containing the point x.(x not), and suppose that the function f:I->R has a continuous third derivative with f'''(x)>0 for all x in I. Prove that if x.+h is in I, there is a unique number % = %(h) in the interval (0,1) such that f(x.+h) = f(x.) + f'(x.)h + f"(x.+%h)(h^2)/2 ...continues

Approximation with Taylor polynomials

Suppose that the function F:R->R has derivatives of all orders and that: F"(x) - F'(x) - F(x) = 0 for all x F(0)=1 and F'(0)=1 Find a recursive formula for the coefficients of the nth Taylor polynomial for F:R->R at x=0. Show that the Taylor expansion converges at every point.

Infinitely Differentiable Function that's not Analytic

Use the given information: the functions g:[a,b]->R and h:[a,b]->R are continuous with h(x) >= 0 for all x in [a,b], and there is a point c in (a,b) such that: the integral from a to b of h(x)g(x)dx = g(c) times the integral from a to b of h(x)dx. to show that the Cauchy Integral Remainder Theorem implies the Lagrang ...continues

Cauchy Integral Remiander Theorem Applied to Newton's Binomial Expansion

Apply the Cauchy Integral Remiander Theorem in the analysis of the expansion (Newton's binomial expansion): ln(1+x) = the sum from k=1 to infinite of (-1)^(k+1) times (x^k/k) if -1

Weierstrass Approximation Theorem

This question is about the Weierstrass Approximation Theorem Show that the Approximation Theorem does not hold if we replace I by R(real number system), by showing that if f(x) = e^x for all x, then f:R->R cannot be uniformly approximated by polynomials.

Weierstrass Approximation Theroem

Note: abs = absolute value Define f(x) = abs(x - 1/2) for 0 <=x <= 1. Use the proof of the Approximation Theorem to find an explicit polynomial p:R->R such that abs(f(x) - p(x)) < 1/4 for all x in [0,1]

Solving for n as a Factorial

Does there exist a natural number n such that [n!/(n-4)!] = 11,880 ? if so find n, if not explain why not (Hint: factor 11,880 into its prime factors)