- #1
cshum00
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Homework Statement
I am given [tex]x^2 + y^2 = r^2[/tex]Need to find [tex]\frac{\partial y}{\partial x}=-\frac{1}{r}sin(\frac{v}{r}t)[/tex]
2. Revelant equations
x parametriztion of circle is:
[tex]x=rcos(\frac{v}{r}t)[/tex]Trigonometric Identity
[tex]sin^2t + cos^2t=1[/tex]
[tex]r^2sin^2t=r^2-r^2cos^2t[/tex]
[tex]1 + \frac{cos^2t}{sin^2t} = \frac{1}{sin^2t}[/tex]
The Attempt at a Solution
so [tex]y = (r^2 - x^2)^\frac{1}{2}[/tex][tex]\frac{\partial y}{\partial x}=\frac{1}{2}(r^2 - x^2)^\frac{-1}{2}(-2x)[/tex]
simplify: [tex]\frac{\partial y}{\partial x}=-x(r^2 - x^2)^\frac{-1}{2}[/tex][tex]\frac{\partial ^2y}{\partial x^2}=-(r^2 - x^2)^\frac{-1}{2}+\frac{-1}{2}(r^2 - x^2)^\frac{-3}{2}(-2x)(x)[/tex]
Simplify: [tex]\frac{\partial ^2y}{\partial x^2} = -\frac{1}{(r^2 - x^2)^\frac{1}{2}}-\frac{x^2}{(r^2 - x^2)^\frac{3}{2}}[/tex]Substitute parametrization to second partial derivative:
[tex]\frac{\partial ^2y}{\partial x^2} = -\frac{1}{(r^2 - r^2cos^2(\frac{v}{r}t))^\frac{1}{2}}-\frac{r^2cos^2(\frac{v}{r}t)}{(r^2 - r^2cos^2(\frac{v}{r}t))^\frac{3}{2}}[/tex]
[tex]\frac{\partial ^2y}{\partial x^2} = -\frac{1}{rsin(\frac{v}{r}t)}-\frac{r^2cos^2(\frac{v}{r}t)}{r^3sin^3(\frac{v}{r}t)}[/tex]
[tex]\frac{\partial ^2y}{\partial x^2} = -\frac{1}{rsin(\frac{v}{r}t)}(1 + \frac{r^2cos^2(\frac{v}{r}t)}{r^2sin^2(\frac{v}{r}t)})[/tex]
[tex]\frac{\partial ^2y}{\partial x^2} = -\frac{1}{rsin(\frac{v}{r}t)}\frac{1}{sin^2(\frac{v}{r}t)}[/tex]
For some reason i ended up with [tex]\frac{1}{sin^2(\frac{v}{r}t)}[/tex] instead of [tex]sin^2(\frac{v}{r}t)[/tex]. Did i do any intermediate steps wrong?