1 / 25

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.

aminia
Download Presentation

AP Calculus BC Wednesday , 06 November 2013

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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:

More Related