Worksheet 16: Differentiation with Parametric Equations 10/29/08 1. First Derivatives of Parametric Equations Consider the curve x 2 + y 2 = 1. a) Sketch a graph of this curve. b) How could we parametrize this curve? c) Which of these forms do you think is easier to work with? For example, of which form is it easier to take derivatives? d) According to your graph, which point(s) on this curve should have a tangent line with slope 0? To compute derivatives of parametric equations, we will need to use the Chain Rule. Recall that the Chain Rule states that dy dt = dy dt. 1
2 e) Use the Chain Rule to prove that as long as x (c) 0, then at the point where t = c, dy is dy dt equal to (c) dt (c) = y (c) x (c). f) Does your answer to part e) give the same result as your answer to part d)? g) Use your parameterization to find an equation for the line tangent to the curve at t = π/4. h) Use your parametrization to find a point on the curve whose tangent line is vertical.
3 2. Second Derivatives of Parametric Equations a) Apply the Chain Rule to dy to obtain d2 y 2 = d ( ) dy = d dt ( ) dy dt. Consider the curve x = 2t 2 + 1, y = 3t 3 + 2. b) Find the equation for the line tangent to the curve at time t = 1. c) Compute d2 y 2 at the point where t = 1 to determine whether the curve is concave up or concave down at that point. d) Compute y (t) x (t) and compare it to d2 y. Are they the same? 2
4 3. Consider the curve x = e t, y = e t. a) What is the slope of the tangent line to this curve at time t? b) Compute d2 y using the parametric equations. 2 c) Eliminate the parameter to describe the curve as an equation of x and y. e) Compute dy and d2 y 2 parts a) and b)? using your answer from part c). Do they match your answers from
5 4. Speed We often use parametric equations to describe objects in motion. The position vector for the motion is (x(t), y(t)) and the velocity vector is given by (x (t), y (t)). We say that x (t) is the horizontal component of velocity and that y (t) is the vertical component. We define the speed to be s(t) = [x (t)] 2 + [y (t)] 2. a) Discuss with your group how the velocity vector, horizontal and vertical components of velocity, and speed are related. A picture may be useful. Suppose that an object follows the path x = sin(4t), y = cos(4t). b) What is the object s speed as a function of t? c) What is the slope of the tangent line to this curve at time t?
6 Suppose further that there is a line segment connecting the object to the origin. d) What is the slope of this line segment at time t? e) What is the relationship between this line segment and the tangent line? 5. Cycloids Suppose a circle with radius r is rolling on the x-axis at speed v > r headed in the positive direction. There is a specific point we are interested in on this circle and that it is at (r, r) at time t = 0. a) Find parametric equations that describe the path taken by the circle s center as it rolls along the x-axis. b) Parametrically describe the movement of our specific point on the edge around the center of the circle.
7 c) Combine your answers to parts a) and b) to obtain parametric equations that describe the movement of this specific point as the circle rolls along the x-axis. This curve is called a cycloid. d) Compute the maximum and minimum speeds of this point and the locations where they occur. e) Graph the cycloid for v = 3 and r = 2.
8 6. Trochoids, Hypocycloids, and Epicycloids a) Now suppose that our specific point is not on the edge of the circle, but rather is inside of it at distance d < r from the center. Find parametric equations that describe the movement of this new point as the circle rolls along the x-axis. This curve is called a trochoid. b) Let s complicate things a bit more. Instead of rolling a circle along the x-axis, consider a smaller circle rolling along the inside of a bigger circle (with radii a < b). Describe the movement of a point on the edge of the smaller circle with parametric equations. This curve is a hypocycloid. c) If we roll the smaller circle along the outside of the larger circle, our resulting curve is an epicycloid. Find parametric equations to describe this curve. d) What will the graphs of these strange curves look like? Try some ideas on your white board.