Q. 9 - Calculus

Question

Suppose you want to approximate the arc length of f(x) = tan x on the interval [0, π]. Would you get a good approximation if you subdivided the interval [0, π] into five subintervals and added up the lengths of the line segments over each subinterval? Why or why not? (Hint: Draw the line segments.)

Step-by-Step Solution

Answer

No, we would not get a good approximation if you subdivided the interval $\left[0,\mathrm{\pi }\right]$ into five subintervals and added up the lengths of the line segments over each subinterval because these will lead the arc length to undefined or extended to infinity.

Step 1. Given Information.

The given function is $f\left(x\right)=\tan x$ and the interval is $\left[0,\mathrm{\pi }\right].$

Step 2. Approximate the arc length.

Let's draw the graph of the  given function and divide the interval into five subintervals at $0,\frac{\mathrm{\pi }}{5},\frac{{\displaystyle 2\mathrm{\pi }}}{{\displaystyle 5}},\frac{{\displaystyle 3\mathrm{\pi }}}{{\displaystyle 5}},\frac{{\displaystyle 4\mathrm{\pi }}}{{\displaystyle 5}},\pi$ and added up the lengths of the line segments over each subinterval. The graph is

From the graph, we can depict that an approximation of the arc length on the interval $\left[0,\mathrm{\pi }\right]$ is the sum of the line segments. But as we know tan x is not defined at $\frac{\mathrm{\pi }}{2}.$ The arc length is undefined on the interval $\left[\frac{2\mathrm{\pi }}{5},\frac{3\mathrm{\pi }}{5}\right].$ The arc length is extended to infinity on the interval $\left[\frac{2\mathrm{\pi }}{5},\frac{\mathrm{\pi }}{2}\right] \mathrm{and} \left[\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{5}\right].$ Thus, the approximation by the line segments is not a good approximation.