Question

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

Answer 1

`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)`

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`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`

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`(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`.