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AP Calculus BC Wednesday , 06 November 2013

AP Calculus BC Wednesday , 06 November 2013. OBJECTIVE TSW (1) find the slope of a tangent line to a parametric curve, and (2) explore parametric vectors. Next Test : Wednesday, 13 November 2013 (next week). This test will cover parametric equations, polar equations, and vectors.

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AP Calculus BC Wednesday , 06 November 2013

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  1. AP Calculus BCWednesday, 06 November 2013 • OBJECTIVETSW (1) find the slope of a tangent line to a parametric curve, and (2) explore parametric vectors. • Next Test: Wednesday, 13 November 2013 (next week). • This test will cover parametric equations, polar equations, and vectors.

  2. Sec. 10.3: Parametric Equations and Calculus

  3. Sec. 10.3: Parametric Equations and Calculus How do you find the derivative of a set of parametric equations?

  4. Sec. 10.3: Parametric Equations and Calculus

  5. Sec. 10.3: Parametric Equations and Calculus Ex: Find dy / dx for the curve given by

  6. Sec. 10.3: Parametric Equations and Calculus For higher order derivatives, use Theorem 10.7 repeatedly. Second derivative Third derivative Notice that the denominator for each higher-order derivative is alwaysdx/dt.

  7. Sec. 10.3: Parametric Equations and Calculus Ex: For the curve given by find the slope and concavity at the point (2, 3).

  8. Sec. 10.3: Parametric Equations and Calculus The second derivative is

  9. Sec. 10.3: Parametric Equations and Calculus We’re given the point (2, 3) & Since x = 2, that means that or t = 4. The slope at (2, 3) is: And the concavity at (2, 3) is: ∴concave up

  10. Sec. 10.3: Parametric Equations and Calculus Ex: The prolate cycloid given by crosses itself at the point (0, 2). Find the equations of both tangent lines at this point.

  11. Sec. 10.3: Parametric Equations and Calculus Ex: The prolate cycloid given by crosses itself at the point (0, 2). Find the equations of both tangent lines at this point.

  12. Sec. 10.3: Parametric Equations and Calculus Ex: The prolate cycloid given by crosses itself at the point (0, 2). Find the equations of both tangent lines at this point.

  13. Sec. 10.3: Parametric Equations and Calculus A point is given; you need only determine the slope, dy/dx. Now you need to determine t. Use the original parametric equations to determine t.

  14. Sec. 10.3: Parametric Equations and Calculus Solve one of these equations for t. The second equation would be the easiest.

  15. Sec. 10.3: Parametric Equations and Calculus When t = /2, and the equation is

  16. Sec. 10.3: Parametric Equations and Calculus When t = –/2, and the equation is

  17. TEST Sec. 3.5 – 3.9

  18. Sec. 10.3: Parametric Equations and Calculus Horizontal Tangents If when t = t0, then the curve represented by has a horizontal tangent at

  19. Sec. 10.3: Parametric Equations and Calculus Vertical Tangents If when t = t0, then the curve represented by has a vertical tangent at

  20. Sec. 10.3: Parametric Equations and Calculus Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined byx = t + 1 and y = t 2 + 3t.

  21. Sec. 10.3: Parametric Equations and Calculus Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined byx = t + 1 and y = t 2 + 3t. never

  22. Sec. 10.3: Parametric Equations and Calculus never Horizontal tangency: Vertical tangency: never NONE

  23. Sec. 10.3: Parametric Equations and Calculus Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined byx = cosθ and y = 2sin2θ.

  24. Sec. 10.3: Parametric Equations and Calculus

  25. Sec. 10.3: Parametric Equations and Calculus Horizontal tangency: Vertical tangency:

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