A). Let M be the set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints).
Let p(x,y) = max_[0,1]|x'(t) - y'(t)|.

B). Let M be the set of continuous functions on [0,1] and define
p(x,y) = integral from 0 to 1 of |x(t) - y(t)|dt. Does this define a metric space? ( Also a proof here please for the yes or no answer).

Solution Preview

A.1
By definition, a metric d(x,y) must satisfy condition d(x, y) = 0 if and only if x = y.

It fails for p(x,y) = max_[0,1]|x'(t) - y'(t)|, because two functions can differ by a constant but still have p(x,y) = 0.

Therefore (M,p) is not a metric space

A.2
The definition of metric contains four conditions:
d(x, y) ≥ 0 (non-negativity) ...

Please see attachment
Determine whether....converge or diverge
...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove...
Using binomial coefficients, derive a formula for the nth derivative of the product of two functions.
Suppose that f(x) has a continuous first

Let f(x) be a continuous function of one variable.
a) Give the definition of the derivative.
b) Use this definition to find the derivative of f(x)=x^2+2x-5
c) Evaluate f'(2)

Please see attachment
All the questions
Determine whether....converge or diverge
...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove...
Using binomial coefficients, derive a formula for the nth derivative of the product of two functions.
Suppose that f(x) has

Please see attachment
Determine whether....converge or diverge
...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove...
Using binomial coefficients, derive a formula for the nth derivative of the product of two functions.
Suppose that f(x) has a continuous fir

Please see attachment
Determine whether....converge or diverge
...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove...
Using binomial coefficients, derive a formula for the nth derivative of the product of two functions.
Suppose that f(x) has a continuous first

Please see attachment
Determine whether....converge or diverge
...derive a necessary condition for the equation...to have a rational root. Then use this condition to prove...
Using binomial coefficients, derive a formula for the nth derivative of the product of two functions.
Suppose that f(x) has a continuous first

Theorem: Let E and E' be metric spaces, with E compact and E' complete. Then the set of all continuousfunctions from E to E', with the distance between two such functions f and g taken to be max {d'( f(p), g(p) ) : p is an element of E} is a complete metric space. A sequence of points of this metric space converges if and only

Suppose that f(x) satisfies the functional equation
f(x + y) = f(x) + f(y)
for all x,y in R (the real numbers). Prove that if f(x) is continuous that f(x) = cx where c is a constant. What can you say about f(x) if it is allowed to be discontinuous?

a) Let M>0, and let f:[a,b]-->R be a function which is continuous on [a,b] and differentiable on (a,b), and such that |f'(x)| <= M for all x belonging to (a,b) (derivative of f is bounded). Show that for any x,y belonging to [a,b] we have the inequality |f(x)-f(y)| <= M|x-y|. *apply mean value theorem.
b) Let f:R-->R be a di