Question

Sketch the curve and find all points on the cardioid

r=a(1+cos(theta)) where the tangent line is horizontal.

Answer 1

To sketch the curve and find all points on the cardioid r = a(1 + cos(theta)), you can plot points by substituting different values of theta into the equation and connect them to create the shape. To find the points where the tangent line is horizontal, you need to find the values of theta where the derivative of the equation is 0.

Step-by-step explanation:

To sketch the curve and find all points on the cardioid, we need to first understand the equation r = a(1 + cos(theta)). The equation represents a cardioid, which is a type of mathematical curve.

1. To sketch the curve, you can plot a series of points by substituting different values of theta into the equation and calculating the corresponding r values. Then, connect the points to create the shape of the cardioid.
2. To find the points on the cardioid where the tangent line is horizontal, we need to find the values of theta where the slope (or derivative) of the equation is 0. Set up the derivative of the equation and solve for theta. The resulting theta values will give you the points on the cardioid where the tangent line is horizontal.

Answer 2

r = a(1 + cos θ)
If the tandent line is horizontal, then dr/dθ = 0
dr/dθ = -a sin θ = 0
sin θ = 0
θ = 0, π

For θ = 0: r = a(1 + cos 0) = a(1 + 1) = 2a
For θ = π: r = a(1 + cos π) = a(1 - 1) = 0

Therefore, the points where the tangent line is horizontal are (2a, 0) and (0, π)