The equation

StartFraction dy Over dx EndFraction vertical line Subscript left parenthesis r comma theta right parenthesis Baseline equals StartFraction f prime left parenthesis theta right parenthesis sine theta plus f left parenthesis theta right parenthesis cosine theta Over f prime left parenthesis theta right parenthesis cosine theta minus f left parenthesis theta right parenthesis sine theta EndFractiondydx$($r$,$$)=$f$($$)$sin$+$f$($$)$cosf$($$)$cos$$f$($$)$sin

gives a formula for the derivative

y primey$$

of a polar curve

requals$=$f$($theta$).$

The second derivative is

StartFraction d squared y Over dx squared EndFractiond$2$ydx$2$equals$=$StartFraction d y prime divided by d theta Over dx divided by d theta EndFractiondy$/$ddx$/$d$.$

Find the slope and concavity of the following curve at the given points.

requals$=33$theta$,$

$$

thetaequals$=$StartFraction $3$ pi Over $2$ EndFraction$3$$2,$

$4$ pi