Real Analysis Answers

Prove that

x<log(1/1-x)< x/1-x, 0<x<1

https://docs.google.com/document/d/1tRJ3QxM6Xb8GOfCGQn1lSnAepw72MDmfJu1iRO1zCxU/edit

please solve the entire question

Find the area of the portion of the paraboloid x=4-y²-z²that lies above the ring 1<=y²+z²<=4in the yz plane.

Define uniform convergence of a sequence of functions. Discuss the uniform convergence of the sequence nx^2+1/nx+1 the interval  [1. 2]

a differentiable real valued function f has at the point (1, 2), directional derivatives +2 in the direction toward (2, 2) and -2 in the direction toward (1, 1). determine the gradient vector at (1, 2) and compute the directional derivative in the direction toward (4, 6).

Show that sinx and cosx are a bound variation on a finite interval

Is f(x)=1/x^2/3 in L₃[1 2] ?

Is 1/x^2/3 continuous on [1 2] ?

a differentiable real valued function f has at the point (1, 2), directional derivatives +2 in the direction toward (2, 2) and -2 in the direction toward (1, 1). determine the gradient vector at (1, 2) and compute the directional derivative in the direction toward (4, 6).

Consider the sequence {xn}, which is defined by

x1 = 1, xn+1 = xn +

1/

x1 + x2 + · · · + xn

, n ∈ N.

Does {xn} converge? Justify your answer.