caseclosed

This is from the book, need help changing it into mathematical model. "Consider a tank used for certain hydrodynamic experiments. After one experiment the tank contains 200 litres of a dye solution with a concentration of 1 g/litre. To prepare for the next experiment, the tank is rinsed with fresh water flowing in at a rate of 2 litres/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank readches 1% of its original value."

ok, I have claried this, apprarently it was for angle between 2 planes... grr those conventional ways

[math]\int{\sec{x}^3dx}=F[/math] [math]u=Sec(x)[/math] [math]du=Sec(x)Tan(x)dx[/math] [math]dv=Sec(x)^2dx[/math] [math]v=Tan(x)[/math] [math]F=[/math][math]Sec(x)Tan(x)[/math] [math]\int{\tan{x}^2sec{x}dx}[/math] [math]F=[/math][math]Tan(x)^2=Sec(x)^2-1[/math] [math]F=[/math][math]Sec(x)Tan(x)[/math] [math]-\int{\sec{x}^3dx}[/math][math]+\int{sec{x}dx}[/math] [math]F=[/math][math]Sec(x)Tan(x)[/math] [math]-F[/math][math]+ln(sec(x)+tan(x))[/math] [math]2F=[/math][math]Sec(x)Tan(x)[/math] [math]+ln(sec(x)+tan(x))[/math] [math]F=[/math][math](Sec(x)Tan(x))/2[/math] [math]+(ln(sec(x)+tan(x))/2[/math] thank you very much for giving me the hint. I so forgot how to do this.

well, I just did the questions in the book and my professor's method doesn't match the answer in the book. He says the book is wrong...... the book is Thomas's Calculus 11th edition. also I see nowhere online where it says absolute value of dot product.

cos(theta)= (A.B)/(lAllBl) OR cos(theta)=(lA.Bl)/(lAllBl) ---->my teacher insist this one is right and other one is wrong

ok, that's why... I should just leave it as sec^3 instead of changing using trig indentity. thank you very much.

yes, I tried setting u=(x^2+1)^(1/2) and dv=dx but I am not getting any log in the answer which means wrong. using the parts on sec(theta)^3 gets me nowhere, I change to (tan(x)^2+1)sec(x) and then set u=tan(x)^2 and dv=sec(x) dx which lead me to very complicated. I know I did this before but I don't remember how I did it.

integrate (1+x^2)^(1/2) I used trig substitution and gotten integrate sec(theta)^3 but no clue what to do ....

find the point of intersection of the lines x=2t+1, y=3t+2, z=4t+3, and x=s+2, y=2s+4, z=-4s-1, and the find the plane determined by these lines. I don't know what to do...

oh, because it looks like the Calculus 2 stuff I did this year, thats why I was wondering.

so what grade are you in since you are doing this?

AH!!!! thanks, why didn't that hit me, heh

how to differentiate sin(x)^x for 0<x<pi/2

ok, now I understand it, basically constant can be thrown out.

I see, I hope the exam does not give mathematica answer as the chice because I have absolutely no clue how to get sec(x)^2/2, anyone know how to get that?

This integral Tan(x)^3+Tan(x) doing it by hang gives Tan(x)^2/2 Mathematica gives Sec(x)^2/2 so why the difference, are both correct, how to get the mathematica answer by hand?

I learned to use this way instead of the others an+1/an so when the ratio get smaller and smaller for n>1 does that mean it is decreasing? and not increasing nor decreasing when the ratio stay same for all n>1 like (x^2+2x)/(x^2+2x+2).

so basically, if there is an upper and lower bound for this type of sequence, it is either a1 and where it converges?

heh, I just found out what I forgot, I forgot to balance out the e by adding natural log. so it would be e^L=e^(ln(e^x+x)*(1/x)) and then by using properties of log and change L to quotient to get (e^x+x)/x so that it is form infinity/infinity and now using L Hopital's rule to get (1+1)/1 so it is 2 but 2=L so the answer would be e^2. woot...

limit of (e^x+x)^(1/x) as x approaches 0 I remember using e^L where L=(e^x+x)^(1/x) for this 1^infinity form what do I do now?

I got (2)/(n!(n+2)) so as n increases to infinity, the thing approaches 0 so that is why it is decreasing right? and less than 1 means it converges. but I don't understand where the upper/lower bound, what is it.... suppose to be a finite number? or is it like lower bound 0 since it is approaching 0 and upper bound unbounded?

how do I find the upper and lower bound of this sequence? (2^(n-1))/(n+1)!

awww, I did parts twice that is why did not work for me.

thank you, I didn't see how the numerator is bigger than the denominator, so when the largest power is same on top and bottom, long division is neccesary? This one looked like a proper fraction to me

woops, I typed it wrong here I meant [math]\frac{x^3+2}{x^3-x}[/math] so why is there extra +1 for mathematica and calculator? when I do it I get [math] \frac{A}{X} + \frac{B}{X-1} + \frac{C}{X+1}[/math] that is how I got my 3 terms.