Homework Statement
Integral of ...
h(x) = x^5 + x^5 + [cos(x)]^6
Homework Equations
The Attempt at a Solution
so it would be
1/6x^6 + 1/6x^6 + 1/7[sin(x)]^7
is this correct?
Yea, I just realized that my H(5) was wrong because I accidently counted 11111, so the pattern here is H(n) = H(n-2) + H(n-1) for n >= 3 or (n > 2)
so this is just fibonnachi sequence, I'll work on getting the closed form of this and get back with what I have. Thank you
Sorry I worded poorly..
I should say that substrings of 1's should be even. (given 0 is a delimeter)
eg 10111 is broken down into 1 and 111 (all odd)
1011 is 1 and 11 (odd / even) but all has to be even so this is not right.
Homework Statement
Given binary string of length n.
substrings of 1's should be even. (given 0 is a delimeter)
eg) 10111 is broken down into 1 and 111 (all odd)
so, for example string with H(n = 4)
0000 1100 0110 0011 1111
there are 5 of them.
H(n = 3)
000 110 011
there are 3 of...
Homework Statement
1. Joshua's afternoon bus is on-time 60% of the time. What is the
probability that it will be on time all five times this week?
State any assumptions that you are relying on.
2. In this trial, a fair 6-sided die will be rolled until two consecutive sixes
have been...
Homework Statement
Question asks to prove that all binary string that begins and ends with the same number(bit) has an even number of occurences of substrings from {0,1}
Hint: you may find useful to combine claims about strings that start and end with different bit
Homework Equations
{0}...
Homework Statement
The induction question is. for all natural n, n4 <= 4n + 17
Base case: 0 Works, since 0 < 1 + 17 then,
I assume that for all n in natural, n4 <= 4n + 17 holds.
Now I believe I need to show that, 4(n4) <= 4(4n + 17)
that is, 4n+1 + 17 >= (n+1)4
To do so, I prove, 4n4 >=...
ok I think i figured out the trick.
I expand both sides, but on the left side i make
n3 + n3 + n3 + 3n and compare it to the right side term by term to prove that its greater for all n > 4.
am I right with this approach?
> 3((n3) + n ) by I.H
I am still unsure if I got it right...
I said in my induction hypothesis that, n3 + n < 3n
but while I am trying to prove that p(k+1) works for all k,
I think I am using assuming this line... which is the p(k+1) that I am trying to prove. 3(n3 + n) < 3(3n) by Induction...
Homework Statement
show n3 + n < 3n for all n >= 4
Homework Equations
The Attempt at a Solution
I.H : n3 + n < n for all n >= 4
3(n3 + n) < 3(3n)
then (3n+1) = 3 x 3 n
> 3((n3) + n ) by I.H...
Homework Statement
If a question states event A and B are "not" independent, does it mean that they are Mutually Exclusive?
My brain is having hard time accepting that if they are not dependent, then they don't neccesarily have to be 'dependent'. Kinda like if its not hot, then its cold. it...
I am currently working on a very very similar question on my own. But, the problem I have is finding the P(Random 2 are good).
Needed Information
P(all are good) = 0.7
P(Random 2 are good | all are good) = 1 (since given all are good, any random ones will be guranteed to be good)
P(Random 2 are...
P(A: No defects in lot) = 0.6 as given
Also would P(B|A: Two pents not defective given no defects in lot) be 1 ?
If we are given that there are no defects in the lot, the probability of two pen being not defective is 100% = 1
I am confused how to calculate P(B: Two pens were not defective)...
I am now tackling the 2nd question. I've read up on Bayem's Rule and starting to get the hang of it.
So we need to find the probability that the 2 pen out from a lot of 20 is not defective right?
This will be related to the equation
P(No defective in the lot|2/20 not defective) = [P(No...
Yes that's what I thought too. Maybe my formula is not right?
1 - P(A) gives everything else other than P(A)?
Oh my mistake, I remember p(A) covers the whole circle and not just moon shape...
so it is. 1 - P(A) + P(A and B)
ok I think I am getting the hang of it.
Can you check just if this is correct. I've drawn the diagram and it looks right to me.
P(~A or B)
The equation for this is 1-P(A)
oh so the overlapping part was where I got confused.
http://en.wikipedia.org/wiki/Symmetric_difference
So this is exactly how the diagram should look. and hence for the reason of 1-P(A or B)
Thanks for your response.
I tried drawing the venn diagram, I guess I tried before but it didn't click for me
P(not A and not B) = 1 - P(A or B)
does not A mean B, and not B mean A?
Because technically venn diagram consists of only 2 group of data. ones in A and the other in B.
Edit: so for...
Homework Statement
1. The question asks for the value of P(~A and ~B) - What equation to use?
The Attempt at a Solution
1. I am confused though, is P(~A and ~B) same as 1 - P(A and B)
or should I use, Pnot(A or B), and this is equivalent to 1 - P (A or B) ?
Extra question...
There is...
Homework Statement
Suppose that n identical planks, each of length 1 metre, are stacked flat on top of each other, with each one shifted a little further to the right, creating an ever-larger overhang. Prove the maximum span of this overhang is Fn = 1/2 [1 +1/2 + 1/3 + 1/4 + ... + 1/(n-1) ]...
it depends where 2nd plank is placed with relative to the base to determine how far 3rd plank has to be inorder to fall right ?
if its
spppaaa---------- something like this would have to occur
s-----------
-----------
centre of mass = (2nd plank distance + 3rd plank distance ) / 2...
if 2nd plank is placed...
space-------- <- any slight shift to the right will cause the 2nd plank to topple over.
0.5m--------
--------
btw when you said "2nd plank is 0.5long" you mean if it has an overhang of 0.5m right ?
cuz all panks are 1m :O
r = 1/2( 1 + 1/2 ) = 3m
if k plank...
thats what i sort of meant. any distance more than 0.5m will shift centre of mass to the right and fall.
But how do we write it in mathematical equation?!
and when do i use this equation to solve the equation using induction?
Prove the maximum span of this overhang is:
1/2(1 + 1/2 + 1/3...
im not how to represent it mathematically.
spa----------- (k+1)
sp----------
s---------
--------
the first k element topple over when the sum of distance shifted to the right from k to (k+1) is > 0.50 right? [distance is s + sp + spa + etc...]
hmm... as obvious it may seem... i don't see what I am suppose to find...
sp ------- (k+2)
s ------- (k+1)
------- k
what am i exactly looking for ?
the point when k+1 topples over?
I know that (k+1)st plank must have less than 0.5m of overhang in comparison to k th plank. Otherwise, the weight will shift and fall.
So...
Im not sure what where i would be using the formula of the maximum distance (k+1) can go without toppling... in this induction problem..
Im througly...
hmm...
Yea I actually tried to solving by using induction method.. so here is where i am at
F[lower subscript(k + 1 )] = 1/2(1 + 1/2 + 1/3 + ... + 1/k)
the above equation is after i substitute k+1 in n.
But how do i represent this F[lower subscript (k +1)] in terms of Fk ? alone ?
Im not...
Homework Statement
Suppose that n identical planks, each of length 1 metre, are stacked flat on top of each other, with each one shifted a little further to the right, creating an ever larger overhang. Prove the maximum span of this overhang is:
1/2(1 + 1/2 + 1/3 + ... + 1/n-1) metre...
Is there some mathematical way of showing that no matter how we distribute it, there's always one pigeonhole with k + 1 objects in it ?
I can represent this with an example, kn + 1 / n gives the number of objects to be placed in each hole to evenly distribute. But the remainder of 1 gives us...
Homework Statement
Pidgeonhole principle: If kn+1 objects are placed in n pigeonholes, then some pigeohoe contains atleast k+1 objects.
Homework Equations
The Attempt at a Solution
I completely understand this problem but I am not sure where or how to start.
If there are n =...
Homework Statement
Water is pumped into a tank. Volume V, is kept constant by continuos flow. The amount of salt S, depends on the amount of water that ahs been pumped in, call it X.
ds/dx = -S/V
Find the amount of water needed to eliminate 50% of the salt. Take v AS 10,000 gallons...
Homework Statement
Question Reads...
A hemispherical bowl 60cm in diameter is filled to within 5 cm of the top. Thirty minutes after the party starts, there are only 5cm of punch left at the bottom of th bowl.
a) how much punch was there at the beginning of the party?
b) How much punch was...
ok so since this is a shell method.
i would have to represent in y integral.
sqrt(y^2 - a ^2) = x
V= integral(0, a) 2pix(sqrt(y^2 - a ^2))dx
is that correct?
Homework Statement
Question Reads: A circular disk x ^2 + y^2 <= a ^ 2 , a > 0 is revolved about the line x = a.
Find the volume of the resulting solid.
Homework Equations
v = integral(a, b) (2pi)y [F(y) - G(y)] dy
The Attempt at a Solution
Im currently confused, should i...
Homework Statement
sin(x) = x - 1
Homework Equations
The Attempt at a Solution
i used the fact that -1 < sin(x) < 1
and set interval to [0, 2pi].
this gave me,
-1 < sin(x) < 1 and -1 < x - 1 < 2pi - 1
so therefore, since sin(x) < 1 < 2pi - 1
there must be x...
Homework Statement
Prove or give a counterexample for each of the following statements.
a) If f(x) is continuous, then the function lf(x)l is continuous.
b) if lf(x)l is continuous, then f(x) is continuous
Homework Equations
Given epsilon > 0 there is delta > 0 such that l x...
Homework Statement
Find an upper bound M for f(x) = abs ( x+2 / x-8 ) if abs(x-7) < 1/2
Homework Equations
The Attempt at a Solution
i first found set of x values using abs(x-7) < 1/2
which is 13/2 < x < 15/2.
Now, i believe i have to find other set of x values to compare to find...