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Calculus/Equation of Tangent Line

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Expert: Alon Mandes - 10/3/2008


Question
x^2 + xy + y^2 = 1

a) find all pts which the tangent line is horizontal (m=0)
b) find all pts in which tangent line is parallel to y=-x (so basically when m = -1)

thanks!

Answer
The best way is to write y as function of x : x^2 + xy + y^2 = 1
--> y^2+y*x+(x^-1)=0 --> y=-x/2ħsqrt[x^2-4(x^2-1)]/2. Let's take
the upper half of the symetric closed curve :
y=-x/2+sqrt[4-3x^2]/2. Let's calculate derivative :
y'=-1/2+(1/2)*(-3x)/sqrt[4-3x^2].
a) y'=0 --> -3x/sqrt[4-3x^2]=1 --> 9x^2=4-3x^2 -->12x^2=4-->x=√3.
b)y'=-1 --> -1=-1/2-(3x/2)*(1/sqrt[4-3x^2])-->1/2=3x/2sqrt[4-3x^2]
& you may continue from here ..

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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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

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

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