Problem 24 - Essential Calculus

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

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

Arc Length in Calculus

In calculus, arc length refers to the distance along a curve between two points. Unlike straight lines, curves require integration to determine this length. For a function, the formula to calculate arc length, from point \(a\) to point \(b\), is given by the integral:
\[ \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
This formula takes into account the slope's rate of change, making it essential for curves like the tangent function. In practical terms, it helps us measure how 'curvy' a path is over a specific interval.

Integral Approximation

When we can't compute an integral exactly, we approximate it. Integrals give us the cumulative 'effect' over a chosen interval, commonly through an area under a curve.
Approximations can use methods like:

• Trapezoidal Rule
• Simpson's Rule
• Rectangular Sums

For the arc length exercise, we visually approximate, ruling out impossible lengths like negatives and choosing the most reasonable positive value. This method is practical, especially in exams, for quick estimations.

Understanding the Tangent Function

The tangent function \(y = \tan(x)\) plays a pivotal role in trigonometry and calculus. It starts at the origin with a slope that increases steeply, especially as \(x\) nears \(\pi/4\).
Key characteristics include:

• Periodic behavior, repeating every \(\pi\)
• Vertical asymptotes at \((\frac{\pi}{2} + k\pi)\)
• Rapid changes in slope around critical points

Understanding how \(\tan(x)\) behaves helps in visualizing and approximating curves, crucial for determining arc lengths.

Sketch Graphing for Visual Analysis

Sketch graphing involves drawing a rough graph of a function to understand its behavior visually. It's a helpful tool for arc length estimation, allowing us to see the curve rather than compute it explicitly.
Steps include:

• Identifying key points (e.g., intercepts, maxima, minima)
• Noting symmetry or periodicity
• Drawing smooth curves between significant points

In our exercise, sketching \(y = \tan(x)\) from \(0\) to \(\pi/4\) involves plotting points like \((0,0)\) and \((\frac{\pi}{4},1)\), and helps us visually assess the arc length.