chris

the question states (x^4+2x^3)/(x^3+4x^2+5x+2) now I'm have difficulties understanding how to transform this to the 1st step in the answer which states that it equals

(x-2)+(3x^2+8x+4)/(x^3+4x^2+5x+2)

if somebody can explain the process I would be grateful

p.s. unless you can be helpful don't bother answering I have an exam coming up in a few days so really want to understand what I am to do so I don't fail (which is more than likely as I am struggling)

thanks in advance

I have an exam coming up and need to understand how this answer has been found ( step by step) as it has me stumped. the equation with the answer is (x^2+3)/(x-3)=x+3+(12)/(x-3)

so if someone could help me understand how the answer is derived I would be sooooo grateful

cheers in advance

I have the graph of sech (x) what I need is

i) use the graph to explain why sech (x) is even, and so the derivative is odd.

ii) what formula gives the derivative of this function.

iii) find points of inflexion on the graph. precisely locate them, by calculation

I'm pretty sure I know the 1st 2 but need to see it from someone that knows what they are doing (unlike me lol) and as for the 3rd not sure what the calculation would be

thanks in advance

in a simple 2 dimensional model for the lift given by a tapered aircraft wing of length h whose cross section shape remains similar, a formula was obtained:

. . . . . . . . . . n . . . . . . . . . . n . . . . . . . . . . . . . h

L=lim(n→∞) ∑ Li =lim(n→∞) ∑ Lo (1-(ξi)/h)δxi = ∫ Lo (1-(x)/h) dx

. . . . . . . . . i=1 . . . . . . . . i=1 . . . . . . . . . . . . o

Lo is the lift at the thick end of the wing (by the fuselage), and Li is the lift from the cross section of the wing at point ξi from the fuselage

i) what mathematical process converts this limiting sum to the integral?

ii) evaluate the lift L in the case that Lo = h = 1

if someone could show me how to do this would be much appreciated.

thanks in advance

this is the problem

Find the centre of mass of S, a solid of revolution obtained by rotating the region R above the interval [1,2] of the x-axis and below the curve y=sqrt(3x-x^2-2). The region R is rotated a full turn about the x-axis. S has density p(x)=3x.

now I was shown this as an answer

Due to the circular symmetry of S the centre of mass will lie on the axis of rotation, namely the x-axis. To find the exact distance (x_c) along the x-axis we need:

x_c * ∫dm = ∫x dm

Where dm = ρ πr² dx

So dm = 3x π (3x-x²-2) dx

So

∫[1,2] dm = ∫[1,2] 3πx(3x-x²-2) dx

= 3π∫[1,2] 3x²-x³-2x dx

= 3π (x³ - x^4/4 - x²) [1,2]

= 3π/4

So

∫[1,2] x dm = ∫[1,2] x 3πx(3x-x²-2) dx

= 3π∫[1,2] 3x³-x^4-2x² dx

= 3π (3x^4/4 - x^5/5 - 2x³/3) [1,2]

= 23π/20

So x_c = (23π/20)/(3π/4)

= 23/15

but I don't think its would be right myself as they have removed the 2 from the equation to get

3π (x³ - x^4/4 - x²) [1,2] in x=1 part

then later on in the x=2 part they have left it in and ended up with

3π (3x^4/4 - x^5/5 - 2x³/3) [1,2]

am I right in thinking they have made an error with the math as in x=1 they have 2x ending as x² but in x=2 they have 2x ending as 2x³

or am I getting my math wrong if so please explain why its different as it doesn't seem right lol

thanx in advance

find the numerical value of the Integral

(NOTE)

2

ʃ is represented by {$} as formatting on here is warped and wont show as i typed it

1

{$} y(x) dx = {$} x ² + 2x dx

-dividing the interval [1,2] into n equal sub-intervals

-summing areas of rectangles based on the sub-intervals, beneath y(x),

-and taking an appropriate limit

if someone can show how to work this out would be most grateful thanks

using a co-factor expansion, find an algebraic expression for

│ 1 a b │

│-a 1 c │

│-b -c 1 │

= 1((1*1) + (-c * c)) - a((1 * 1) + (-c * c)) - b((1 * 1) + (-c * c)) = 1(1 - c²) - a(1 - c²) - b(1 - c²) = 1 - c² - a + ac² - b + bc²

have I done this correctly and have I got right answer

thanx in advance

just want to know if I have done this right and if have the right answer

│ 2 0 0 0 0 0│

│13 -2 0 0 0 0│

│ 0 0 2 0 5 61│

│ 0 0 0 1 7 17│

│ 0 0 0 0 -1 -11│

│ 0 0 0 0 0 -2 │

= (0)(0)(5)(17)(0)(0) + (0)(0)(0)(7)(-11)(0) + (2)(-2)(2)(1)(-1)(-2) – (0)(0)(0)(0)(0)(0) – (0)(0)(1)(5)(0)(2) – (0)(0)(7)(61)(13)(0)

=(0) + (0) + (-12) – (0) – (0) – (0) = -12 so determinant = -12

I have a pc with no wireless adapter and will be getting a laptop soon. I would like to know if there is a setup that will allow me to plug in the 3G dongle which connects via usb so It can transmit as a wireless connection so I don't have to keep removing the dongle and also so I can connect my Wii to the internet. I don't have a wireless adapter in pc YET but will get one if it can be done. don't want to have pc running while laptop is on hence the question. hope what I am asking makes sense to someone besides me lol.

thanx in advance

An integer unimodular matrix has all entries integers, and determinant one.

1) an identity matrix I2 is integer unimodular explain why?

2) is a product of two integer unimodular matrices also integer unimodular?

3) a homogeneous linear transformation has matrix A ( integer unimodular ). what can be said about the image of a point with integer co-ordinates? find the image of the unit square ( vertices (0,0), (1,0), (0,1) and (1,1)).

I am unsure of what this question is actually asking me to do if anyone can decipher I'd be grateful

Let A and B be non-singular matrices of the same order Aˉ¹=C and Bˉ¹=D, write down the inverse of the matrix C ͭ B (note ͭ is the transpose), in terms of A and D.

is it asking me to write if Aˉ¹=C and Bˉ¹=D then C ͭ B = A ͭ D or to show the matrices

∞

∑((3k+5)/(k^2-2k))^p

k=1

now I am required to consider and for each p ϵ R show this series {converges if p>1, diverges if p<=1

so can someone please explain what he question is actually asking and how to solve this I was told to use Integration by substitution but not sure how to go about it

thanks in advance

I need to find the numerical sum of the series

∞

∑(6k+3)/(k^2(k+1)^2)

k=1

I also have to find the error involved in approximating this sum by truncating the 5th term.

I have the hint of {-3k^-2} and telescopic sums

now for what I have so far (not sure if this is right or not)

(6k+3)/(k^2(k+1)^2) = 6k/k^2 + 3/(k+1)^2 = a/k^2 + B/(k+1)^2

so (6k+3)/(k^2(k+1)^2) = (A(k+1)^2)/(k^2) + B(k^2)//(k+1)^2

.: 6k+3 = A(k+1)^2 + B(k^2)

so if k=0 then 0+3 = A(0+1)^2 +b(0^2) therefore A=3

if k=-1 then -6+3 = A(-1+1)^2 + B(-1^2) therefore B = -3

so if A=3 and B= -3 then 3/k^2 - 3/(k+1)^2

so that's what I have so far can someone please tell me if this is right to this point and if so where to go from here and if not help lol. and with the 5th term am I right in thinking that's the telescopic sum 5th term being a1+a2+a3+a4+a5....

consider the convergence of the series

∞

∑(-1)ⁿ/(5n+9)

n=1

i) show this series is not absolutely convergent

ii)does this series converge? justify use of convergence tests or general theory

as Im new to this I just need to see it done correctly I think Im SLOWLY getting the idea but not sure

thanks in advance

I need help with understanding the following

∞

∑(3n+7)/(2n²-n)

n=1

I have been shown with aid of a program that The limit of (3n+7)/(n(2n-1) as n approaches ∞ does not equal 0.

what I would like to know so I can understand this answer is, what it would look like as a sum so I know what I'm actually doing to prove it is divergent as apposed to convergent

using the definition of null sequence, show { ͣ n}= (-1)ⁿ / n² is a null sequence.

so (-1)ⁿ / n² = 1/n²

now I know this is a null sequence but as this is an assignment I need to show the proof, what should I show in my answer

use the method of telescopic sums to confirm

999

∑(2k+1)/((k^2)(k+1)^2)=.000099

K=100

with the hint consider difference k^-2 -(k+1)^-2

what I would like is help with is, in the process (2k+1)/((k^2)(k+1)^2) expands to 1/(k^2) - 1/((k+1)^2). so why does 2k/k^2 transform into 1/k^2 and if k^-2 -(k+1)^-2 is the difference wouldn't that cancel out k^2(k+1)^2 or am I thinking to logical lol

I need help, I need to use mathematical induction to show the following

n

∑3^k=(3^(n+1)-3)/2 for al n>=1

k=1

would like to see how it is done.

thanks in advance.