Q 37. - Calculus

Question

Find the smallest interval necessary to draw a complete graph of the functions in Exercises $33-38$ and then graph each function using a graphing calculator or computer algebra system.

$r ={ e}^{ \sin \theta }- 2 \cos 4\theta + {\sin }^5(\theta /12)$

Step-by-Step Solution

Answer

$[0,24\pi ]$

Step 1: Given information

$r ={ e}^{ \sin \theta }- 2 \cos 4\theta + {\sin }^5(\theta /12)$

Step 2: Explanation

Consider the polar equation $r=e^{\sin \theta }-2\cos 4\theta +{\sin }^5\left(\frac{\theta }{12}\right)$

The goal is to draw the graph of the polar equation supplied. We will draw the graph here using a graphing calculator. 

If we observe the graph, the values of the graph are repeated after an interval of $24\pi$ so the smallest interval required for the graph is $[0,24\pi ]$ Therefore, the smallest interval for the graph is $[0,24\pi ]$