Explain parameter elimination for complex curves
I'm reading Stewart's Calculus and in "Calculus with Parametric Curves" he shows the formula which comes from the Chain rule:
$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
But what I don't understand is that for some complex curves for example:
$x = 2\sin\left(1+3t\right), y = 2t^{3}$
If I try to retreive y(x) I will get
$y\ =\ 2\left(\frac{\arcsin\left(\frac{x}{2}\right)-1}{3}\right)^{3}$
Which will represent only small part of the parametric curve(graph).
And only for some small range of t $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. For example when t = 1, $y(x)$ is not defined.
So the question is: Am I missing something, or $\frac{dy}{dx}$ can't always represent $\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$?
![Babaduras](https://matchmaticians.com/storage/user/105178/thumb/AAcHTtdbRJSP7BZBgbm_Fb-ZH-NdLwp21VO9f_y2JqAJ7td8-avatar-512.jpg)
Answer
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
-
That’s not exactly what I’ve asked. As I’ve said if we take a look into graph of parametric equation it has complex shape and many negative derivatives(negative angles of tangent lines). But y(x) is just a small part of the graph of parametric equation, and it doesn’t have those negative derivatives(angles of tangent lines), since it grows from left to right. That’s why I said that dy/dx can’t even have similar values to dy/dt/dx/dt for some of the t values( for example 1).
-
I will tip you additional 5$. Just give me some better explanation please.
-
I see you accepted. Do you still need aditional help? I explained above that dy/dx always equals (dy/dt)/(dx/dt). Not sure what you mean by when t = 1, y(x) is not defined.
-
I'm preparing another question, will be ready in couple minutes.
-
- answered
- 494 views
- $5.00
Related Questions
- Prove that $\int _0^{\infty} \frac{1}{1+x^{2n}}dx=\frac{\pi}{2n}\csc (\frac{\pi}{2n})$
- Find $\lim _{x \rightarrow 0} x^{x}$
- A function satifying $|f(x)-f(y)|\leq |x-y|^2$ must be constanct.
- In what direction the function $f(x,y)=e^{x-y}+\sin (x+y^2)$ grows fastest at point $(0,0)$?
- Find the volume of the solid obtained by rotating $y=x^2$ about y-axis, between $x=1$ and $x=2$, using the shell method.
- Measure Theory and the Hahn Decomposition Theorem
- Beginner Question on Integral Calculus
- Two short calculus questions - domain and limits