Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.

(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933

d. Plot the curve with various values of k. How many fingers can you produce?

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).

e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.

63–74. Arc length of polar curves Find the length of the following polar curves.

The curve r = sin³(θ/3), for 0 ≤ θ ≤ π/2