Problem 9 - Calculus Early Transcendentals

Question

The point \(P(1,0)\) lies on the curve \(y=\sin (10 \pi / x)\). $$\begin{array}{l}{\text { (a) If } Q \text { is the point }(x, \sin (10 \pi / x)) \text { , find the slope of the secant }} \\ {\text { line } P Q \text { (correct to four decimal places) for } x=2,1.5,1.4 \text { , }} \\\ {1.3,1.2,1.1,0.5,0.6,0.7,0.8, \text { and } 0.9 . \text { Do the slopes }} \\\ {\text { appear to be approaching a limit? }}\end{array}$$ $$\begin{array}{l}{\text { (b) Use a graph of the curve to explain why the slopes of the }} \\ {\text { secant lines in part (a) are not close to the slope of the tan- }} \\ {\text { gent line at } P .}\end{array}$$ $$\begin{array}{l}{\text { (c) By choosing appropriate secant lines, estimate the slope of }} \\ {\text { the tangent line at } \mathrm{P} .}\end{array}$$

Step-by-Step Solution

Answer

The secant line slopes fluctuate without approaching a limit due to curve oscillations.

Step 1: Find Secant Line Formula

The secant line between points \(P(1,0)\) and \(Q(x, \sin(\frac{10\pi}{x}))\) can be calculated by using the formula for the slope of a secant line: \(m = \frac{f(x) - f(1)}{x - 1}\), thus \(m = \frac{\sin(\frac{10\pi}{x}) - 0}{x - 1}\).

Step 2: Calculate the Slope for Each Given x-value

Compute the slopes of the secant line at each given value of \(x\): - For \(x=2\), \[ m = \frac{\sin(\frac{10\pi}{2}) - 0}{2 - 1} = 0 \] - For \(x=1.5\), \[ m = \frac{\sin(\frac{10\pi}{1.5}) - 0}{1.5 - 1} \approx -0.8660 \] - For \(x=1.4\), \[ m = \frac{\sin(\frac{10\pi}{1.4}) - 0}{1.4 - 1} \approx -1.4998 \]

Step 3: Continue Calculations for All x-values

Continue calculating the slope for each remaining \(x\) value: - For \(x=1.3\), \[ m \approx -1.9284 \] - For \(x=1.2\), \[ m \approx -1.4695 \] - For \(x=1.1\), \[ m \approx 0.5451 \] - For \(x=0.9\), \[ m \approx -0.5451 \] - For \(x=0.8\), \[ m \approx 1.4695 \] - For \(x=0.7\), \[ m \approx 1.9284 \] - For \(x=0.6\), \[ m \approx 1.4998 \] - For \(x=0.5\), \[ m \approx 0.8660 \]

Step 4: Analyze If Slopes Approach a Limit

Reviewing all slopes: Given \(x\) values present fluctuations, showing no clear tendency to converge towards any particular limit as \(x\) approaches 1.

Step 5: Graphical Explanation

Plot the curve \(y = \sin(\frac{10\pi}{x})\). Notice that as \(x\) approaches 1, the oscillations intensify due to the nature of the sine function with variable frequency. This erratic behavior explains why the slopes fluctuate strongly and do not match the tangent's expected slope at \(x=1\).

Step 6: Estimate the Tangent Line Slope

Given the intense fluctuation of values, specifically examining smaller intervals may give a closer estimate. Using the overall fluctuating pattern, small-scale guesses may not be valid; reference local trends close to \(x=1.1\) and \(x=0.9\) might infer a range of slope expected around 0, but no precise estimate is conclusively recognized due to the oscillation.

Key Concepts

Trigonometric Functions

Trigonometric functions, like sine, cosine, and tangent, are foundational elements in mathematics. These functions are defined based on a circle with a radius of one, known as the unit circle. The sine function, which we're dealing with in this exercise, is related to the y-coordinate on the unit circle. As angles change, these functions oscillate between -1 and 1 because of the circular path.

Let's consider the function in this problem: \( y = \sin(\frac{10\pi}{x}) \). It includes a variable frequency which is not fixed like typical sine graphs that depend on just one variable. This transformation creates unique and frequent oscillations as the value of \(x\) varies, especially when it approaches values close to 1. The challenge comes from these oscillations, affecting how we interpret slopes of lines on such graphs.

Understanding how such functions behave helps in analyzing patterns and trends, such as the frequent up and down movements in this problem. This high frequency indicates that the function's value switches rapidly as \(x\) changes, leading to difficulties in calculating secant and tangent line slopes that would otherwise stabilize with simpler functions.

Limit of a Function

A limit helps us understand how a function behaves as it approaches a certain point or as it goes to infinity. For example, if you want to determine how a function behaves as \(x\) gets closer to a particular value, you compute the limit.

In the context of this exercise, although we aim to find the slope of the tangent line at \(x = 1\), the rapid changes due to the sine function lead to fluctuating secant slopes. Ideally, if the secant slopes converge, they point to the tangent line's slope—the value the function approaches as \(x\) gets closer. Here, fluctuations prevent us from easily observing convergence, indicating the limits aren't straightforward.

This behavior can be better understood by considering how in trigonometric functions, frequency alterations (as with \(10\pi / x\)) significantly impact the limit as \(x\) changes. Understanding limits extends beyond just trigonometric functions, as it is crucial in studying calculus and general mathematics.

Tangent Line

A tangent line is a straight line that touches a curve at just one point and has the same slope as the curve does at that point. Finding this slope is key in many calculus problems because it represents the function's instantaneous rate of change at a particular point.

In our exercise, we are trying to estimate the tangent line's slope at point \(P(1,0)\) on the curve \(y = \sin(\frac{10\pi}{x})\). However, because of the curve's rapid oscillations near \(x = 1\), obtaining an accurate tangent slope is complex. In simpler functions, calculating secant line slopes (connecting two points) generally helps estimate the tangent slope by letting one point approach the other. Here, the oscillating nature results in diverse secant slopes that don't settle to a constant value, explaining the difficulty.

Even though the fluctuations make it hard to pinpoint the tangent slope exactly, we can aim for an average understanding by examining local behavior with smaller intervals between known points, but bear in mind, oscillations still pose a challenge in exact determination.