1. AP Calculus BC Monday, 16 November 2015
OBJECTIVE TSW (1) find the slope of a tangent line to a parametric curve, and (2) find the concavity of a parametric curve.
TODAY’S ASSIGNMENT (due tomorrow)
Sec. 11.2: Problems given on the next slide.
TEST: Parametric Equations, Polar Equations, and Vectors is this Friday, 20 November 2015.

2. Sec. 11.2: Calculus with Parametric Equations
Find (a) dy/dx, (b) d 2y/dx2, (c) the slope, (d) the equation of the tangent line, and (e) concavity (if possible) at the given value of the parameter.
Due tomorrow, Tuesday, 17 November 2015.
Find all points (if any) of horizontal and vertical tangency to the curve defined by the given parametric equations.

3. Sec. 11.2: Calculus with Parametric Equations3

4. Sec. 11.2: Calculus with Parametric Equations
How do you find the derivative of a set of parametric equations? 4

5. Sec. 11.2: Calculus with Parametric Equations
11.1 5

6. Sec. 11.2: Calculus with Parametric Equations
Ex: Find dy / dx for the curve given by 6

7. Sec. 11.2: Calculus with Parametric Equations
For higher order derivatives, use Theorem 11.1 repeatedly.
Second derivative
Third derivative
Notice that the denominator for each higher-order derivative is always dx/dt. 7

8. Sec. 11.2: Calculus with Parametric Equations
Ex: For the curve given by
find the slope and concavity at the point (2, 3). 8

9. Sec. 11.2: Calculus with Parametric Equations
The second derivative is 9

10. Sec. 11.2: Calculus with Parametric Equations
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

11. Sec. 11.2: Calculus with Parametric Equations
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. 11.2: Calculus with Parametric Equations
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. 11.2: Calculus with Parametric Equations
Ex: The prolate cycloid given by
crosses itself at the point (0, 2). Find the equations of both tangent lines at this point.

14. Sec. 11.2: Calculus with Parametric Equations
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.

15. Sec. 11.2: Calculus with Parametric Equations
Solve one of these equations for t.
The second equation would be the easiest.

16. Sec. 11.2: Calculus with Parametric Equations
When t = /2,
and the equation is

17. Sec. 11.2: Calculus with Parametric Equations
When t = –/2,
and the equation is

18. Sec. 11.2: Calculus with Parametric Equations
Horizontal Tangents If
when t = t0, then the curve represented by
has a horizontal tangent at

19. Sec. 11.2: Calculus with Parametric Equations
Vertical Tangents If
when t = t0, then the curve represented by
has a vertical tangent at

20. Sec. 11.2: Calculus with Parametric Equations
Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined by x = t + 1 and y = t 2 + 3t.

21. Sec. 11.2: Calculus with Parametric Equations
Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined by x = t + 1 and y = t 2 + 3t.
never

22. Sec. 11.2: Calculus with Parametric Equations
never
Horizontal tangency:
Vertical tangency:
never
NONE

23. Sec. 11.2: Calculus with Parametric Equations
Ex: Find all points (if any) of horizontal and vertical tangency to the curve defined by x = cosθ and y = 2sin2θ.

24. Sec. 11.2: Calculus with Parametric Equations
Find (a) dy/dx, (b) d 2y/dx2, (c) the slope, (d) the equation of the tangent line, and (e) concavity (if possible) at the given value of the parameter.
Due tomorrow, Tuesday, 17 November 2015.
Find all points (if any) of horizontal and vertical tangency to the curve defined by the given parametric equations.