Union of countable set

Prove that union of countable sets is countable. See attached file for full problem description.

Determine derivative

(See attached file for full problem description) Determine the derivative: 1) d/dx

Proof involving integral

(See attached file for full problem description) Assume that f is continuous on Reals and periodic with p. Show that for any a

Proof Involving Integral, Derivative and Double Derivative

See the attached file for full problem description. Assume that f, f’ and f’’’ are continuous on [a,b] and f(a)=f(b)=0. Then S b--> a f(x)f''(x)dx = -S b--> a (f'(x))^2 dx

Finding Derivatives : Fundamental Theory of Calculus and Chain Rule

Please see attached file for full problem description: Using the fundamental theorem of calculus and chain rule, For example, letting the expression equal F(x), and G(u) Determine the derivative:

Using theorems to determine derivative

(See attached file for full problem description) Using the fundamental theorem of calculus and chain rule, For example, letting the expression equal F(x), and G(u) So for 1) we would have F(x)=G(x ) By the chain rule, dF/dx(x)=(dG/du)|u= x (du/dx) =(1/ | u=x ) (2x) =2x// Determine the derivative: 1) d/dx ...continues

Proof that integral is increasing on interval

(See attached file for full problem description)

Proof function is increasing and express in different form.

The function is defined by: erf(x) = for each x 1. Show that erf is strictly increasing on (- ,+ ) 2. Express the integral in terms of the values of erf. Hint(May want to do a substitution)

Real Anaylsis : Metrics Proof

Let d be a metric in X. Prove that p(x,y)=(d(x,y))/(1+d(x,y)) is also a metric in X.

Let (X,d) be a metric space. Define a closed ball with center x and radius r to be the set B[x;r]={y:d(x,y)<=r}. Prove that B[x,r] is a closed set.

Let (X,d) be a metric space. Define a closed ball with center x and radius r to be the set B[x;r]={y:d(x,y)<=r}. Prove that B[x,r] is a closed set.