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janhardo

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@acer ThanksIts all about using in ...

janhardo 695
May 25 2020
0 0

@acer 

Thanks

Its all about using in the :  exercise set 2-task 3..  array's and how to come up with them (invent) and to get used of them   
In the book the use is widespread 

Your approach of the task 3 is a different one then is  done in book. 
That is the problem here: i don't have the answers from the author in Maple from the book and try to follow the book    CD2-5.MWS

Note: in enclosed worksheet the tangentlines in maxs or mins of function are not 0 (horizontal) in the  plot :  y' = 0 , strnge that this is not done in the example  -> no it was done for a serie of x-values ( not for x where f '=0 ), so its correct the positions of the tangentline

@acer I am following the book where...

janhardo 695
May 25 2020
0 0

@acer 

I am following the book where are array used  and not vector or matrix ( it s not vector calculus or matrix calculus what's written in book at the moment ) 

The array is used to store different type of data (types) in Maple : set of points, a equation, a plot  like that.
The next time i upload a worksheet with array questions 

@janhardo  It is difficult to fill...

janhardo 695
May 25 2020
0 0

@janhardo 

It is difficult to fill the arrays with the right content ..the reasoning
 

> 

 

> 

 

 

Inflction point : buigpunt

Tangent Lines (Examples 2.1 and 2.3)

   
> 

restart;
> 

with(Student:-Calculus1):
with(plots):
> 

f:=x->x^sin(x);intvx:= x =0.1..16;

proc (x) options operator, arrow; x^sin(x) end proc

  

x = .1 .. 16

 (1)
> 

plot(f(x), intvx, gridlines= false);
> 

C:=plot(f(x), intvx, gridlines= false):

 
> 

 

i see 6 inflection points in the graph

> 

df:=D(f);

proc (x) options operator, arrow; x^sin(x)*(cos(x)*ln(x)+sin(x)/x) end proc

 (2)
> 

df_expr:=x^sin(x)*(cos(x)*ln(x)+sin(x)/x);

x^sin(x)*(cos(x)*ln(x)+sin(x)/x)

 (3)

Uitzoeken van functievoorschrift naar expressie ivm solve argument-(niet meer nodig ?)

 

> 

 

> 

X:= solve( df_expr =0, x); # standaard solve is niet werkend

0, RootOf(tan(_Z)+ln(_Z)*_Z)

 (4)

This is not working this solve for  fprime , so i asked on Mapleprimes forum : needed with(Student:-Calculus1):

> 

Roots(df(x), x=0.1..16, numeric);

[.3522153993, 2.127615825, 4.842558340, 7.914977694, 11.03330637, 14.16379619]

 (5)
> 

# Second derivative
ddf := D(D(f));

proc (x) options operator, arrow; x^sin(x)*(cos(x)*ln(x)+sin(x)/x)^2+x^sin(x)*(-sin(x)*ln(x)+2*cos(x)/x-sin(x)/x^2) end proc

 (6)
> 

 

==============================================================================

> 

Roots(ddf(x), x=0.1..16, numeric);

[1.395288666, 2.916095860, 7.258616748, 8.576145756, 13.57205647, 14.75675949]

 (7)
> 

InflectionPoints(f(x), x=0.1..16, numeric);

[1.395288666, 2.916095860, 7.258616748, 8.576145756, 13.57205647, 14.75675949]

 (8)
> 

Inflpts := [ fsolve(ddf(x), x=0..16, maxsols=6) ];

[1.395288666, 2.916095859, 7.258616747, 8.576145755, 13.57205647, 14.75675948]

 (9)
> 

 

=======================================================================

> 

 

Roots / InflectionPoints / Inflpts  give the same answer.
Now in the spirit of Tangentline example  drawing the tangentlines int he infliction points  

 

The idea is to use two arrays : G and  P for storing inflection points, equation of tangentline and plot tangentline in function f.
The dimension for array G (matrix) is : 6 infliction points = 12 cells  and  6 cells for equation tangentline   
Dimension array G:. 6 points + equation tangentline   ( 6 rows and 3 colums ) : every row can store 3 datatypes.

 

-The inflection points  and the equation of the tangent lines to the graph of f

 are to be stored in a 6 by 3 array called G.

-The plot of f and  the plots of the tangent lines are to be stored in an array P.  First, we declare these arrays.

 

The first row of array G stores the x-value of the first point, the y-value of the first point,  and the equation of the tangent line at the first point.  The second row of G will contain the same information for the second point, the third row the information for the third point, and so on.  The element P[1] of array P will contain the plot of the tangent line at the first point, P[2] will contain the plot of the tangent line at the second point, and so on.  All these assignments can be made easily with the following do loop.

 

> 

 for i from 1 to 6 do
     G[i,1]:= 0.5+(i-1);
     G[i,2]:= f(G[i,1]);
     G[i,3]:= G[i,2]+fprime(G[i,1])*(x-G[i,1]);
     P[i]:=plot(G[i,3],x=G[i,1]-0.8..G[i,1]+0.8,thickness=2):
 od:

You must have asked yourself the question with a given function f  : what do i need for plotting the tangentlines in the inflectionpoints  of graph f ?
Looking at the for statement  ...

 

> 

G:=array(1..6,1..3): # definition of 6 rows and 3 colums array
> 

P:=array(0..6):

The inflection points  for function f are called by:

> 

Q := map(p->[p,f(p)], Inflpts);

[[1.395288666, 1.388167079], [2.916095859, 1.270355627], [7.258616747, 5.161057836], [8.576145755, 5.015577540], [13.57205647, 9.048000408], [14.75675948, 8.947326153]]

 (10)
> 

 

To define the equations of the tangent lines we need their slopes.  Thus, we need the derivative function of f.

 

> 

df:=D(f);

proc (x) options operator, arrow; x^sin(x)*(cos(x)*ln(x)+sin(x)/x) end proc

 (11)
> 

 


 

Download betounes_ex_set2_task_3.mw

 

@acer ThanksIts primarily for Maple...

janhardo 695
May 25 2020
0 0

@acer 

Thanks

Its primarily for Maple programming and doing recreational math. 
Basic calculus i know something : function of two variables , double integrals and some more 

But i am a B ed in math focussed on teaching a not a bachelor or master in math who do have stronger math knowledge/skills 

Indeed i gave y' at the start of the thread, because i thought that inflection points also were mins and maxs y-values in the english language for y


 

@janhardo  For finding real roots ...

janhardo 695
May 25 2020
0 0

@janhardo 

For finding real roots there was also another command used: it can be added to the list 

RootFinding:-Analytic(diff(fprime_expr,x,x)=0, re=0..10, im=-1..1);

ThanksNice workAh i do see the use now o...

janhardo 695
May 25 2020
0 0

Thanks

Nice work

Ah i do see the use now of the fsolve command: 

The enclosed example uses array, because they are versatile for programming to use as it seems  

The bookexample draws a tangentline on mins and maxs of y, while this task is for the infliction points of y 

Thanks Peter Stone i knew already...

janhardo 695
May 25 2020
0 0

Thanks

Peter Stone i knew already

@acer   ......

janhardo 695
May 25 2020
0 0

@acer 


 

> 

 

 

 

I thought i could do the task step by step (learning step by step)

The calculations made on reaction on question are very useful for me to see basis calculus with plots to learn more

Here is the task where i am working on , could be interesting to see if there are more answer are possible opposite  the direction of the book is aiming for. 

But that is a lot work and that's why i tried to do it step by step and come up with a own solution 

 
 

Download betounes_ex_set2_task_3_vraagforum.mw

CD2-5.aangepast.mwsCD2-5.aangepast.mws

@acer ThanksImpressive..The task is...

janhardo 695
May 25 2020
0 0

@acer 

Thanks

Impressive..

The task is draw in y the tangentlines in all 6 inflection points of y on  interval  0.1..16  ( so the y '' is needed) 
So inspection by sight shows then 6 infllection points for ( your calculation use 0..10 ) 

Yes, change sign must then be shown in  y '' to conclude were y  has inflectionpoints with a particular direction.

https://www.hhofstede.nl/modules/buigpunten.htm

@tomleslie  Thanks  I could c...

janhardo 695
May 25 2020
0 0

@tomleslie 
Thanks 

I could chance in the procedure the functions : it is y and y ' , so the y plot with the inflection point depicted then

@tomleslie  Thanks...

janhardo 695
May 25 2020
0 0

@tomleslie 

Thanks

@tomleslie  Thanks  The star...

janhardo 695
May 25 2020
0 0

@tomleslie 

Thanks 

The start function is f(x) = x^sinx  and the prime : x^sin(x)*(cos(x)*ln(x)+sin(x)/x  ,  and has the form  x^n
The task ask me to use fsolve command  

I think for polynomial the exponent must be a integer number ..i must read more about this 

ThanksI noticed already dat y ''...

janhardo 695
May 25 2020
0 0

Thanks

I noticed already dat y '' = 0 was calculated ( that's good too for learning) : what gives all special points for y ( maxs,mins, inflection points), but i need only the inflection points of y for now. 

RootFinding:-Analytic(diff(fprime_expr,x,x)=0, re=0..16, im=-1..1);

If a replace fprime_expr (= y ' ) for y then i get the maxs and mins, right?  

y

y ' =0 :  max or min y

y '' = 0  : max or min +inflection points -> gives all points of direction chance of graph y

 

note: i must try to keep this simple, because the task was draw in  y the tangentlines in all 6 inflection points of y ( so the y '' is needed) 

ThanksAmazimg with Maple Now i must...

janhardo 695
May 24 2020
0 0

Thanks

Amazimg with Maple 

Now i must this relate with the fprime graph in my worksheet.

@acer  When the 2-dimensional arra...

janhardo 695
May 24 2020
0 0

@acer 

When the 2-dimensional array is evaluated you can see the content. 
Now with a one-dimensional array it seems to be not possible to see the content inside as a whole  ?..why is that ?

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