The function

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This was asked before: https://www.mapleprimes.com/questions/233343-Drawing-Of-Complex-Numbers-On-A-Complex-grid-#comment283684

It really showed me what was going on. But now there is something else, that really does not look the like, i tried. But there is nothing that really looks like an answer for the b. question. "Draw the following collection in the complex plane".

There really is no explanation on how to handle the numbers in the |z| part and (z) part of the argument of question a. Also, adding imaginary numbers to the argument en the |z| part is really odd. Do i multiply those numbers with the x and y components in there?  

a.

plots:-inequal([1 <= 2*sqrt(x^2+y^2), 2*sqrt(x^2+y^2) < 2, (1/4)*Pi <= arctan(2*y, 2*x), arctan(2*y, 2*x) < 3*Pi*(1/4)], x = -3 .. 3, y = -3 .. 3, optionsopen = [linestyle = dash, color = red, thickness = 2])

 

b.

plots:-inequal([1 < (4*I)*sqrt(x^2+y^2), (4*I)*sqrt(x^2+y^2) <= 1, (1/2)*Pi <= arctan((2*I)*(y, x)), arctan((2*I)*(y, x)) <= Pi], x = -3 .. 3, y = -3 .. 3, optionsopen = [linestyle = dash, color = red, thickness = 2])

 

``

This really helps out a lot if someone could explain this to me. 

Thank you!

Greetings,

The Function

Download Mapleprimes_Book_2_Question_3.1.mw

Hello everybody! Happy new year! 

Im back this year. I bought a flatbedscanner to scan some books. Already scanned 3 whole books in 2 days, its a blast!

Ive got a question about some complex numbers that i have to draw on a grid. 

The translated question is: "draw in the complex grid the following collections:"

Here is what ive got. But the book does not give any solutions for me to check on. So its not great at all to have to learn like that. 

Download Mapleprimes_Book_2_Question_3.mw

a.

-2+I = 2

sqrt((x-2)^2+(y+1)^2) = sqrt(2)

((x-2)^2+(y+1)^2)^(1/2) = 2^(1/2)

(1)

smartplot(sqrt((x-2)^2+(y+1)^2) = sqrt(2), sqrt((x-2)^2+(y+1)^2) = 1)

 

b.

-1+I*2

-1+2*I

(2)

(1/2)*sqrt((x-4)^2+(y+2)^2) = 0

(1/2)*((x-4)^2+(y+2)^2)^(1/2) = 0

(3)

smartplot(sqrt((x-1)^2+(y+2)^2) = sqrt(3))

 

c.

Error, `;` unexpected

 

NULL

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#d.

sqrt(x^2+y^2) = 1

(x^2+y^2)^(1/2) = 1

(4)

sqrt(x^2+y^2) = 3

(x^2+y^2)^(1/2) = 3

(5)

smartplot(sqrt(x^2+y^2) = 1, sqrt(x^2+y^2) = 3, x = -(1/4)*Pi, x = 3*Pi*(1/4))

 

-(1/4)*Pi

-(1/4)*Pi

(6)

evalf(%)

-.7853981635

(7)

3*Pi*(1/4)

(3/4)*Pi

(8)

evalf(%)

2.356194490

(9)

e.

everything under y=0 and everything to the right of x=0

plot(x = 0 .. 10, y = -10 .. 0)

 

f.

sqrt(x^2+y^2) = 0

(x^2+y^2)^(1/2) = 0

(10)

sqrt(x^2+y^2) = 2

(x^2+y^2)^(1/2) = 2

(11)

smartplot(sqrt(x^2+y^2) = 0, sqrt(x^2+y^2) = 2, x = -(1/4)*Pi, x = (1/3)*Pi)

 

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The book gave an example, the translations says:"Draw in the complex grid the collection:
Solution

This we can read as: draw all complex numbers for which the distance to i is similar to 2. This collection is a circle in the complex grid with the center (0,1) and a radius of 2. See the figure on the left.

Draw in the complex grid the collection:

Solution

We can wright     This we can read as: Draw all complex numbers for which the distance to -3+2i is simular to 1. This collection is a circle in the complex grid, see the figure on the right. Another method is z=x+iy, we can substitute this into. This substitution delivers a relation between x and y, after which we can draw each z=x+iy that complies with |z+3-2i|=1. The substitution gives us:  frown which leads,   so that  . This is again a formula of a circle in the complex grid with the middlepoint (-3,2) and a radius of 1." 

Some more examples:

I also added a PDF of the chapter, so you can see what a flatbed scanner can do (a canon lide 400) and adobe DC (to combine the scans and perform "OCR" on the text so the computer can read the text so you can copy and paste it and search the text with CTRL+F. A must have if you are serious in doing studies in my opinion, its way way faster to look up things like that than to go to a register or glossary at the back of the book that may not even have the topic listed what you are looking for.). It is very impotant that you get good scans of the PDF program will not read your numbers and letters in the scanned file well. The solutions (after having scanned half a book already, so i had to do that part again) was to lay the book on the right side of the scanner, with half the book on top of the scanner. Press on the back of the book, and lay 2 hands on top to press the page (lightly) against the glass. When you are dont with that page, you invert the book and keep half the book on the right side again. The PDF program (adobe DC) will flip the scans for you with the OCR recognition function. 



I thought id share this with you, while this costed me 2 books to find this out, and costed me a lot of time. Time a valuable, that is why i started the scanning in the first place. It better be good on the first run.  

Looking at the google translate results the "grid" probably is a "plane" just as in a 3d drawing program. 

Any way, i hope i was specific enough. Im having a bit of trouble with this planing of complex numbers. 

Greetings,

The Function. 

Maple_question_3.pdf

Download Mapleprimes_Book_2_Question_3.mw

Hello everyone,

The "rows and series" chapter is coming to an end. But im not getting this question. Ive got a feeling they are not really specific with this book. But that could just be me.

Any way here is the question:

"In classical physics there is the kinetic energy of a body with the mass m0 and the speed v given by E1=1/2*m0*v^2. According to Einstein the kinetic energy E2=(m*c^2)-(m0*c^2)=((m0*c^2)/sqrt(1-(v/c)^2))-m0*c^2, at which m is the relativistic mass with a speed v, and m0 the mass in rest. Further c is the speed of light. Wright down E2 as a linear function of v^2 and show that E2=E1 when v is small compared to c." 

Now i cant see what they did to get this answer:

A taylor series was probably used, the question before it also used a taylor series. 

If someone knows what they did. What did they do to get the the answer the book gave? 

Thank you!

Greetings,

The Function

Hello everybody,

I managed to get to the point that i could start part 2 of the book series of applied Dutch math. 

This paragraph is about Taylor series.

Now i am being asked to find a solution for the taylorseries with a certain x value. That is all fine, Maple can spit it out. But to put in into something that is there with a sum sign in front of it is something else. I have to convert the solution into something that is written with "x to the k-ed, something something". 

Now i did find to solution to a not so complex one. I will add that one on the maple sheet. But there is this one that is really complex, and i cant get my head around how to get to the final solution that makes the sum go and work well. It does have some series to it. But i dont know how to find that one out. Is there some trick to make it work? 

Without further ado:

a.

taylor(1/(3-x), x = 2)

series(1+x-2+(x-2)^2+(x-2)^3+(x-2)^4+(x-2)^5+O((x-2)^6),x = 2,6)

(1)

Sum((x-2)^k, k = 0 .. infinity)

Sum((x-2)^k, k = 0 .. infinity)

(2)

b.

taylor(sqrt(x), x = 1, 16)

series(1+(1/2)*(x-1)-(1/8)*(x-1)^2+(1/16)*(x-1)^3-(5/128)*(x-1)^4+(7/256)*(x-1)^5-(21/1024)*(x-1)^6+(33/2048)*(x-1)^7-(429/32768)*(x-1)^8+(715/65536)*(x-1)^9-(2431/262144)*(x-1)^10+(4199/524288)*(x-1)^11-(29393/4194304)*(x-1)^12+(52003/8388608)*(x-1)^13-(185725/33554432)*(x-1)^14+(334305/67108864)*(x-1)^15+O((x-1)^16),x = 1,16)

(3)

"Sum(((x-1)^(k))/(???????),k=0..infinity)"

8*(1/2); 16*(1/8); 128*(1/16); 256*(1/128); 1024*(1/256); 2048*(1/1024); 32768*(1/2048); 65536*(1/32768); 262144*(1/65536); 524288*(1/262144); 4194304*(1/524288); 8388608*(1/4194304); 33554432*(1/8388608); 67108864*(1/33554432)

4

 

2

 

8

 

2

 

4

 

2

 

16

 

2

 

4

 

2

 

8

 

2

 

4

 

2

(4)

``

Thank you!

Greetings,

The Function

Download Mapleprimes_Book_2_Question_1.mw

Hello everybody. Ive got some questions. 

As the title states: Draining a shaped container from the top, and the amount of Joule it takes to do it. That is what the questions are about. 

I did make the first one, and some attempts at the sphere questions b, and c. I left them out because i want to see what you guys come up with. I think ive got the right answers, but we will see if our answers compare. :) 

The Dutch text translated: Calculate in questions a, b, and c the amount of work (joule, J) it takes to drain a tank filled with water from the top with a pump. Each question has a discription of the shape of the tank. Take for gravity g=9.81m/s^2. For water take rho=1000kg/m^3.

a.
A coneshaped tank, the topangle is down, with a height of 2.0meters and a radius of 0.5meters. 

b.
A sphere with a radius of 1.0meters.

c.
The lower half of a sphere with a radius 2.0meters. 

Greetings,

The Function. 

"f(x):=(Pi*(x/(4))^(2)*x)/(3)*9.81"

proc (x) options operator, arrow, function_assign; (1/48)*Pi*x^2*x*9.81 end proc

(1)

1000*(int(f(x), x = 0 .. 2))

2568.251995

(2)

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Download Question_for_maple_primes_8.mw

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