Suppose a curve is defined in polar form by

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

r=f(\theta)

, with

f

differentiable on an interval containing

\theta_0

. At

\theta=\theta_0

it is known that

\dfrac{dy}{d\theta}=0

while

\dfrac{dx}{d\theta}\neq0

, and at a different value

\theta=\theta_1

it is known that

\dfrac{dx}{d\theta}=0

while

\dfrac{dy}{d\theta}\neq0

. Let

(x_0,y_0)

and

(x_1,y_1)

be the corresponding Cartesian points on the curve. (a) Explain why the curve has a horizontal tangent line at

(x_0,y_0)

. Describe geometrically what feature of the derivatives leads to this conclusion. (b) Explain why the curve has a vertical tangent line at

(x_1,y_1)

. Justify why the equation of this tangent line can be written in the form

x=c

and state the value of

c

in terms of

(x_1,y_1)