Let
r e-1/x2 i(x) = i 0
x740 x = 0
Show that the nth derivative of 1(x) exists for all n E N. Please justify all steps and be rigorous because it is an analysis problem. (Note: The problem falls under the chapter on Differentiability on IR in the section entitled The Mean Value Theorem.)

It is observed that: "The function g is non-zero, infinitely often differentiable, and any derivative of g at x = 0 equals zero." Now, how?:

This function is special, because when dealing with power series representations: if one wants to find a power series representation, one could apply Taylor's formula to find the coefficients of the power series, say, centered at zero. That requires that the function under consideration has to be infinitely often differentiable. This function is. Taylor's formula involves derivatives of the function at the origin. In this case, they are all zero. Hence, the power series associated with this function would be identically equal to zero. But then it does not represent the original function.
In other words, there are functions for which you can use Taylor's theorem to find a convergent power series, ...

Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and the binomial

Find the derivative;
G(v)= (v^3-1)/(v^3+1)
Find the limit;
lim(sin3x)/(sin5x)
x->0
Find the derivative;
R(w)= (cosw)/(1-sinw)
H(o)=(1+seco)/(1-seco)
Find the derivative;
F(x)= cos(3x^2)+{cos^2}3x
N(x)=(sin5x-cos5x)^5
"Assume that the equation determines a differentiable function f such that y=f(x),

At 2:00 pm a car's speedometer reads 30 mi/h. At 2:10 pm it reads 50 mi/h. Use the meanvaluetheorem to show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h^2. Please show line by line work and be as clear as possible.

Using the Fundamental Theorem of Calculus I need to find the solution of the following problems. Can you explain how?
Please see the attached file for the fully formatted problems.

Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

Set up a short problem related to your work environment to calculate the probability(ies) of an event happening. Then use Bayes' Theorem to revise the probability. Show all your work.