Question

Find the points on the curve where the tangent is horizontal or vertical. You may want to use a graph from a calculator or computer to check your work. (If an answer does not exist, enter DNE.) x = e^sin(), y = e^cos()

Answer 1

To find the points on the curve where the tangent is horizontal or vertical, we need to find the values of t that make the derivative of y with respect to x equal to zero or undefined. The curve is given by the parametric equations x = e^sin(t) and y = e^cos(t). By differentiating both x and y with respect to t, and setting the derivative to zero or finding values of t that result in an undefined derivative, we can find the points where the tangent is horizontal or vertical.

Step-by-step explanation:

To find the points on the curve where the tangent is horizontal or vertical, we need to determine the values of t that make the derivative of y with respect to x equal to zero or undefined.

The curve is given by the parametric equations x = e^sin(t) and y = e^cos(t). To find the derivative, we can use the chain rule:

dy/dx = (dy/dt) / (dx/dt)

By differentiating both x and y with respect to t, we get:

dx/dt = e^sin(t) cos(t)

dy/dt = -e^cos(t) sin(t)

Setting the derivative to zero or finding values of t that result in an undefined derivative will give us the values where the tangent is horizontal or vertical.

Setting dy/dt = 0, we have:

-e^cos(t) sin(t) = 0

This equation is satisfied when sin(t) = 0 or cos(t) = 0. This occurs at t = 0, t = π, t = 2π, etc.

Therefore, the points on the curve where the tangent is horizontal or vertical are given by (x, y) = (e^sin(t), e^cos(t)) at

t = 0, t = π, t = 2π, etc.