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Calculus/AP Calculus Derivative Question

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Expert: Alon Mandes - 12/11/2008


Question
I have these following questions I need answered.
We were asked to find the derivative of:
x^2 - xy + y^2 = 28
I believe I was able to solve this..
I received the answer of:
dy/dx= -2x / y-x

Then.. we are asked to find the equation(s) of the
tangent line(s) for the above equation when x=4.

AND find where it has an undefined slope.

PLEASE HELP!?!? :)

Answer
First of all dy/dx is not -2x/(y-x) it's (y-2x)/(2y-x). & here's
why :
d/dx {x^2 - xy + y^2 = 28} = d/dx {28}
2x-d/dx(xy)+d/dx(y^2)=0
2x-y-xy'+2yy'=0
y'(2y-x)=y-2x
y'=(y-2x)/(2y-x).

Now, remember that x^2 - xy + y^2 = 28 is an ellipsoid, so we will
have 2 tangent line for point x=4. Let's calculate the y value
corresponding to x=4 :
4^2-4y+y^2=28 -> 16-4y+y^2=28 -> y^2-4y-12=0 ->
y1=6 & y2=-2. Let's take y=-2. So our point will be (4,-2).
y'(4,-2)=(-2-8)/(-4-4)=5/4.
The equation for the tangent line at point (4,-2) is :
(y+2)/(x-4)=5/4.

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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