Polar Curves Flashcards
How to convert from polar to Cartesian
You must sub in r, x or y in
If can’t , either SQAURE BOTH SIDES
- or multiply everything by r and then try
- (dint be afraid to sub in sqaure root x2 + y2…
Limacons of form r = a +b cos theta, what happens for different values of a and b and how to tell?
1) if a less then b, there will be a negative part, as max value of cos theta is b, and so goes negative
- can find rhis angle setting r to 0, will be same both sides
2) id a = b then it’s a CARDIOID, with a CUSP
3 if a >b it’s a NEST, where gradient is infinite, don’t lack and draw it closing in!
What happens limacon r = a + b cos theta when a becomes much more bigger and why?
The effect of the b cos theta becomes more negliblfe as a increase, thus becomes more like a circle shape!
Draw a cardioid , a + a cos theta what are the features!
At theta 0 max value, so 2a, theta pi, 0
Theta is pi / 2, intercepts are a!
I’d ensure try yourself then draw out!
What about the Lima son when it’s a +b sin theta?
Here it just changed ordination as the max will now be at PI/2! SO that’s how you can tell
Same results still apply as sin still bounded between -1 and 1 etc, a<b then negative rigging , a = b cardioid cusp, a>b infinite region
If liamcons is a-bcostheta?
Same concept in terms of carodiod etc, just REFLECTED, so make sure graoh still drawn good
How to differntiate between sin and cos petal curves and how many petals do they have?
Petals always = 2 x coeffeicmt if theta, sometimes the petals are hidden due tonegsrive
If a petal is on x axis = cos
If y axis = sin
For p sec formulas?
Domt get cknfused for half line!
Convert to Cartesian and see its a straight line !
Half line lit half, but p sec is a FULL LINE
How to draw r2 curves?
Sqaure root but make sure to draw both + AND - versions
It will be undefined for some area as it equasl -1 here and yiu can’t sqaure root this
- the number of petals are Conserved, and max value = sqaure root!
How to find MAXIMUJ value for r, and theta too? IMPORTANT
Just diffenrjttwe with respect to theta, max value is = to 0 and min too apparently as always, solve for this, see which values of theta give the MAX POSITIVE VALUE
How to find the polar equation of something easily
Draw it, and then try show, r theta and a known value
And relate these
Else write the Cartesian equation EXPMAD, and try to separate in terms of x2 + y2, and x and y etc!
What’s the formula for area for polar?
1/2 r 2
Remmeber if they asking for area where r >0, and itsabout petals then…
Take HALF THE PETALS, find total area , divide by 2!
How to find exactly when r < 0?
Write this inequality, and simialr to solving quadratics we make it = and plot poitnnsand see below 0, do the SAME HERE but draw the curve, and then see where beliw 0
Now integrate betwene thesr two points!
How to integrate BETWEEN AREAS!
the area from pole to any point is just from 0 to angle
So split the area up fro, 0 to that angle and angle to angle
Find these angles by finding the INTERSECTIONS of angle!
Separately integrate AND ADD
- make sure to choose the right shapes properly, draw a GOOD BIG SKETCH
