10.6B and 10.7 Calculus of Polar Curves

Try graphing this on the TI-Nspire

Graphing Polar Equations Recognizing Common Forms Circles Centered at the origin: r = a radius: a period = 360 Tangent to the x-axis at the origin: r = a sin  center: (a/2, 90) radius: a/2 period = 180 a > 0  above a < 0  below Tangent to the y-axis at the origin: r = a cos  a > 0  right a < 0  left r = 4 sin r = 4 cos Note the Symmetries

Graphing Polar Equations Recognizing Common Forms Flowers (centered at the origin) r = a cos n or r = a sin n radius: |a| n is even  2n petals petal every 180/n period = 360 n is odd  n petals petal every 360/n period = 180 cos  1st petal @ 0 sin  1st petal @ 90/n r = 4 sin 2 r = 4 cos 3 Note the Symmetries

Graphing Polar Equations Recognizing Common Forms Spirals Spiral of Archimedes: r = k |k| large  loose |k| small  tight r =  r = ¼ 

Graphing Polar Equations Recognizing Common Forms Heart (actually: cardioid if a = b … otherwise: limaçon) r = a ± b cos  or r = a ± b sin  r = 3 + 3 cos  r = 2 - 5 cos  r = 3 + 2 sin  r = 3 - 3 sin  Note the Symmetries

Graphing Polar Equations Recognizing Common Forms Leminscate a = 16 Note the Symmetries

To find the slope of a polar curve: We use the product rule here.

To find the slope of a polar curve:

Example:

Area Inside a Polar Graph:

Tangent lines at the pole The line is tangent to the graph of at the pole if Ex. Graph and find the tangent(s) at the pole

Example: Find the area enclosed by:

Notes: To find the area between curves, subtract: Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

When finding area, negative values of r cancel out: Area of one leaf times 4: Area of four leaves:

To find the length of a curve: Remember: For polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: So:

There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: p