MAT 271 Recitation. MAT 271 Recitation. Sections 10.1,10.2. Lindsey K. Gamard, ASU SoMSS. 22 November 2013

Transcription

1 MAT 271 Recitation Sections 10.1,10.2 Lindsey K. Gamard, ASU SoMSS 22 November 2013

2 Agenda Today s agenda: 1. Introduction to concepts from 10.1 and Example problems 3. Groupwork

3 Section 10.1 Introduction to Parametric Equations - Motivation One application of parametric equations is to provide a way to model movement in two-dimensional space as a function of time. Suppose you were to record a video of your movement on your phone s navigation app. Plotting a parametric equation could reproduce this video! There are three variables involved in the type of parametric equations just described: x (a horizontal coordinate), y (a vertical coordinate), and t (time). Parametric equations have more applications that just the one mentioned above. Warning: do NOT assume that x, y, and t always represent locations and times!

4 Section 10.1 Introduction to Parametric Equations - Formal Definition Definition. A parametric equation is an ordered pair of real-valued functions (x(t), y(t)) such that x and y each depend on a third variable t, called the parameter.

5 Section 10.1 Example 3, page 637 Problem. A turtle walks with constant speed in the counterclockwise direction on a circular track of radius 4ft centered at the origin. Starting from the point (4, 0), the turtle completes one lap in 30 minutes. Find a parametric description of the path of the turtle at any time t 0.

6 Section 10.1 Example 3, page 637 Problem. A turtle walks with constant speed in the counterclockwise direction on a circular track of radius 4ft centered at the origin. Starting from the point (4, 0), the turtle completes one lap in 30 minutes. Find a parametric description of the path of the turtle at any time t 0. Answer. A circle of radius 4 can be represented by the functions x(t) = 4 cos(bt) and y(t) = 4 sin(bt), where x represents the horizontal coordinate in feet, y represents the vertical coordinate in feet, and t represents time in minutes. The unknown constant b depends on how quickly the turtle moves. He completes one lap in 30 minutes, or in other words, he travels 2π radians in 30 minutes, so b = 2π 30 = π 15.

7 Section 10.1 Example 3, page 637 Exercise. Compute (x(t)) 2 + (y(t)) 2 for x(t) = 4 cos(bt) and y(t) = 4 sin(bt) as in the previous example problem (recall that b = π ). What do you notice about your answer? 15

8 Section 10.1 Example 3, page 637 Exercise. Compute (x(t)) 2 + (y(t)) 2 for x(t) = 4 cos(bt) and y(t) = 4 sin(bt) as in the previous example problem (recall that b = π ). What do you notice about your answer? 15 It is worth noting that this is a special case in a particular problem. You should not expect results like this with every parametric equation you encounter. You should also notice that y is not a function of x in this problem: y and x are individually functions of t, but a single value of x is related to more than one value of y.

9 Section 10.2 Visualizing polar coordinates

10 Section 10.2 Visualizing polar coordinates

11 Section 10.2 Visualizing polar coordinates

12 Section 10.2 Converting between Cartesian and polar coordinates The basics. Here are the equations for converting between Cartesian and polar coordinates: x = r cos θ y = r sin θ, where r is the distance (possibly negative) from the origin, and θ is the angle as measured counterclockwise from the positive x-axis.

13 Section 10.2 Example 2, page 647 Problem. Convert the following: (a) The point ( 2, 3π ) 4 in polar coordinates to the point (x, y) in Cartesian coordinates. (b) The point (1, 1) in Cartesian coordinates to the point (r, θ) in polar coordinates.

14 Section 10.2 Example 2, page 647 Problem. Convert the following: (a) The point ( 2, 3π ) 4 in polar coordinates to the point (x, y) in Cartesian coordinates. (b) The point (1, 1) in Cartesian coordinates to the point (r, θ) in polar coordinates. Answer. (a) ( 2, 2) (b) There are many possible answers, but we restrict ourselves to 0 θ < 2π to get two of them: ( 2, 3π ) ( ) 4 and 2, 7π 4.