(a)
(b)
then
.To find the slope of a polar curve r = f (θ), we must first express the curve in
parametric form. Since
x = r cos θ and y = r sin θ,therefore,
x = f (θ) cos θ and y = f (θ) sin θ.
If f (θ) is differentiable, so are x and y; then
,.Also, if , then
.In doing an exercise, it is far easier simply to express the polar equation
parametrically, then find dy/dx, rather than to memorize the formula.
Example 37 __
Find the slope of the cardioid r = 2(1 + cos θ) at θ = . See Figure N4–24.
Where is the tangent to the curve horizontal?