What is the slope of the polar curve #f(theta) = theta - sec^2theta+costheta # at #theta = (7pi)/12#?

1 Answer
Mar 21, 2017

#dy/dx = -6.968 if theta = (7pi)/12#

Explanation:

The slope of the polar curve at #theta = (7pi)/12# is #dy/dx# evaluated in terms of #theta# at #theta = (7pi)/12#.

To find #dy/dx#, we can first find #(dy)/(d theta)# and #(dx)/(d theta)# and use the chain rule for derivatives to determine that:

#dy/dx = (dy)/(d theta) times (d theta)/(dx) = (dy)/(d theta) div (dx)/(d theta)#


#y=r sintheta = (theta - sec^2theta+costheta)(sintheta)#
#(dy)/(d theta) = (1 - 2sec^2(theta)tan(theta) - sintheta)(sintheta) + (theta-sec^2theta+costheta)(costheta)#

Evaluating this at #theta = (7pi)/12# gives that #(dy)/(d theta) = 111.12 if theta = (7pi)/12#

#x=r costheta = (theta - sec^2theta + costheta)(costheta)#
#(dx)/(d theta) = (1 - 2sec^2(theta)tan(theta) - sintheta)(costheta) + (theta-sec^2theta+costheta)(-sintheta)#

Evaluating this at #theta = (7pi)/12# gives that #(dx)/(d theta) = -15.948 if theta = (7pi)/12#


#(dy)/(d theta) div (dx)/(d theta) = 111.12/(-15.948) = -6.968#

Therefore, the slope of the line tangent to #f(theta)# at the point where #theta = (7pi)/12# is #-6.968#.