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Consider the equation x^2 - 2xy + 4y^2 = 84

 

1) Write an expression for the slope of the curve at any point (x, y).

 

2) Find the equation of the tangent lines to the curve at the point x=2.

 

3) Find (d^2 y)/(dx^2) at (0, sqrt(21))

 Apr 25, 2019
 #1
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1. Slope at any point ...using implicit differentiation....  2x - 2y - 2xy ' + 8y y '  = 0

 

y' ( 8y - 2x ] = 2y - 2x

 

y '  =   [ 2y - 2x ] / [ 8y - 2x ]

 

2.  At x  = 2....we can find y as

 

2^2 - 2(2)y  + 4y^2  = 4

 

4 - 4y + 4y^2  = 84

 

4y^2 - 4y - 80 = 0

 

y^2 - y - 20  = 0       factor

 

(y - 5) (y + 4)  = 0     .....so   y = 5  or y = -4

 

So....we have the points   ( 2, 5) and (2, -4)

 

So  the slope at  (2, 5)  =  [2(5) - 2(2) ] / [ 8(5) - 2(2) ]  =   6/36  = 1/6

So...the equation is  y  = (1/6) (x - 2) + 5   

 

The slope at (2, -4) = [ 2(-4) - 2(2)] / [ 8(-4) -2(2)]  =  -12/ -36  =  1/3

So...the equation is  y  = (1/3)(x - 2) - 4

 

See the graph here : https://www.desmos.com/calculator/q55m709tum

 

I don't know the answer to the last one.....sorry  !!!

 

 

cool cool cool

 Apr 26, 2019

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