9.1 Parametric Curves

Transcription

1 Math 172 Chapter 9A notes Page 1 of Parametric Curves So far we have discussed equations in the form. Sometimes and are given as functions of a parameter. Example. Projectile Motion Sketch and axes, cannon at origin, trajectory Mechanics gives and. Time is a parameter. Given parameter. Then, are parametric equations for a curve in the -plane. Example., Draw the curve in the -plane. t x y Sketch axes and line in -plane Eliminate Example where

2 Math 172 Chapter 9A notes Page 2 of 20 -, -, plot points, draw portion of circle with arrow, show Parameter is the angle. Eliminate, Gives the first quadrant portion of a circle of radius. Example. hyperbola with asymptotes sketch -axes, asymptotes, hyperbola parametric equations describe the top branch of the hyperbola A cycloid is a curve traced by a point on the rim of a rolling wheel. sketch wheel, wheel rolled about a quarter turn ahead, portion of cycloid Find parametric equations

3 Math 172 Chapter 9A notes Page 3 of 20 circle has radius a point on the cycloid length of arc Parametric equations for cycloid Table 0 sketch -axes, plot points, draw curve through them one arch of the cycloid Drawing Graphs of Parametric Equations using Maple Command form: plot([ -expression, -expression, parameter range],scaling=constrained); Show graph of parametric equations

4 Math 172 Chapter 9A notes Page 4 of Calculus with parametric curves Tangents Curve in -plane described by parametric equations Chain rule This gives the slope of curve Let This gives the concavity of the curve Example (a) Find the equation of the line tangent to the curve at. At equation of tangent line

5 Math 172 Chapter 9A notes Page 5 of 20 or (b) Sketch the curve and the tangent line sketch -axes, asymptotes, plot points, draw upper branch of hyperbola, sketch tangent line (c) Find and discuss the concavity of the curve. Since, for The top branch of the hyperbola is concave up. Example. Cycloid, Table with additional row

6 Math 172 Chapter 9A notes Page 6 of , -, plot points, draw cycloid (a) Discuss the slope at and in the limit as At add segment with slope 0 At add segment with slope 1 As where L Hospital s rule has been used. The cycloid has a vertical tangent in the limit. add (b) Discuss concavity of the cycloid

7 Math 172 Chapter 9A notes Page 7 of 20 for The cycloid is concave down over the entire arch, except for the cusp points defined. where it is not Areas sketch -,, curve above Area under the curve Suppose, where and Example. Find the area of the circle,, sketch -axes, unit circle, angle CCW from positive -axis Notice the last integral integrates over a full period of cosine. Example. Find the area of the asteroid

8 Math 172 Chapter 9A notes Page 8 of 20 sketch -, -, astroid therefore so Notice The first two integrals are seen to be zero by symmetry because the integrands are odd powers of cosine and the argument varies over a full period. The value of the last integral can be seen from the fact that the average value of sin 2 or cos 2 over their period is ½. It is also an immediate consequence of the half angle identities.

9 Math 172 Chapter 9A notes Page 9 of 20 Arc Length Symbolically -, -, curve over, triangle Suppose is described by parametric equations then where and. Example Find the length of the curve Example Find the total length of the astroid

10 Math 172 Chapter 9A notes Page 10 of 20 -, -, astroid For Therefore and where we used 9.3 Polar Coordinates Cartesian or rectangular coordinates is associated with a unique ordered pair

11 Math 172 Chapter 9A notes Page 11 of 20 Polar Coordinates 0, 0, indicate the polar axis, ray, distance from origin angle measured CCW from the polar axis Example. Plot ray, Representation in polar coordinates is not unique. axes,, indicate backward extension of ray through origin may be represented by,, Example. May represent by Sometimes restrict every point except has a unique representation. anything always represents the origin Conversion from polar to Cartesian coordinates, ray,, projection from tip to -axis

12 Math 172 Chapter 9A notes Page 12 of 20 Example. Convert to Cartesian coordinates Conversion from Cartesian to polar coordinates Example. Convert to polar coordinates with and., [1a] [1b] Solutions of [1] with and :, Notice that But is not the same point as Equations [1] are not sufficient, we must also choose to be in the correct quadrant.

13 Math 172 Chapter 9A notes Page 13 of 20 Polar Equations General form Common form Example. axes, circle of radius circle, center at origin, with radius To find equation in Cartesian coordinates, square both sides: giving Example. Find the polar equation for the curve represented by [2] Let and, then Eq. [2] becomes Solutions are or [2] is an equation for a circle. To see, complete squares sketch axes, circle centered at with radius circle with radius and center.

14 Math 172 Chapter 9A notes Page 14 of 20 Symmetry of solutions of graph axes, and ray in first quadrant, and ray, and ray, and ray (1) If whenever, the solution is symmetric about the -axis. (2) If whenever, the solution is symmetric about the -axis. (3) If whenever, the solution is symmetric about the origin. Example... Solution is symmetric about the -axis. Example. Solution is symmetric about the -axis Sketch sketch points in first quadrant, draw smooth curve through them, complete in fourth quadrant This is a cardioid. Tangents to Polar Curves Common form of a polar equation

15 Math 172 Chapter 9A notes Page 15 of 20 where and. Consider as a parameter, then from the results of section 9.2 Let Suppose the graph of passes through the origin at an angle axes, ray, curve passing through origin tangent to ray slope = add projection down from ray to -axis to complete a right triangle Example. Find the slope of the line tangent to at. At,,,, Thus

16 Math 172 Chapter 9A notes Page 16 of 20 Sketch sketch axes, ray, plot points, draw curve 9.4 Areas and Lengths in Polar Coordinates Area of a sector of a circle circle of radius, sector subtended by, area Area bounded by a polar curve : dashed polar axis, ray, ray, curve between these angles, small sector subtended by, its angle Area of slice with angle : Total area bounded by and the rays and Example. Find the area enclosed by the loop of between and. Use the half-angle identity

17 Math 172 Chapter 9A notes Page 17 of 20 with the substitution. Example. Find the area of the region that lies inside the graph of but outside the graph of plot points for cardioids and draw curve (lower half plane by symmetry), plot points for circle and draw curve. Area has components in the first and second quadrants. Area between curves is. Find intersections between curves This misses the intersection at the origin! Why? Graphs have different values of at the origin.

18 Math 172 Chapter 9A notes Page 18 of 20 Then. Area between the curves. Arc Lengths in Polar Coordinates sketch, ray, ray, curve Symbolically where Keeping in mind that depends on

19 Math 172 Chapter 9A notes Page 19 of 20 Thus Example. Find the length of the cardioid. where consider Then for

20 Math 172 Chapter 9A notes Page 20 of 20