CPM Homework Help : APCALC Problem 7-219

A particle moves along the curve $y = 2^x$ so that its $x$ -coordinate increases at the constant rate of $5$ units per second. How fast is its angle of inclination $θ$ changing when $x = 2$ ?

(As shown in the diagram at right, $θ$ is the angle between the segment joining the point to the origin and the $x$ -axis.)

Hint:

$\tan(\theta) =\frac{\text{opposite side}}{\text{adjacent side}}$

What are the opposite and adjacent sides here?

Step 1:

$\tan(\theta) =\frac{y}{x}=\frac{2^{x}}{x}$

Step 2:

To obtain $θ$ , use implicit differentiation in terms of $t$ , since the $x$ -coordinate and $θ$ are changing over time.

Increasing exponential curve, with point on the curve in first quadrant, segment connects the point to the origin, angle between segment and positive x axis, labeled theta.