10.3 Parametric Equations. 1 Math 1432 Dr. Almus

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Math 1432 DAY 39 Dr. Melahat Almus almus@math.uh.edu OFFICE HOURS (212 PGH) MW12-1:30pm, F:12-1pm. If you email me, please mention the course (1432) in the subject line.

1 Math 1432 DAY 39 Dr. Melahat Almus OFFICE HOURS (212 PGH) MW12-1:30pm, F:12-1pm. If you me, please mention the course (1432) in the subject line. Check your CASA account for Quiz due dates. Don t miss any online quizzes! Be considerate of others in class. Respect your friends and do not distract anyone during the lecture Parametric Equations 1 Math 1432 Dr. Almus

Finding Parametric Equations for a graph 1) Line Segments or Lines To parameterize a line SEGMENT from x y to, 0, 0 x y : 1 1 0 1 0 x

2 Finding Parametric Equations for a graph 1) Line Segments or Lines To parameterize a line SEGMENT from x y to, 0, 0 x y : x t x t x x yt y t y y 0t For a LINE, use: t Example: Parameterize the line segment from (3, 6) to ( 2, 5). 2 Math 1432 Dr. Almus

3 Example: Parameterize the line through (2, 1) and (8, 4). 3 Math 1432 Dr. Almus

4 2) Circles or Ellipses You can use: x() t acos() t and yt () bsin() t ; 0 t 2 (where a and b 0 ) to parametrize the ellipse x a 2 2 y b If a b, then these equations will trace the circle x y a. Note that these sets would trace the ellipse counterclockwise. To change the orientation, you can switch the trig functions. Example: Parametrize x 2 2 y 4. Example: Parametrize 2 2 x y Math 1432 Dr. Almus

5 2 2 Example: Parametrize x1 y ) A general function: Example: Parametrize the curve: 3 y x x. 5 Math 1432 Dr. Almus

6 10.4 Derivatives of Curves Given Parametrically To make sure that at least one tangent line exists at each point of C, we will make the additional assumption that 2 2 x' t y' t 0 This assumption is equivalent to requiring that x (t) and y (t) are not simultaneously equal to 0. 6 Math 1432 Dr. Almus

7 If y is a function of t and t is a function of x, then dy dy dy dt dx dt, 0 dx dt dx dx dt dt Slope of tangent line for parametric curves: dy y' t m dx x ' t So, the equation of the tangent line to a parametric curve at t = t 0 is y' t y y t x x t x' t0 Example: Find dy dx and evaluate at the given value for x t y t t t 2 2, 3 2, 1 7 Math 1432 Dr. Almus

8 Example: Find the slope and equation of the tangent line when t=1; 2 3 F() t 2 t, t 4t. Exercise: Find an equation for the tangent line when t 3 /4; xt 4sin( t), y( t) 2tan( t). 8 Math 1432 Dr. Almus

9 Finding Horizontal and Vertical Tangent Lines: Example: 2 3 x t t, y( t) t 3t Find the values of t for which the curve has a) horizontal tangent line(s): b) vertical tangent line(s): 9 Math 1432 Dr. Almus

10 Exercise: xt sin(2 t), y( t) 4sin( t) Find the values of t for which the curve has a) horizontal tangent line(s): b) vertical tangent line(s): 10 Math 1432 Dr. Almus

11 Exercise: Find the points where the curve has horizontal and vertical tangents: x t 4t 6 t, y( t) t 12t 11 Math 1432 Dr. Almus

12 10.5 Arc length for Parametric Curves Recall: Formula for finding the arc length of a curve in rectangular form: b 2 L() c 1 f '() x dx a Formula for finding the arc length of a curve in polar form: 2 2 L() c r r'() d New: Formula for finding the arc length of a curve in parametric form: b 2 2 L() c x' t y'() t dt a x t 3 t, y( t) 2 t, 0 t 1 Example: Find the arc length of the curve Math 1432 Dr. Almus

13 Example: Give an integral which represents the length of the curve given s t 2cos t,3sin t for 0 t 2. parametrically by 13 Math 1432 Dr. Almus

14 Example: Give an integral which represents the length of the curve given r t 2cos 3 t,3sin 4t for 0 t 2. parametrically by 14 Math 1432 Dr. Almus

15 Velocity and Speed (magnitude of velocity) If the position of a particle at time t is given by s() t x(), t y() t then the velocity is given by vt ( ) x'( t), y'( t) and the speed is given by speed = vt x' t y' t 2 2 Acceleration: at ( ) x''( t), y''( t) 15 Math 1432 Dr. Almus

16 Example: A particle is traveling on an elliptic path in the xy-plane so that its position at time t is s t 2cos t,3sin t. given by Give the position, velocity and speed of the particle at time t = /4. 16 Math 1432 Dr. Almus

17 Exercise: Suppose a particle is traveling on a path represented by a portion of the xy-plane. In this case, assume that the y-axis represents the up/down direction. Given this information, suppose the position of the particle is given at time t in seconds by 2, st t t t for t > 0 until the particle strikes the ground. Give the time when the particle strikes the ground, and determine the velocity and acceleration of the particle at this time. 17 Math 1432 Dr. Almus

18 18 Math 1432 Dr. Almus

19 Example: Consider the curve: x() t 2cos(), t y() t 2sin(); t 0t. Find the area of the surface formed when this curve is rotated about the x-axis. 19 Math 1432 Dr. Almus

20 Example: Consider the curve in the first quadrant determined by the curves: x() t 3cos(), t y() t 4sin() t ; If this curve is rotated about the x-axis, set up an integral that gives the area of the surface of revolution. NEXT: REVIEW FOR FINAL. THE END!!!! 20 Math 1432 Dr. Almus

Math 1431 DAY 16. Dr. Melahat Almus. OFFICE HOURS: MWF 11-11:30am, MW 1-2:15pm at 621 PGH

Math 1431 DAY 16. Dr. Melahat Almus. OFFICE HOURS: MWF 11-11:30am, MW 1-2:15pm at 621 PGH Math 1431 DAY 16 Dr. Melahat Almus almus@math.uh.edu OFFICE HOURS: MWF 11-11:30am, MW 1-:15pm at 61 PGH If you e-mail me, please mention your course (1431) in the subject line. Check your CASA account