Problem 57 - Precalculus Mathematics for Calculus

Question

The beam from a lighthouse completes one rotation every two minutes. At time \(t,\) the distance \(d\) shown in the figure on the next page is $$d(t)=3 \tan \pi t$$ where \(t\) is measured in minutes and \(d\) in miles. (a) Find \(d(0.15), d(0.25),\) and \(d(0.45)\) (b) Sketch a graph of the function \(d\) for \(0 \leq t<\frac{1}{2}\) (c) What happens to the distance \(d\) as \(t\) approaches \(\frac{1}{2} ?\)

Step-by-Step Solution

Answer

(a) \(d(0.15) \approx 1.3908, d(0.25) = 3, d(0.45) \approx 13.458\). (b) Graph shows \(d(t)\) increasing. (c) \(d(t)\) approaches infinity as \(t\to 0.5\).

Step 1: Evaluate d(t) at Specific Times

We need to evaluate the function \(d(t) = 3 \tan \pi t\) at \(t = 0.15, 0.25,\) and \(0.45\). Let's start with these one by one:- For \(d(0.15)\): \[ d(0.15) = 3 \tan (\pi \times 0.15) = 3 \tan (0.15\pi) \approx 3 \times 0.4636 = 1.3908 \]- For \(d(0.25)\): \[ d(0.25) = 3 \tan (\pi \times 0.25) = 3 \tan (0.25\pi) = 3 \times 1 = 3 \]- For \(d(0.45)\): \[ d(0.45) = 3 \tan (\pi \times 0.45) = 3 \tan (0.45\pi) \approx 3 \times 4.486 = 13.458 \]

Step 2: Graph the Function for a Specific Interval

The function to graph is \(d(t) = 3 \tan \pi t\) for the interval \(0 \leq t < \frac{1}{2}\). In this interval, the graph is a piece of the tangent function.Start at \(t = 0\), where \(d(0) = 3 \tan(0) = 0\), and follow the increasing behavior of the tangent function up to just before \(t = 0.5\). At \(t = 0.5\), the function is undefined, as tangent reaches its vertical asymptote at this point.

Step 3: Analyze Behavior as t Approaches 0.5

As \(t\) approaches \(\frac{1}{2}\), the value of \(d(t) = 3 \tan \pi t\) grows without bound. This is because tangent of an angle approaches infinity as the angle approaches \(\frac{\pi}{2}\) (or its multiples), corresponding to the vertical asymptotes in the tangent function.

Key Concepts

Tangent Function

The tangent function, often seen in trigonometry, is part of the set of trigonometric functions, which also includes sine and cosine. It is expressed as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). The value of the tangent function depends on the angle, which in this context is given in radians, and it repeats periodically every \( \pi \) radians (180 degrees). This means that the function has a period of \( \pi \) and repeats its values every \( \pi \) units.
Understanding the tangent function's values is essential to grasp its nature. As it increases, it passes through its mid-point of zero, the x-axis, at multiples of \( \pi \). At other points, particularly where \( \cos(\theta) = 0 \) (multiples of \( \frac{\pi}{2} \)), the function is undefined, leading to what are known as vertical asymptotes. These asymptotes occur at these intervals because dividing by zero is undefined in mathematics.

Graphing Functions

Graphing functions involves representing the functional behavior of equations pictorially on a coordinate plane. For the function \( d(t) = 3 \tan \pi t \) explained in the exercise, graphing helps visualize how \( d \) changes with different values of \( t \) within the given interval \( 0 \leq t < \frac{1}{2} \).
When graphing a function like the tangent, you would look for key characteristics such as:

• Intercepts: Points where the graph crosses axes (here, it crosses the d-axis at 0).
• Asymptotes: Lines the graph approaches but never touches, occurring here at \( t = \frac{1}{2} \).
• Periodicity and Repetition: The graph shows repeating patterns every \( \pi \) units.

Sketching this particular function in the specified range shows a section of the tangent curve moving upwards, beginning from zero and increasing rapidly as it nears \( t = 0.5 \). Graphing provides an effective way to interpret the infinite growth of \( d \) as \( t \) closes in on the asymptote.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as inputs approach certain critical values, especially undefined ones. For the tangent function, these behaviors are tied to vertical asymptotes, which occur for \( \tan \), specifically, at odd multiples of \( \frac{\pi}{2} \).
In our model \( d(t) = 3 \tan \pi t \), the asymptotic behavior comes into play as \( t \) nears \( \frac{1}{2} \). Here, the function tends towards infinity, which means \( d(t) \) grows larger and larger without any upper limit as \( t \) gets closer to \( 0.5 \). This behavior is typical of hyperbolic branches that shoot off towards infinity on a graph.
Recognizing asymptotic behavior helps in understanding the limits and long-term trends of functions. It signals rapid changes or tendencies toward an extreme, offering critical insight into the nature of the function and its graph. This understanding is pivotal when predicting function behavior in contexts like navigation using a lighthouse, as in the original problem.