graph of tangent line

Question

Find all points on the graph of  where the tangent line is horizontal.

https://ibb.co/Mfqtb60

Answer 1

A key tip of calculus is to remember that the derivative, the slope of the tangent line and the instant rate of change are all three the same thing. Many "word" problems ask for one of these three; and you need one of the other 2 to find the answer. The slope of a horizontal line is zero, so the tangent line is horizontal when its slope is zero and, therefore, when the derivative of the curve, or relation, is zero.

The general form of an ellipse is

(x-h)²/a²+(y-k)²/b²=1

Differentiating the equation:

2(x-h)/a²+[2(y-k)/b²]dy/dx=0 recalling that the derivative with respect to x of an expression in y is the derivative of that expression, with respect to y, multiplied by the derivative of y with respect to x:

dg(y)/dx=[dg(y)/dy](dy/dx) where g(y) is an expression in y and the dy's cancel.

Then solve for dy/dx.

[2(y-k)/b²]dy/dx=-2(x-h)/a²

dy/dx=[-2(x-h)/a²]/[2(y-k)/b²]

AND

dy/dx=0 when x=h

So we go back to the original equation of an ellipse, and "plug in" h for x.

(x-h)²/a²+(y-k)²/b²=1

(h-h)²/a²+(y-k)²/b²=1

(y-k)²/b²=1

(y-k)²=b²

(y-k)=+/-SQRT(b²)

y=k +/-SQRT(b²)

y=k+SQRT(b²), k-SQRT(b²),

y=k+|b|, k-|b|

THEREFORE, THE POINTS WHERE THE TANGENT LINE IS HORIZONTAL ARE:

(h, k+|b|) and (h, k-|b|)