Chapter 9 Math

When graphing in a polar coordinate system : Ex) graph (3,180 degrees) find 180 degrees and count 3 linesEx) graph (3, -30 degrees) find 330 degrees and count 3 linesEx) graph (-3,120 degrees) find 120 degrees, count 3 lines, and reflect over the x-axis(look across the circle)

From polar to rectangular formulas : x=rcos thetay=rsin theta

Graph each polar equation : Make a chart with 3 columns of theta, the part with sin/cos (Ex:r=1+2sintheta so put sintheta in the center column), put the original function in the last column.Do all theta of 0 degrees to 90 degrees in factors of 15 degrees. The second quadrant should be the reflection of the first quadrant.Do the degrees of factors of 15 for the 3rd quadrant in the table as well.***Test out degrees from all quadrants just to be sure they are symmetrical or not.If it is cos theta it will be that the 1st and 4th quadrant are symmetrical because cos theta in the second quadrant is negative so would reflect to the 4th quadrant.SIN THETA IS SYMMETRICAL WITH 1st/2nd QUADRANT

Polar form : z = r(cos theta + isin theta)

Theorem 4 : The sine and cosine functions are continuous everywhere.

Theorem 5 : 1. Lim x->0 sinx/x = 12. Lim x->0 (1-cosx)/x = 0

Find min and max values of f. : Ex) f(x) = 2sinx - cos2xf'(x) = 2 d/dx sinx - d/dx cos2x= 2cosx -(-sin2x) d/dx 2x= 2cosx + 2sin2x= 2cosx + 4sinxcosx0 = 2cosx(1 + 2sinx)Cosx = 0 or sinx = -1/2x = 90,270 or x=210,330Plug in these degrees to the original function and show the max and min

Rates and extrema problems : dx/dt = ratedx/dt = dx/dtheta x dtheta/dt

Graph of a polar equation is symmetric with respect to the polar axis : Replace (r,theta) by (r,-theta) or (-r,pi-theta)Graph looks like a reflection over the x-axis with 1st/4th quadrant

Graph of a polar equation is symmetric with respect to the pole : Replace (r,theta) by (r,pi+theta) or (-r,theta)Graph looks like a 180 degree rotation with 1st/3rd quadrant