11–14. Working with parametric equations Consider the following parametric equations.

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).

x=−t+6, y=3t−3; −5≤t≤5 

Area of roses Assume m is a positive integer.

a. Even number of leaves: What is the relationship between the total area enclosed by the 4m-leaf rose r=cos(2mθ) and m?

Channel flow Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point P(r,θ) in the channel, the flow is in the tangential direction (counterclock wise along circles), and it depends only on r, the distance from the center of the semicircles.

a. Express the region formed by the channel as a set in polar coordinates.

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).

Area of a sector of a hyperbola: Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus

a. What is the area of R?

Navigating A plane is 150 miles north of a radar station, and 30 minutes later it is 60 degree east of north at a distance of 100 miles from the radar station. Assume the plane flies on a straight line and maintains constant altitude during this 30-minute period.

a. Find the distance traveled during this 30-minute period.

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

a. x = 1 + s, y = 2s and x = 1 + 2t, y = 3t

The ellipse and the parabola: Let R be the region bounded by the upper half of the ellipse x²/2 + y² = 1 and the parabola y = x²/√2

a. Find the area of R

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 = 2 + 4t, y = 4 − 8t; t = 2

Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus.

a. What is the volume of the solid that is generated when R is revolved about the x-axis?

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

a. The point with Cartesian coordinates (−2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4), (2√2, −5π/4), and (−2√2,−π/4).  

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

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 = t + 1/t, y = t − 1/t; t = 1

(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.

Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).

a. Find the area Aₙ of Rₙ.

11–14. Working with parametric equations Consider the following parametric equations.

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).

x=2 t,y=3t−4;−10≤d≤10 

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).

b. Explain why tan θ = y/x.

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

b. An object following the parametric curve x=2cos 2πt, y=2 sin 2πt circles the origin once every 1 time unit.

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t

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

b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.

Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus.

b. What is the volume of the solid that is generated when R is revolved about the y-axis?

67–72. Derivatives Consider the following parametric curves.

b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t.

x = 2 + 4t, y = 4 − 8t; t = 2

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

c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².

Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).

c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.

11–14. Working with parametric equations Consider the following parametric equations.

c. Eliminate the parameter to obtain an equation in x and y.

d. Describe the curve.

x=−t+6, y=3t−3; −5≤t≤5 

Spiral arc length Consider the spiral r=4θ, for θ≥0.

c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.

11–14. Working with parametric equations Consider the following parametric equations.

c. Eliminate the parameter to obtain an equation in x and y.

d. Describe the curve.

x=2 t,y=3t−4;−10≤t≤10 

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t

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

d. The point on a parabola closest to the focus is the vertex.

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

d. The point (3,π/2) lies on the graph of r=3 cos 2θ.