A rotating platform lifts a point vertically according to the height function h( )=…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

A rotating platform lifts a point

P

vertically according to the height function

h(\theta)=\frac{\theta+2}{\csc \theta}, \qquad 0<\theta<\frac{\pi}{2},

where

h

is measured in meters and

\theta

(in radians) is the angle of rotation measured from the starting position. (a) Determine

\dfrac{dh}{d\theta}

. Express your answer in simplest exact form. (b) Evaluate

\dfrac{dh}{d\theta}

\theta=\dfrac{\pi}{6}

and include the correct units. (c) Using the sign of

\dfrac{dh}{d\theta}

\theta=\dfrac{\pi}{6}

, describe whether the height of

P

is increasing or decreasing at that instant and explain your conclusion. (d) Discuss what the height function suggests will happen to

h(\theta)

\theta\to 0^{+}

. Justify your answer using limits or derivative information.

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