Question

Find the points on the curve x = cos t,   y = 2 sin t,    0 leq t leq 2 pi . at which the tangent is a) horizontal, b) the tangent is vertical. What is the name of this curve?

Answer 1

a) horizontal: t=π/2, 3π/2; points (0, 2), (0, -2)

b) vertical: t=0, π; points (1, 0), (-1, 0)

c) the curve is an ellipse

Explanation:

The derivative of y with respect to x is ...

dy/dx = (dy/dt)/(dx/dt) = (2cos(t))/(-sin(t))

The derivative will be zero (horizontal tangent) when cos(t)=0, at t=π/2 and 3π/2.

The derivative will be undefined (vertical tangent) when sin(t)=0, at t=0 and t=π.

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The curve is an ellipse.