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:

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.

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

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

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

Hyperbola example

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.

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

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

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

Example 4: t x y p/2 p

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

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

Example 6: Graphs from appropriate settings

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

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

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

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

Example 3: Arc length f(x)

Example 3: continued

Lecture 34 – Polar Coordinates Cartesian Coordinates: Polar Coordinates:

Example 1: Graphing Polar Points

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

Converting Coordinates Switch between polar and Cartesian coordinates.

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

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

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:

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

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

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

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

Example 4: Graphs from appropriate settings

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:

Example 1: Find the slope at q = 0 for

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:

Example 2(cont.): the angles :

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

Example 3(cont.):

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

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

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.

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

Example 1 Find the area for the bounded region.

Example 2 Find the area inside the cardioid.

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

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

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.

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