Question

BREAKING NEWS Eyeballing Ellipses Evidence that Implicit Differentiation Works The graph of x2 -xy y2 = 4 is a tilted ellipse, as shown below. Sketch the approximate tangent lines on the ellipse where the slopes of the tangent lines are (a) zero and (b) undefined by "eyeballing". Label the coordinates with your best estimate to two decimal places. Then, use implicit differentiation to determine the exact values of the coordinates. Express your answers as proper square root ratios. Use a calculator to round your exact answers to two decimal places to CN compare with your original "eyeball" estimates. 1

Answer 1

Given the ellipse equation:

2 y-ry(x-4) 0

By eyeballing the graph we can say that tangent with slope zero is approximately at (1.2,2.3) and (-1.2,-2.3).

The tangent whose slope is undefined is approximately at (2.3,1.2) and (-2.3,-1.2).

Now by solving the above equation,

we get

tV 4r16 y = 2

t16 3x y = 2

By differentiating;

3.x y' 2 2 16 3r2

Here y' is the slope of the tangent.

For slope to be zero;y'=0

3x 2 2 2 16 3x2

2 =土1.15 V3
    
y=±2.31]

For slope to be undefined; denominator must be zero.

16 320

r= 土-=土2.31 V3
      
=t1.15

So both eyeballed values and derived values are almost the same.