show all the work 1) Use implicit differentiation to find dz/dx and dz/dy. assum
ID: 3289199 • Letter: S
Question
show all the work
1) Use implicit differentiation to find dz/dx and dz/dy. assume that equation defines z as a differentiable function near each (x,y)
Ln(x^2+y^2)-z=tan^1(x+z)
2) Given that r=(x^2+y^2)^1/2 show that dr/dx= x/(x^2+y^2)^1/2 = x/r =cos theta. starting from r=x/cos theta , does it follow that dr/dx = 1/ cos theta? Explain why its not possible for both calculations to be correct
3)locate all critical points and classify them using the second derivative test
a) 2x^2+y^3-x^2y-3y
b) f(x,y)= xsiny
4) Use Lagrange multipliers to find the closest point on the given curve to the indicated point.
y=x^2,(3,0)
Explanation / Answer
1)Differentiating wrt x gives
yz + xy(dz/dx) = -sin(x+y+z)
=> (dz/dx) = -(yz + sin(x+y+z))/xy
Similarly
(dz/dy) = - (xz + sin(x+y+z)/xy
2)
When you make the change of variablesx=rcos?,y=rsin?, the integral becomes
whereD?is the changed domain, wherer,?belong, andJ(r,?)=??????x?r?y?r?x???y???????is the jacobian matrix.
The formula(F)stays true even when you make different change of variables forx,y. This is the theory behinddxdy=rdrd?.
For a proof of(F)you need to use Jordan measurable sets (I think ) and the definition of the double integral. Of course, this works in higher dimensions, with more intricate computations.
Changing coordinates in multiple integrals requires adding the Jacobian as a factor. The Jacobian in this case isr(cos2?+sin2?)=r.
3)
down vote
When you make the change of variablesx=rcos?,y=rsin?, the integral becomes
?Df(x,y)dxdy=?D?f(rcos?,rsin?)J(r,?)drd?(F)
whereD?is the changed domain, wherer,?belong, andJ(r,?)=??????x?r?y?r?x???y???????is the jacobian matrix.
The formula(F)stays true even when you make different change of variables forx,y. This is the theory behinddxdy=rdrd?.
For a proof of(F)you need to use Jordan measurable sets (I think ) and the definition of the double integral. Of course, this works in higher dimensions, with more intricate computations.
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