Calculus II Polar Function

Question

Let r = 1 + 2sinθ

(a) Find the slope of the tangent line when θ = pi/4.

(b) Find the angles for which the graph has a vertical tangency.

(c) Graph the polar function. Draw a dashed tangent line to the graph at each point where the graph has a vertical tangency. Is your drawing consistent with the result of part (b)? Use an online graphing utility to graph the function.

Answer 1

Since nobody else has answered this question, I will take a shot.

(r-1)/2 = sin θ; then y=r(r-1)/2

Substitute for r: r²=x²+y²

The you get a messy implicit equation and you find you need to use BOTH signs for the radical to get from 0 to 2π on the graph:

x²+y²-2y+sqrt(x²+y²)=0

Now you can use implicit differentiation to get a messy expression containing dy/dx.

You can now use θ=π/4 and r=1+2sinθ to get values for x and y.

Plug these values into the implicit derivative expression and solve out the derivative dy/dx.

That gives you the required slope.

The rest of this problem you will have to puzzle out for yourself!

I would not even try to do this in closed form...do it numerically using your calculator.