Worksheet 3 - Math 53: Multivariable Calculus Worksheets 7^th Edition

Curves Defined by Parametric Equations-

Questions-

1. (a) Check that the graph of the function y = x² is the same as the parametrized curve x = t, y = t².

(b) Using (a) as a model, write parametric equations for the graph of y = f(x) where f(x) is any function.

2. Consider the circle C = {(x, y) ∈ R² |x² + y² = 1}.

(a) Is C the graph of some function? If so, which function? If not, why not?

(b) Find a parametrization for C. (Hint: cos²θ + sin²θ = 1.)

3. Consider the parametric equations x = 3t, y = t, and x = 6t, y = 2t.

(a) What curves do the two sets of equations describe?

(b) Compare and contrast the motions for the two sets of parametric equations by interpreting each set as describing the motion of a particle.

(c) Suppose that a curve is parametrized by x = f(t), y = g(t). Explain why x = f(2t), y = g(2t) parametrize the same curve.

(d) Show that there are an infinite number of different parametrizations for the same curve.

Problems-

1. Consider the curve parametrized by x(θ) = a cosθ, y(θ) = b sinθ.

(a) Plot some points and sketch the curve when a = 1 and b = 1, when a = 2 and b = 1, and when a = 1 and b = 2.

(b) Eliminate the parameter θ to obtain a single equation in x, y, and the constants a and b. What curve does this equation describe? (Hint: Eliminate θ using the identity cos²θ + sin²θ = 1.)

2. Consider the parametric equations x = 2 cost - sin t, y = 2 cost + sin t.

(a) Eliminate the parameter t by considering x + y and x - y.

(b) Your result from part (a) should be quadratic in x and y, and you can put it in a more familiar form by substitution x = u + v and y = u - v. Which sort of conic section does the equation in u and v describe?

3. Let C be the curve x = t + 1/t, y = t - 1/t.

(a) Show that C is a hyperbola. (Hint: Consider (x + y)(x - y).)

(b) Which range of values of t gives the left branch of the hyperbola? The right branch?

(c) Let D be the curve x = t² + 1/t², y = t² - 1/t². How does D differ from C? Explain the difference in terms of the parametrizations.

Additional Problem-

1. A helix is a curve in the shape of a corkscrew. Parametrize a helix in R³ which goes through the points (0, 0, 1) and (1, 0, 1).