## Calculus assignment...

I am doing a Calculus assignment and I can't find the commands for certain things.

1.Given the function f(x) = ((x+1)^2) / (1+x^2)

i) The domain of continuity of f(x)

ii) The intervals of increase and decrease of f(x) by using test points.

2. Use the IVT to prove existence of a root to the equation x^3 +10x^2 -100x +50=0 in the interval [-20,10]. Use again the IVT to show that there is a 1st root in [-17,-15], a 2nd toot in [0,1] and a 3rd root in [ 5,6]. Find or approximate those roots with Maple. (the bolded is what I need help).

## How to plot function using the solution of ODE?...

I want to plot E with respect to t.

I typed the sys of this equation and try to plot it with DEplot but failed.

This is the code

DEplot([sys], [x(t), y(t), E(t)], t = -25 .. 25, [[x(0) = -.3, y(0) = .2]], scene = [E(t), t], stepsize = 0.5e-1, linecolor = red, arrows = none);

The error is Error, (in DEtools/DEplot/CheckDE) derivatives must be given explicitly.

## Equation of motion rocket...

Im working on an assignment involving a rocket being launched vertically from rest.
I have a eq. of motion that looks like this: v(t)=vs*ln(-m0/(r*t-m0))-g*t

Now Im supposed to implement the air resistance, and write an expression for the total force on the rocket. After that creating a plot displaying the speed v(t) as a function of time for both, with and without air resistance.

All constants are known including CD, rho and A for the dragforce.

I would be grateful if anyone could help me with this!

Thanks

## Convert an equation with ln...

Im working on a assignment on maple. I have an equation of motion that looks like this:  v(t):= -g*t-vs*ln(r*t-m)+vs*ln(-m)
Im supposed to use this equation and solve it for t an later integrate it. Since the constant inside the ln is negative I end up with a annoying imaginary part. Is there any way to covert this equation so that the ln disappear, so that I can get a result whitout a imaginary part?

## Inflection Points?...

How would I find inflection points? I believe it would the same as if I was finding critical points for f '.

Function is:(7-x)*sin(x^2-7)

So would it be:

a:=fsolve(f2,-2.4..2.4);
b:=fsolve(f2,-2.4..-0.06);
c:=fsolve(f2,-0.06..2.4);
d:=fsolve(f2,1.49..3);
e:=fsolve(f2,-3 ..-1.52);

Plz and thanks!

## Solving Critical Points on Maple?...

Hi,

Ive been trying to solve critical points for maple but i keep getting this werid equation:

(1/7)*RootOf(tan(_Z)^2+4*_Z*tan(_Z)+4*_Z^2+28*tan(_Z)-140*_Z-1176)+1+(1/14)*tan(RootOf(tan(_Z)^2+4*_Z*tan(_Z)+4*_Z^2+28*tan(_Z)-140*_Z-1176)).

What I have typed

f:=x->(7-x)*sin(x^2-7);

f1:=D(f);

When I try putting:

solve(f1(x)=0);

I get that werid equation

## Euler's Method on Aircraft...

Hi, I am working on an assignment and I am having difficulties with this Eurler method since I am working with vector.

The assignment:

If the aircraft's engine transmits power P to the air, the force of the propellar, Fp, satisfies P=Fpv. Since the force Fp acts in the direction of motion, it can be written in vector form as: (what I have so far)

with(linalg);

v:=Vector(2,1,[vx,vy]); velocity vector

nu:=v->Norm(Vector(v),2); (length of v)

Fd:=-1/2*C*rho*A*nu*v # air resistance

Fl:=1/2*Cl*rho*A*nu*v # liftforce in y direction here v=[-vy,vx]

Fp:=(P/nu^2)*v^2 # power from aircraft

F_T:=v->Vector(evalm(Fp/nu(v)^2*v+Fd*nu(v)*v+Fl*nu(v)*zip((x,y)->x*y,v([2,1]),Vector([-1,1]))+Vector(2,1,[0,-m*g]))); # I have tried it and it works

so up til here I am good. It is this next part that I can't seam to get.

Write a routine which uses Euler's method to solve the equation of motion numerically for velocity v(t):

m*dv/dt=F_T(v)

Test the program by trying a plane with intitial velocity [v0,0] such that it maintains this speed when the engines power is P=P0

We have been given an Esolve we can use:

Esolve:=proc(f::procedure,h,x0,y0,N::integer) # Calculates y(x0+n*h) for n=1..N given y0=y(x0) and f(x,y)

local n,y,x;

y[0]:=y0; # Start Value
x:=x0;

for n from 1 to N do
y[n]:= y[n-1]+h*f(y[n-1],x); # Euler's formula
x:=x+h; # Next x
end do;

y;

end proc;

If anyone could help me understand how it works and how to get this to work with vectors I would be very gratefull.

## math problem!! (Newton's method)...

 Consider the following function.f (x) = ( x3 − 442x2 + 65107x − 3196058) e1/x f:=x->(x^3-442*x^2+65107*x-3196058)*exp(1/x);
 Use Newton's method to find all 3 roots (correct to at least 6 decimal places) of f (x). (Note that you might have to increase the 'Digits' variable to 15, i.e., Digits:=15, in order to get the required accuracy.)

but, it is wrong. can anyone tell me where is the wrong part?

## plot a 2D arc in a coordicates which is in 3D coor...

In a local coordivates frame {B}, an arc whith center "coc" ,radius "radius" and angle between “alpha1” and “alpha2” can be plotted in XY plane of {B}. The coord {B} is in a Ref frame {O}. The homogeneous transformation matrix is “H”. How can I plot this arc in coordinate {O}？

## How to use Verlet algorithm to derive the orbit of...

Derive the orbit of the Moon around the Earth by doing a Verlet algorith of Molecular Dynamics simulation. Use one hour for your step τ. Place the stationary Earth at the origin of the Cartesian system. For initial conditions, use the position and the speed of the Moon when it is at its apogee (furthest from Earth). Plot the orbit.

## How to solve Cauchy sequences...

(a) Show that if {an} ∞ n=1 is Cauchy then {a 2 n} ∞ n=1 is also Cauchy. (b) Give an example of a Cauchy sequence {a 2 n} ∞ n=1 such that {an} ∞ n=1 is not Cauchy

## How to compute direct exponentiation of the matrix...

Hi EveryOne!

In the the answer of the question "How to find k^th root of the given matrix over finite field (at URL: http://www.mapleprimes.com/questions/203997-How-To-Find-Kth-Root-Of-The-Given-Matrix#comment215683). Carl Love helped to find k^th root of the given matrix M over GF(2m)/f(x).

Now, I need to compute direct exponentiation of the given matrix M in finite field. (If M = [mi,j], we say Direct Exponent (element-wise exponent matrix), Mdk of M is a matrix whose each element is the result of exponentiation of corresponding elements of M. If k=2, then we say Md2 is a direct square matrix of M )

## sequence convergence and limit...

Determine wether the sequence below converges or diverges, and if it converges, find the limit.

and,

Thanks

## varying initial conditions...

Restrict calculation to real numbers.

Using y' = u, express the oscillator equation: y" + 3y' + 2y = cos(t) as a first order system.

Plot an approximate solution curve for the specified initial conditions.

[x0=5, y0=1],[x0=-2, y0=-4],[x0=0, y0=.1],

This is what i have so far but i am not sure if its correct.

Eulers modified method:

with(RealDomain);

x[0] := 0;

y[0] := 5;

t[0]=0

h := .1;

for n to 100 do

x[n] := x[n-1]+h*(x[n-1]+y[n-1]);

k1 := x[n-1]+y[n-1];

k2 := h*k1+x[n]+y[n-1];

k := 1/2*(k1+k2);

y[n] := h*k+y[n-1]

end do;

data := [seq([x[n], y[n]], n = 0 .. 100)];
G1 := plot(data, style = point, color = "blue");
G1;

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