1. Lecture 31 – Conic Sections
Parabola: the collection of all points that are equidistant from a point(focus) and a line(directrix)
Distance from A to focus:
Distance from B to focus:
Distance from C to focus:

2. Parabola standard forms
Vertex at (h, k), opens up if a > 0, down if a < 0.
Vertex at (h, k), opens left if a > 0, right if a < 0.

3. Ellipse: the collection of all points whose sum of distances from two fixed points (foci) is constant.

4. Ellipse standard form if a > b, then major-axis vertices:
if a = b, circle with radius = a.

5. Hyperbola standard form
Vertices:
Vertices:
Draw a box using all four vertices
Draw the two diagonals
Create curves from the major axis only, treating the diagonals as asymptotes

6. Hyperbola example

7. Lecture 32 – Parametric Curves
Ability to describe functions and non-functions in 2-D by how vertical and horizontal movement changes according to some parameter, t.
Create a table of points and if possible eliminate the parameter to arrive at an equation in two variables.
Curve should note direction and start/end points if available.

8. Example 1: t x y
p/2 p
3p/2 2p

9. Example 2: t x y
p/2 p
3p/2

10. Example 3: t x y
p/4
p/2-

11. Example 4: t x y
p/2 p

12. Example 5:
Find the parametric equations for the curve shown.

13. Graphing Parametric – on the calculator, TI 83/84
Change mode:
MODE, highlight Par
Enter curves at: Y=
input each pair under X1T= , Y1T=, X2T= , Y2T=,
To set window settings: WINDOW
enter Tmin, Tmax, Tstep, xmin, xmax, ymin, ymax
To see plot: GRAPH

14. Example 6:
Graphs from appropriate settings

15. Lecture 33 – Parametric and Calculus
Goal: find slopes on a parametric curve
1st derivative:
2nd derivative:

16. Example 1:
Find slope and concavity at t = 4.

17. Example 2:
Find a tangent line at (0, 2).

18. Example 2: continued
Find a tangent line at (0, 2).

19. Example 3: Arc length
f(x)

20. Example 3: continued

21. Lecture 34 – Polar Coordinates
Cartesian Coordinates:
Polar Coordinates:

22. Example 1: Graphing Polar Points

23. Representations
Infinite number of ways to represent a point using polar coordinates.

24. Converting Coordinates
Switch between polar and Cartesian coordinates.

25. Example 2: Converting Coordinates
Plot and find the Cartesian coordinates for

26. Example 3: Converting Coordinates
Plot and find two Polar coordinates
(one with r>0 and one with r < 0) for

27. Lecture 35 – Polar Curves x-axis: y-axis: origin:
Similar to graphing in Cartesian functions, to graph r = f(q) find all points (r, q) with a representation that satisfies the equation by using:
symmetry, intercepts, table of values
x-axis:
y-axis:
origin:

28. Example 1:
Graph r q
p/6
p/4
p/3

29. Example 2:
Graph r q
0. 16
p/2 p
3p/2

30. Example 3:
Graph r q
p/6
p/2
3p/2

31. Graphing Polar – on the calculator, TI 83/84
Change mode:
MODE, highlight Pol
Enter polar functions: Y=
Input under r1, r2 , r3 , … : r1 =
To set window settings:
WINDOW, qmin, qmax, qstep, xmin, xmax, ymin, ymax
To see plot: GRAPH

32. Example 4:
Graphs from appropriate settings

33. Lecture 36 – Polar Curves and Slopes
Goal: find slopes on a polar curve
Treat (x, y) as a parametric curve with parameter q.
Then, the derivative:

34. Example 1:
Find the slope at q = 0 for

35. Example 2: OR At largest r value:
Find the slopes of the tangent lines at the tips of the leaves for
Halfway through leaf
OR At largest r value:

36. Example 2(cont.):
the angles :

37. Example 3:
Find the locations of the horizontal and vertical tangent lines for the cardioid

38. Example 3(cont.):

39. Example 3(cont.):
Horizontal(num=0)
Vertical(denom=0)

40. Example 3 (concluded):
horizontal
vertical
At q = p?

41. Lecture 37 – Polar Area
Similar to Cartesian Area – divide region into subintervals
Instead of rectangles, circle sectors are used to approximate
the area in each subinterval.
Instead of dealing with rectangular areas:
need to look at circular areas. In particular, sectors of a circle.

42. Sector Area:
ith Sector Area:
Riemann Sum with n subintervals yields:
As n goes to infinity, Riemann Sum goes to:

43. Example 1
Find the area for the bounded region.

44. Example 2
Find the area inside the cardioid.

45. Example 3
Find the area inside the rose.
1 leaf:
Entire rose:

46. Example 4 Find the area outside the circle but inside the leaves.

47. Example 5
Find the area: A1 = inside the cardioid but outside the circle. A2 = inside the intersection of the two curves. A3 = inside the circle but outside the cardioid.

48. Example 5
Set up but do not solve the integrals that represent the areas.