partial fractions & parametric equations

[xcrush]

hi, I'm having trouble with just three problems..

1. A particle moves along a straight line with its position at any time "t" given by:

x = 2t^3 - 15t^2 + 20t - 8
y = 2

Find the start & end positions.
How many times did the particle change directions? Explain
Approximate to one decimal place, the position (or the x-value) of when the particle changes direction.


2. Write the partial fraction decomposition of:

x^2 + x + 2 / (x^2+2)^2


3. Write the parametric equations of the top half of an ellipse having the equation:

(x-3)^2 / 16 + (y+1)^2 / 4 = 1

Also state the restrictions of t.


I know how to do partial fraction decomposition, but it's not working out for me in #2 and I don't know what I'm doing wrong. For the other two problems, I just have no idea...

 

[pka]

Get yourself a good CAS (computer Algebra Syatem).
Doing partical fractions is so out dateded.

 

[xcrush]

ahhh I meant x^2 + 2, not x^2 + x. sorrryyy.
i suppose that makes the problem a bit more complicated

 

[xcrush]

i got #2 and 3, but I still need help with #1. anyone..?

 

[stapel]

Without limits on the time "t", how can one find the "start" and "end" positions?

Eliz.

 

[soroban]

Hello, xcrush!

2. Write the partial fraction decomposition of: \(\displaystyle \L\,\frac{x^2\,+\,x\,+\,2}{(x^2\,+\,2)^2}\)

This has a repeated quadratic factor . . .

\(\displaystyle \L\frac{x^2\,+\,x\,+\,2}{(x^2\,+\,2)^2}\;=\;\frac{Ax\,+\,B}{x^2\,+\,2}\,+\,\frac{Cx\,+\,D}{(x^2\,+\,2)^2}\)

We have: \(\displaystyle \,x^2\,+\,x\,+\,2\;=\;Ax(x^2\,+\,2)\,+\,B(x^2\,+\,2)\,+\,Cx\,+\,D\)

Let \(\displaystyle x\,=\,0:\;\;2\;=\;A\cdot0\,+\,B\cdot2\,+\,C\cdot0\,+\,D\;\;\Rightarrow\;\;2B\,+\,D\:=\:2\;\;\;[1]\)

Let \(\displaystyle x\,=\,1:\;\;4\;=\;A\cdot3\,+\,B\cdot3\,+\,C\cdot1\,+\,D\;\;\Rightarrow\;\;3A\,+\,3B\,+\,C\,+\,D\:=\:4\;\;\;[2]\)

Let \(\displaystyle x\,=\,\)-\(\displaystyle 1:\;\;2\;=\;A(\)-\(\displaystyle 1)(3)\,+\,B\cdot3\,-\,C\,+\,D\;\;\Rightarrow\;\;\)-\(\displaystyle 3A\,+\,3B\,-\,C\,+\,D\:=\:2\;\;\;[3]\)

Let \(\displaystyle x\,=\,2:\;\;8\;=\;A\cdot2\cdot6\,+\,B\cdot6\,+\,C\cdot2\,+\,D\;\;\Rightarrow\;\;12A\,+\,6B\,+\,2C\,+\,D\:=\:8\;\;\;[4]\)

Add [2] and [3]: \(\displaystyle \,6B\,+\,2D\:=\:6\;\;\Rightarrow\;\;3B\,+\,D\:=\:3\)
\(\displaystyle \;\;\;\)Subtract [1]: . . . . . . . . . . . . . . . . . . .\(\displaystyle 2B\,+\,D\:=\:2\)

And we have: \(\displaystyle \,B\,=\,1,\;D\,=\,0\)


Substitute into [2]: \(\displaystyle \,3A\,+\,3\,+\,C\,+\,0\:=\:4\;\;\Rightarrow\;\;3A\,+\,C\:=\:1\;\;\;[5]\)

Substitute into [4]: \(\displaystyle \,12A\,+\,6\,+\,2C\,+\,0\:=\:8\;\;\Rightarrow\;\;6A\,+\,C\:=\:1\;\;\;[6]\)

Subtract [5] from [6]: \(\displaystyle \,3A\,=\,0\;\;\Rightarrow\;\;A\,=\,0\) . . . then: \(\displaystyle C\,=\,1\)


Therefore: \(\displaystyle \L\.\frac{x^2\,+\,x\,+\,2}{(x^2\,+\,2)^2}\;=\;\frac{1}{x^2\,+\,2}\,+\,\frac{x}{(x^2\,+\,2)^2}\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I was congratulating myself for a job well done
\(\displaystyle \;\;\)when I saw something that chilled me to the bone!

\(\displaystyle \L\frac{x^2\,+\,x\,+\,2}{(x^2\,+\,2)^2} \;= \;\frac{(x^2\,+\,2)\,+\,x}{(x^2\,+\,2)^2}\)

\(\displaystyle \L\;\;\;\;= \;\frac{x^2\,+\,2}{(x^2\,+\,2)^2}\,+\,\frac{x}{(x^2\,+\,2)^2} \;=\;\frac{1}{x^2\,+\,2}\,+\,\frac{x}{(x^2\,+\,2)^2}\)

Scary, isn't it?

3. Write the parametric equations of the top half of an ellipse having the equation:

\(\displaystyle \L\frac{(x\,-\,3)^2}{16}\,+\,\frac{(y\,+\,1)^2}{4}\;=\;1\)

Also state the restrictions of \(\displaystyle \theta\).

There is a standard procedure for circles and ellipses.

Let: \(\displaystyle x\,-\,3\:=\:4\cos\theta,\;\;y\,+\,1\:=\:2\sin\theta\)

So we have: \(\displaystyle \,\begin{array}{cc}x\,=\,4\cos\theta\,+\,3\\y\,=\,2\sin\theta\,-\,1\end{array}\)

For the top half of the ellipse: \(\displaystyle \,0\,\leq\,\theta\,\leq\,\pi\)

In general: \(\displaystyle \,(2n)\pi\,\leq\,\theta\,\leq\,(2n+1)\pi\;\) for any integer \(\displaystyle n\)

 

[soroban]

Hello, xcrush!

1. A particle moves along a straight line with its position at any time \(\displaystyle t\) given by:

\(\displaystyle x\;=\;2t^3 \,-\,15t^2\,+\,20t\,-\,8,\;\;y\,=\,2\)

(a) Find the start & end positions.
(b) How many times did the particle change directions? Explain
(c) Approximate to one decimal place, the position (or the x-value) of when the particle changes direction.

(a) As Eliz. Stapel pointed out, we need the start and end times.


(b) The particle changes direction when its velocity is zero.
\(\displaystyle \;\;\;\)That is, when: \(\displaystyle \,x'\;=\;6t^2\,-\,30t\,+\,20\:=\:0\)

The Quadratic Formula gives us: \(\displaystyle \,x\:=\:\frac{15\,\pm\,\sqrt{105}}{6}\:\approx\:\begin{Bmatrix}0.7922 \\ 4.2078\end{Bmatrix}\)

The particle changes direction twice.


(c) When \(\displaystyle t\,=\,0.7922:\;x\;=\;2(0.7922)^3\,-\,15(0.7922)^2\,+\,20(0.7922)\,-\,8\:\approx\:-0.6\)

\(\displaystyle \;\;\,\)When \(\displaystyle t\,=\,4.2078:\;x\;=\;2(4.2078)^3\,-\,15(4.2078)^2\,+\,20(4.2078)\,-\,8\;\approx\;-40.4\)