Find the slope of the curve r=cos(2theta) at theta=pi/6.. must this be solved parametrically? if so, why?

By converting into parametric equations,

x(θ)=r(θ)cosθ=cos2θcosθ
y(θ)=r(θ)sinθ=cos2θsinθ

By Product Rule,

x'(θ)=−sin2θcosθ−cos2θsinθ

x'(π/2)=−sin(π)cos(π/2)−cos(π)sin(π/2)=1

y'(θ)=−sin2θsinθ+cos2θcosθ

y'(π/2)=−sin(π)sin(π/2)+cos(π)cos(π/2)=0

So, the slope m of the curve can be found by

m=dy/dx∣θ=π2
= y'(π/2)/x'(π/2)
=0/1

=0

I hope my answer has come to your help. Thank you for posting your question here in .

step-by-step explanation:

i found the life tables for the year 2012. 

life table of males at age 30. probability of dying between ages x and x+1

total population:  

29–30 . . . . . . . . . . . . 0.001030

30–31 . . . . . . . . . . . . 0.001061

males:  

29–30 . . . . . . . . . . . . 0.001420

30–31 . . . . . . . . . . . . 0.001448

white population

29–30 . . . . . . . . . . . . 0.001005

30–31 . . . . . . . . . . . . 0.001037

white males

29–30 . . . . . . . . . . . . 0.001372

30–31 . . . . . . . . . . . . 0.001399

non hispanic white population

29–30 . . . . . . . . . . . . 0.001083

30–31 . . . . . . . . . . . . 0.001116

non hispanic white males

29–30 . . . . . . . . . . . . 0.001477

30–31 . . . . . . . . . . . . 0.001503

jacob, non hispanic white male. 30 years old.

probability that jacob lives to be 30 years old. 

since the given probabilities are probabilities of dying, i think we simply subtract it from 1 to get the probability of living. we will use the probability of non hispanic white male.

                                          probability of living to be 30 y.o.

29 - 30 : 1 - 0.001477 = 0.998523

30 - 31 : 1 - 0.001503 = 0.998497