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Derivatives of Parametric Equations

Derivatives of Parametric Equations. Lesson 10.2. Studying Graphs. Recall that a function , y = f(x) is intersected no more than once by a vertical line Other graphs exist that are not functions We seek to study characteristics of such graphs

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Derivatives of Parametric Equations

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  1. Derivatives of Parametric Equations Lesson 10.2

  2. Studying Graphs • Recall that a function, y = f(x) is intersected no more than once by a vertical line • Other graphs exist that are not functions • We seek to study characteristicsof such graphs • How do we determine the slopeat a point on the graph (for aparticular value of t)? We will use parametric equations

  3. Derivative of Parametric Equations • Consider the graph ofx = 2 sin t, y = cos t • We seek the slope, thatis • For parametric equations • For our example

  4. Try It Out • Find dy/dx for the given parametric equations • x = t + 3y = t2 + 1 • What is the slope of the line tangent to the graph when t = 2? • What is the slope of the line tangent to the graph when x = 2?

  5. Second Derivatives • The second derivative is the derivative of the first derivative • But the first derivative is a function of t • We seek the derivative with respect to x • We must use the chain rule

  6. Second Derivatives • Find the second derivative of the parametric equations • x = 3 + 4cos ty = 1 – sin t • First derivative • Second derivative

  7. Try This! • Where does the curve described by the parametric equations have a horizontal tangent? • x = t – 4y = (t 2 + t)2 • Find the derivative • For what value of t does dy/dx = 0?

  8. Human Cannonball • Let's model this motion with parametrics • http://www.dailymail.co.uk/news/article-465214/The-human-cannonball-mastered-art-flying.html

  9. Human Cannonball • Using some software we can get ordered pairs of various locations on the image

  10. Human Cannonball • We know

  11. Human Cannonball • Say his height is 6 ft • We have (412, 121) and (421, 132) as ordered pairs for his height

  12. Human Cannonball • Now how far did he travel • We have 314 units

  13. Human Cannonball • Let's use the parametric equations to determine the velocity

  14. Assignment • Lesson 10.2A • Page 412 • Exercises 1 – 21 • Lesson 10.2B • Page 413 • Exercises 22 – 26 all

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