Implicit differentiation

Rumor

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Here's the problem:

"Consider the curve defined by y^2+xy+x^2=15.

a) Calculate dy/dx.

b) Show that at the two points where y=0, the tangent lines at those points are parallel.

c) Find the coordinates of all points at which there is a horizontal tangent line. Show your work."

For dy/dx, I got dy/dx= -(2x-y)/(2y+x). Is this correct?

Also, could someone show me how to do parts b) and c)?
 
Rumor said:
Here's the problem:

"Consider the curve defined by y^2+xy+x^2=15.

a) Calculate dy/dx.

b) Show that at the two points where y=0, the tangent lines at those points are parallel.

c) Find the coordinates of all points at which there is a horizontal tangent line. Show your work."

For dy/dx, I got dy/dx= -(2x-y)/(2y+x). Is this correct? <<< Almost - you have problem with "sign"
Also, could someone show me how to do parts b) and c)?

b) Show that at the two points where y=0, the tangent lines at those points are parallel.

at y =0 , ? x^2 = 15 ? x = ±sqrt(15)

Now find slopes of tangents at (sqrt(15), 0) and (-sqrt(15), 0).

c) What is the value of dy/dx when the tangent line is horizontal?
 
Hm, okay.

When there's a horizontal tangent line, that means the numerator = 0, right?
 
Rumor said:
Hm, okay.

When there's a horizontal tangent line, that means the numerator = 0, right? <<< Numerator of what? Why?
 
Ah, never mind that. I'm getting concepts mixed up.

It's asking for the points where the derivative equals 0?
 
y2+xy+x2=15, Implicit differentation yields: 2yy+y+xy+2x=0\displaystyle y^{2}+xy+x^{2}=15, \ Implicit \ differentation \ yields: \ 2yy'+y+xy'+2x=0

2yy=xy = 2xy, y=2x+yx+2y\displaystyle 2yy'=xy' \ = \ -2x-y, \ y'=-\frac{2x+y}{x+2y}

Horizontal Tangents: 2x+y = 0      y = 2x (black lines)\displaystyle Horizontal \ Tangents: \ 2x+y \ = \ 0 \ \implies \ y \ = \ -2x \ (black \ lines)

(2x)2+x(2x)+x2 = 15, 4x22x2+x2 = 15, 3x2 = 15,      x = ±5\displaystyle (-2x)^{2}+x(-2x)+x^{2} \ = \ 15, \ 4x^{2}-2x^{2}+x^{2} \ = \ 15, \ 3x^{2} \ = \ 15, \ \implies \ x \ = \ \pm\sqrt5

(5,25), m =0 and (5,25), m = 0 gives y = 25 and y = 25\displaystyle (\sqrt5,-2\sqrt5), \ m \ =0 \ and \ (-\sqrt5,2\sqrt5), \ m \ = \ 0 \ gives \ y \ = \ 2\sqrt5 \ and \ y \ = \ -2\sqrt5

Vertical Tangents: x+2y = 0      x = 2y (green lines)\displaystyle Vertical \ Tangents: \ x+2y \ = \ 0 \ \implies \ x \ = \ -2y \ (green \ lines)

y2+(2y)y+(2y)2 = 15, y22y2+4y2 = 15, 3y2 = 15      y = ±5\displaystyle y^{2}+(-2y)y+(-2y)^{2} \ = \ 15, \ y^{2}-2y^{2}+4y^{2} \ = \ 15, \ 3y^{2} \ = \ 15 \ \implies \ y \ = \ \pm\sqrt5

Hence, x = 25 and x = 25, see graph.\displaystyle Hence, \ x \ = \ -2\sqrt5 \ and \ x \ = \ 2\sqrt5, \ see \ graph.

[attachment=0:esmgwyzi]good.jpg[/attachment:esmgwyzi]
 

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