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#1 Exercises » Integration of inverse functions » 2012-03-06 15:06:46

Kryptonis
Replies: 1

Could use a hint on how to begin this problem:

∫cos^-1 (x) dx

#2 Re: Exercises » Integration with Inverse Trig function » 2012-03-06 14:46:24

Thanx a bunch, was totally going in the wrong direction on this

#3 Exercises » Integration with Inverse Trig function » 2012-03-04 20:49:50

Kryptonis
Replies: 10

I'm having a bit of a time with this. I have tried integration by parts and just can't seem to get it right....

∫(3-3x)/(sqrt(64-9x^2)) dx

#4 Help Me ! » Length of a curve » 2011-03-11 22:06:12

Kryptonis
Replies: 2

x = t / (3+t) y = ln(3 + t) 0 ≤ t ≤ 4

Here's the problem, my real question lies in the answer...

-sqrt(u^2 + 9)/(u) + ln(u + sqrt(u^2 + 9)) with the lower @ 3 and the upper @ 7
where u = t + 3 and du = dt

When I integrate, I'm at a loss as to how i should be coming up with -sqrt(u^2 + 9)/(u) + ln(u + sqrt(u^2 + 9)).
As I keep coming up with -sqrt(u^2 + 9)/(u) + ln(sqrt(u^2 + 9).

#5 Re: Help Me ! » set help » 2011-03-10 16:05:12

Ty, got this one shortly after i posted. Thanx for the help though. Much appreciated!

#6 Re: Help Me ! » functions » 2011-03-10 16:02:58

ty sir, much appreciated!

#7 Re: Help Me ! » set help » 2011-03-04 02:13:14

Figured it out...
    A − B ⊆ C ≡ (A ∩ ¬B) ⊆ C                Def of Diff
    ≡ {x | (x ∈ A ∧ x  ¬∈ B) → x ∈ C}            Def of Diff, Def of subset
    ≡ {x | (x ¬∈  A ∨ x ∈ B) ∨ x ∈ C}            Log Equiv, De Morg
    ≡ {x | (x ∈ A → x ∈ B) ∪ x ∈ C}            Log Equiv, Def of Union
    ≡ A ⊆ B ∪C                                Set Builder Notation

#8 Help Me ! » set help » 2011-03-04 01:25:35

Kryptonis
Replies: 1

Let A, B and C be arbitrary sets taken from the positive integers.
    Prove the following statement: If A − B ⊆ C , then A ⊆ B ∪C

#9 Help Me ! » functions » 2011-03-04 01:24:29

Kryptonis
Replies: 3

a) Determine which of the followings are functions with domain X.
        i) (3 pts) X = {1, 3, 5, 7, 8} and R ={(1,7), (3,5), (5,3), (7, 7), (8,5)}
       
        ii) (3 pts) X = {-2, -1, 0, 1} and R = {(-2, 6), (0, 3), (1, -1)}
       
        iii) (3 pts) X is the set of real numbers and, for x ∈ X,
            g(x) = x^2 − 3x + 2, assume that the codomain is also X
       
        iv) (3 pts) X is the set of real numbers and, for x ∈ X,
            g(x) = sqrt(x^2 − 3x + 2) , assume that the codomain is also X
       
        v) (3 pts) X is the set of real numbers and, for x ∈ X, g(x) = log2 x , ,             assume that the codomain is also X
   
    b) Let Z = {...−2, −1, 0, 1, 2, ...} denote the set of integers. Suppose f :             Z→Z is a function, defined by:

        f (n) = {2 if is odd
               n/ 2 if is even

        i) (5pts) Prove or disprove that f is one-to-one (injective)

        ii) (5pts) Prove or disprove that f is onto (surjective).

#10 Help Me ! » set help » 2011-03-04 01:22:41

Kryptonis
Replies: 2

Prove the following assertions for sets A and B from an universe U without     using     Venn Diagrams or membership tables:
        a) (10 pts) A ⊆ B if and only if A ∩ ¬B = ∅.

        b) (10 pts) A ⊆ B if and only if  ¬A ∪ B = U.

#11 Help Me ! » Set operations, proof » 2011-03-03 16:27:07

Kryptonis
Replies: 1

Let A, B, and C be sets such that C  ⊂ B (i.e., C is a proper subset of B, or possibly C = B). Use appropriate set theoretic laws and theorems to prove     that (A – B) ∪ (B – C) = ¬C ∩ (A ∪ B). Be sure to explain each step of your proof.

This is what i have, and i have tried several ways just can't quite seem to get it right... sad   Any help would be great, ty in advance!

(A – B) ∪ (B – C)
    ≡{x | x ∈ A ∧ x   B}∪{x | x ∈ B ∧ x   C}            Def of diff
    ≡{x | (x ∈ A ∧ x   B) ∨ ( x ∈ B ∧ x   C)}            Def of union
    ≡{x | (x ∈ A ∧ x   B) ∨ ( x   B ∨ x ∈ C)}            De Morgan
    ≡{x | x ∈ A ∧ (x   B ∨ x ∈ C)}                       Idem & Assoc
    ≡{x | x ∈ A ∧ (x ∈ B ∧ x   C)}                    De Morgan
    ≡{x | x   C x ∧ (∈ A ∧ x ∈ B)}                    Assoc
    ≡{x | x   C x ∩ (∈ A ∩ x ∈ B)}                    Def of Inter
    ≡ ¬C ∩ (A ∩ B)                        Def of Set Build Notation

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