Question

Approximation In Exercises 23 and \(24,\) determine which value best approximates the length of the arc represented by the integral. (Make your selection on the basis of a sketch of the arc and not by performing any calculations.) \(\int_{0}^{\pi / 4} \sqrt{1+\left[\frac{d}{d x}(\tan x)\right]^{2}} d x\) (a) 3 (b) -2 (c) 4 (d) \(\frac{4 \pi}{3}\) (e) 1

Short answer

The best approximation for the length of the arc represented by the given integral, based on the sketch and visual estimation is (e) 1.

Step-by-step solution

Function Sketching

Create a sketch of the function \(y = \tan(x)\) from \(0\) to \(\pi/4\). This curve starts from the origin (0, 0) and tilts right upwards as x increases up to \(\pi/4\). Remember, \(\tan(0) = 0\) and \(\tan(\pi/4) = 1\). Therefore, plot the points (0,0) and \((\pi/4, 1)\) and sketch a smooth curve connecting these points.

Length estimation

Now you have a visually drawn arc of the curve from 0 to \(\pi/4\). You can easily observe that the length of this curve will be greater than the straight line from 0 to \(\pi/4\) (which is 1) and less than 2 (as it does not reach to 2 in our drawing). So approximate the length of the arc accordingly. Comparing with the given values (3, -2, 4, \(\frac{4 \pi}{3}\), and 1), we can easily rule out -2 (as length cannot be negative) and 1 (as the length of curve is definitely greater than 1). The values 3, 4, and \(\frac{4 \pi}{3}\) all seem to be larger than the visual length of the arc.

Final Selection

We, therefore based on the length approximation, can conclude that none of these options perfectly fit the visually estimated length. However, the value that best approximates is 1 from the given choices since it is closest to the visually estimable length of the arc.