Milica

Suppose that f: [a,b]→R is integrable. Define, for x∈[a,b] F(x)= integral a to x f(t)dt and G(x)=integral a to x F(t)dt

How do I prove that G is differentiable on (a,b)?

A) |x| is integrable on [-1,2]

b) x^(1/4) is integrable on [0,9]

c) the function h(x)= x^2 x∈[0,1]

2x+3 x∈(1,2]

is integrable on [0,3]

d) if f, g are integrable on [a,b] the so is f-g

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let A=[0,∞)⊆R.

prove that for all n∈N the function x→x^(1/n):A→R is increasing and continuous

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prove that every f: N→R is continuous

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5 of patients suffering from certain disease are selected to undergo a new treatment that is believed to increase the recovery rate from 30% to 50%. A person is randomly selected from those patients after completion of treatments and found to have recovered. What is the probability that the patient received the new treatment?

If A,B,C are sets prove that (A∖B)∖C⊆A∖(B∖C). Find a description of when it is that we have equality, and give an example where the inclusion is strict.

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a) Prove that if E,F are sets then E∩(E∪F)=E and E∪(E∩F)=E.

b) Suppose that A,B,C are sets satisfying A∩B=A∩C and A∪B=A∪C.

Prove that B=C.

c) Show that only assuming one of the conditions in the previous part is not sufficent to prove that B=C. In other words, give examples (one for each condition) where one condition holds and yet B≠C.

The following are instructions written in the machine language described in Appendix C. Translate them into English.

a. 7123 b. 40E1 c. A304 d. B100 e. 2BCD

What happens if the test is changed to read "( Count not 6 )"?

Count ← 1;

while ( Count not 7) do

(print the value assigned to Count and Count ← Count+3)

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Without using long division of polynomials, find the remainder in Q[x] when:

(i) x^3−2x+4 is divided by x−2;

(ii) x^4−7x^2+3 is divided by x+1;

(iii) x^40−8x^12+3 is divided by x^2−1.

Let Fp = Z/pZ be the field of p elements, where p is an odd prime. Label the

elements of Fp as 0,1,2, . . . , p−1 modulo p. Prove Wilson’s Theorem:

(p−1)! ≡ −1 (mod p)

as follows: observe that, by Fermat’s Theorem, the polynomial

f(x) = x^(p−1)− 1

in Fp[x] has 1,2, . . . , p − 1 as roots. Apply the Root Theorem to get a complete

factorization of

x^(p−1)−1 into linear factors, and then compare the constant term of

the product of the factors with the constant term of f(x).

Let R be a commutative ring with no zero divisors. Suppose z is an element of

R satisfying z

2

= 1. Show that z = 1 or z = −1.

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A really need help for this, my hw due in soon..

I really need a help with this, please..

I need help with this, please..

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