Help Needed: Find Points on x+xy+2y^2=6 with Slope at (2,1)

[ashleyk]

I'm new at this, so I hope someone can help me. I have AP Calculus and need some help on a problem...

Given the curve x+xy+2y^2 = 6

Find the coordinates of all points on this curve with slope equal to the slope at (2,1)

Any help is much appreciated!

 

[Muzza]

First find the derivative (you could, for example, solve for x and calculate dx/dy, or you could perform implicit differentiation on the relation you were given). Then calculate the derivative at (2, 1), etc.

 

[Gokul43201]

Let me add this, to help you with implicit derivatives :

Example : xy^2 + 2x = 1

then (d/dx)(xy^2) + (d/dx)(2x) = (d/dx)1

So, using the product rule on the first term, x(d/dx)(y^2) + y^2(d/dx)(x) + (d/dx)(2x) = (d/dx)1

Now evaluating the derivatives : x(2y)(dy/dx) + y^2(1) + 2 = 0

Separating out dy/dx gives, (dy/dx) = (-2 - y^2)/(2xy)

Now substituting any pair of values (x1,y1) gives you the value of the slope at (x1,y1).