The function

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Hello! 

I dont like cursing, but this had got me going. 

How are there no solutions for the intersection points, i am only getting some, but not all of them, but i did ask Maple to produce an answer, but it however failed to produce all the answers. Its not that i did not try different techniques. 

I really wonder what went wrong. How could this be?! Just Maple... Answers between zero and 2 Pi... plz.... (!!!)

The answers i get with the suggested method from my math book dont give any straight answer. How is that an answer, i dont want imaginary numbers, i dont want to have  Z values (what is with that anyways?)..

Well here is what i found. Also if i were to ask for solutions between x values closer together i got more answers that were not there before. That is not what i want, that is like manual labor, that is not what i want at all, i want to be faster not slower. My graphing calculator was even a faster and that did get all the answers right away.. 

Any way, it missed an answer between 1/2Pi and Pi, answers that were not there before when looking for values between 0 and 2Pi. 

So, so odd..


It says in english: Also with finding solutions for solving goniometric equations with Maple, we should not forget that there are more solutions than just one, most of the times. We give you two methods to find all the solutions for solving goniometric equations. 

 

#first attempt

evalf(solve({x > 0*Pi, sin(x) = cos(4*x-(1/6)*Pi), x < 2*Pi}, x, allsolutions, explicit)); smartplot(sin(x), cos(4*x-(1/6)*Pi)); plot([sin(x), cos(4*x-(1/6)*Pi)], x = 0 .. 2*Pi, y = 0 .. 1); evalf(solve({x > (1/2)*Pi, sin(x) = cos(4*x-(1/6)*Pi), x < Pi}, x, allsolutions, explicit))

{x = 4.188790204}, {x = 5.445427267}

 

 

 

{x = 1.675516082}, {x = 2.932153141}

(1)

#second attempt

"f(x):=sin(x); g(x):= cos(4 x-1/(6)*Pi);fsolve([f,g]);"

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

 

proc (x) options operator, arrow, function_assign; cos(4*x-(1/6)*Pi) end proc

 

[6.8067840827778841, -9.1035796650957585]

(2)

#third attempt;

_EnvAllSolutions := true; solve(sin(x) = cos(4*x-(1/6)*Pi), x)

-(2/3)*Pi+2*Pi*_Z4, -arctan((1/6)*3^(1/2)*(2*((1/2)*(4+(4*I)*3^(1/2))^(1/3)+2/(4+(4*I)*3^(1/2))^(1/3))^2-(1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)-4)/((1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)))+2*Pi*_Z5, -arctan((1/6)*3^(1/2)*(2*(-(1/4)*(4+(4*I)*3^(1/2))^(1/3)-1/(4+(4*I)*3^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)))^2+(1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))-4)/(-(1/8)*(4+(4*I)*3^(1/2))^(1/3)-(1/2)/(4+(4*I)*3^(1/2))^(1/3)+((1/4)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))))-Pi+2*Pi*_Z5, -arctan((1/6)*3^(1/2)*(2*(-(1/4)*(4+(4*I)*3^(1/2))^(1/3)-1/(4+(4*I)*3^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)))^2+(1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))-4)/(-(1/8)*(4+(4*I)*3^(1/2))^(1/3)-(1/2)/(4+(4*I)*3^(1/2))^(1/3)-((1/4)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))))+Pi+2*Pi*_Z5, arctan((1/3)*3^(1/2)*(2*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 1)^3-7*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 1)+2)/RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 1))+2*Pi*_Z6, arctan((1/3)*3^(1/2)*(2*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 2)^3-7*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 2)+2)/RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 2))+2*Pi*_Z6, arctan((1/3)*3^(1/2)*(2*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 3)^3-7*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 3)+2)/RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 3))+Pi+2*Pi*_Z6, arctan((1/3)*3^(1/2)*(2*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 4)^3-7*RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 4)+2)/RootOf(_Z^4-_Z^3-4*_Z^2+4*_Z+1, index = 4))+Pi+2*Pi*_Z6

(3)

#fourth attempt;

RootOf(sin(x) = cos(4*x-(1/6)*Pi), x); allvalues(%)

RootOf(sin(_Z)-sin(4*_Z+(1/3)*Pi))

 

-(2/3)*Pi+2*Pi*_Z7, -arctan((1/6)*3^(1/2)*(2*((1/2)*(4+(4*I)*3^(1/2))^(1/3)+2/(4+(4*I)*3^(1/2))^(1/3))^2-(1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)-4)/((1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)))+2*Pi*_Z8, -arctan((1/6)*3^(1/2)*(2*(-(1/4)*(4+(4*I)*3^(1/2))^(1/3)-1/(4+(4*I)*3^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)))^2+(1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))-4)/(-(1/8)*(4+(4*I)*3^(1/2))^(1/3)-(1/2)/(4+(4*I)*3^(1/2))^(1/3)+((1/4)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))))-Pi+2*Pi*_Z8, -arctan((1/6)*3^(1/2)*(2*(-(1/4)*(4+(4*I)*3^(1/2))^(1/3)-1/(4+(4*I)*3^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3)))^2+(1/4)*(4+(4*I)*3^(1/2))^(1/3)+1/(4+(4*I)*3^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))-4)/(-(1/8)*(4+(4*I)*3^(1/2))^(1/3)-(1/2)/(4+(4*I)*3^(1/2))^(1/3)-((1/4)*I)*3^(1/2)*((1/2)*(4+(4*I)*3^(1/2))^(1/3)-2/(4+(4*I)*3^(1/2))^(1/3))))+Pi+2*Pi*_Z8, arctan((1/6)*3^(1/2)*(2*(1/4+(1/4)*5^(1/2)+(1/4)*(30-6*5^(1/2))^(1/2))^3+1/4-(7/4)*5^(1/2)-(7/4)*(30-6*5^(1/2))^(1/2))/(1/8+(1/8)*5^(1/2)+(1/8)*(30-6*5^(1/2))^(1/2)))+2*Pi*_Z9, arctan((1/6)*3^(1/2)*(2*(1/4+(1/4)*5^(1/2)-(1/4)*(30-6*5^(1/2))^(1/2))^3+1/4-(7/4)*5^(1/2)+(7/4)*(30-6*5^(1/2))^(1/2))/(1/8+(1/8)*5^(1/2)-(1/8)*(30-6*5^(1/2))^(1/2)))+Pi+2*Pi*_Z9, arctan((1/6)*3^(1/2)*(2*(1/4-(1/4)*5^(1/2)+(1/4)*(30+6*5^(1/2))^(1/2))^3+1/4+(7/4)*5^(1/2)-(7/4)*(30+6*5^(1/2))^(1/2))/(1/8-(1/8)*5^(1/2)+(1/8)*(30+6*5^(1/2))^(1/2)))+2*Pi*_Z9, arctan((1/6)*3^(1/2)*(2*(1/4-(1/4)*5^(1/2)-(1/4)*(30+6*5^(1/2))^(1/2))^3+1/4+(7/4)*5^(1/2)+(7/4)*(30+6*5^(1/2))^(1/2))/(1/8-(1/8)*5^(1/2)-(1/8)*(30+6*5^(1/2))^(1/2)))+Pi+2*Pi*_Z9

(4)

``


​​​​​​​Download Question_for_maple_primes.mw

Ive got a question about the Dutch book again. They sure did their part on finding questions with answers you cannot find in the book (maybe in later chapters or the 2nd book "part 2"). Maybe to make sure you go to the teacher, so he sees you, and he is sure you did your part, and to get some interaction between the student and the teacher.

I guess you guys/girls are the teacher... Hahah :)

It is the 3rd question. It says: "given is the funtion y=f(x) ( see the photo for the function, "voor" means "for")

a. Draw the graph of y=f(x)
b. Draw with the help of the graph of y=f(x), a graph y=f(-x),y=-2f(x),y=f(x-1), and y=(2x)"

This paragraph is about shifting graphs of functions, and how to alter them. I know how it works, ive done that quite some time ago, but now in maple it is a new challange. Is there a way to get the funtion f(x) to adhere to the rules imposed on it by the text in the book? If that can be done, the answering of the questions will be a lot easier!

Thank you!

Greetings,

the function 

I have some more questions, now i get some more odd situations where an obvious answer cannot be given by solve. And a funtion cannot be plotted somehow. 

Here is the file. I tried to understand why, but it is not happening. They did it on purpose (i guess) to get simple questions, and show you that Maple sometimes does not have the answer right away. So you run into the "limitations" of the program right away so you are aware what could go wrong.

Find x with the help of a graph from: 2^x<1/2*x^2+2  

smartplot([2^x, (1/2)*x^2+2])

 

solve(2^x = (1/2)*x^2+2, x)

Warning, solutions may have been lost

 

RootOf(_Z^2-2*2^_Z+4)

(1)

"f(x):=2^(x)"

proc (x) options operator, arrow, function_assign; 2^x end proc

(2)

"g(x):=1/(2)x^(2)+2"

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

(3)

evalf(solve(f(x) = g(x), x))

Warning, solutions may have been lost

 

.8841764608-2.067784075*I

(4)

While the answer is obviously x=2 to be equal, and thus for values of x= (infinity;2> g(x) is larger, even if you would manually fill in the equetions,you get a value of 2. Both sides would get a solution of y=4. Somehow Maple does not give a straight answer.

``

2nd question
I have to plot this function, and it wont let me plot it..

"f(x):=(-2)^(x)"

proc (x) options operator, arrow, function_assign; (-2)^x end proc

(5)

plot(proc (x) options operator, arrow, function_assign; (-2)^x end proc)

 

I am not seeing a line, it is so odd.

``

Download Applied_Math_Part_1_questions_part_two.mw

The function

Hello, ive got some trouble calculating this problem. Ive been looking at it for quite some time. I suspect you first need to differentiate to get the minimum value and then make it back to p. Time is ticking away, and i can just cant get it right. Its 2 questions out of some Dutch mathbook from 2007. It has a lot of Maple in it too. Just so to get a broad a view on maple as possible. I must say these maplebooks have quite the repertoirs. The other book "Advanced Problem Solving Using Maple" book turns you into some British chap. And this book:"toegepaste Wiskunde voor het hoger beroepsonderwijs deel 1"(translated applied math for higher job education part 1), turns you into the odd obnoxious Dutch mathematician... Well what can i say??

Here are the questions: 

1st Question:
For which values of p does the graph of the function: y=f(x)=(p*x^2)+3*p*x+1 have one intersection with the x-axis? When does it have two intersections with the x-axis? When does it have no intersections with the x-axis. 

2nd Question:
Given functions are: y=f(x)=(x^2)-6x+p+3, and y=g(x)=(4x^2)-(p-8)x+7. When do these functions have the same minimum value. Calculate p and the minimum value.

Greetings,

1st Question:
For which values of p does the graph of the function: y=f(x)=(p*x^2)+3*p*x+1 have one intersection with the x-axis? When does it have two intersections with the x-axis? When does it have no intersections with the x-axis.  

f(x) = p*x^2+3*p*x+1

f(x) = p*x^2+3*p*x+1

(1)

NULL

restart

2nd Question:
Given functions are: y=f(x)=(x^2)-6x+p+3, and y=g(x)=(4x^2)-(p-8)x+7. When do these functions have the same minimum value. Calculate p and the minimum value.

f(x) = x^2+p-6*x+3

f(x) = x^2+p-6*x+3

(2)

g(x) = 4*x^2-(p-8)*x+7

g(x) = 4*x^2-(p-8)*x+7

(3)

``

Download Applied_Math_Part_1_questions.mw

the function

Okey, here is something for you people: the command "pointplot" does not seem to work, however "plot" command does seem to work. Plot command with "style=point" in the syntax seems to give the same result as the books example. Hoever the books example does not give the same results as displayed in the book. 

How is that? Where did i go wrong? 

Could you please help me out? It really feels dumb to do what the book suggests and not getting the same results is a disappointment IMO.. 

k, M, init := 0.9e-3, 670, 30.0

biomass := proc (n::integer) option remember; piecewise(0 < n, biomass(n-1)+k*biomass(n-1)*(M-biomass(n-1)), init) end proc

pts := [seq([n, biomass(n)], n = 0 .. 30)]

pointplot(pts, view = [0 .. 30, 0 .. 700], title = "Biomass")

pointplot([[0, 30.0], [1, 47.280000], [2, 73.77798144], [3, 113.3672328], [4, 170.1607576], [5, 246.7084793], [6, 340.6951260], [7, 441.6684350], [8, 532.4305955], [9, 598.3521395], [10, 636.9357251], [11, 655.8895612], [12, 664.2189618], [13, 667.6748495], [14, 669.0720496], [15, 669.6308287], [16, 669.8533163], [17, 669.9417472], [18, 669.9768706], [19, 669.9908171], [20, 669.9963543], [21, 669.9985526], [22, 669.9994254], [23, 669.9997719], [24, 669.9999094], [25, 669.9999640], [26, 669.9999857], [27, 669.9999943], [28, 669.9999977], [29, 669.9999991], [30, 669.9999996]], view = [0 .. 30, 0 .. 700], title = "Biomass")

(1)

plot(pts, style = point, view = [0 .. 30, 0 .. 700], title = "Biomass")

 

``


So the 2nd line trying to make the plot does seem to work, however i would like to use the "pointplot" command, which does not work. :( 

Greetings,

 

The Function 

Download Discrete_Dynamical_Models_3.mw

 

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