Question

Sketch the curve and find all points on the cardioid
r=a(1+cos(theta)) where the tangent line is horizontal.

Answer 1

totally have to get my notes for this one...

Answer 2

according to Leithold's Calculus...\[{dy\over dx}={{{dr\over d\theta}\sin\theta+r\cos\theta}\over{{dr\over d\theta}\cos\theta-r\sin\theta}}=0\]is what we are looking for

Answer 3

\[\frac{dr}{d\theta}=-a\sin\theta\]so this is\[\frac{dy}{dx}={-a\sin^2\theta+r\cos\theta\over-a\sin\theta\cos\theta-r\sin\theta}=0\]sub in our expression for r and solve for theta
looks like it's gonna get a bit ugly; probably we'll have to use some trig identities

Answer 4

after subbing in\[r=a(1+\cos\theta)\]into the above and playing with some trig identities and simplifying I got\[2\cos^2\theta+\cos\theta-1=0\]\[(2\cos\theta-1)(\cos\theta+1)=0\]which I'm gonna let you take from here
did I mess up phi? is there a better way to do it?

Answer 5

not too bad if you sub (r-a)/a for cos(theta)

Answer 6

of course, you can solve for theta.

Answer 7

ah, that's what would've made this problem prettier.
I knew there was an easier way :D

Answer 8

theta will be nice, ± pi/3 and pi

Answer 9

ok let me see if i can work this out

Answer 10

so i should plug in r-a/a for all cos(theta)?

Answer 11

Are you asking how to sketch the curve?

Once you do, the horizontal tangents are easy to spot.

Answer 12

The tangents occur when theta = ±60º and at 180º