Eliminate the parameter in the parametric equations x=1+sin t, y=3+2 sin t, for 0≤t≤π/2, and describe the curve, indicating its positive orientation. How does this curve differ from the curve x=1+sin t, y=3+2 sin t, for π/2≤t≤π?

Reflection property of parabolas: Consider the parabola y = x²/(4p) with its focus at F(0, p). The goal is to show that the angle of incidence (α) equals the angle of reflection (β).

a. Let P(x₀, y₀) be a point on the parabola. Show that the slope of the tangent line at P is tan θ = x₀/(2p).

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.

a. For what value of p is P tangent to H?

Cartesian conversion Write the equation x=y ² in polar coordinates and state values of θ that produce the entire graph of the parabola.

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The segment of the curve x=y ³ +y+1 that starts at (1, 0) and ends at (11, 2).

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.

x = cos t, y = 8 sin t; t = π/2

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

a. Show that f(0) = f(2π) and find the point on the curve that corresponds to θ = 0 and θ = 2π.