Problem 31 - Applied Calculus

Question

Table 1.43 gives values for \(g(t),\) a periodic function. (a) Estimate the period and amplitude for this function. (b) Estimate \(g(34)\) and \(g(60)\). Table 1.43 $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\\\\hline g(t) & 14 & 19 & 17 & 15 & 13 & 11 & 14 & 19 \\\\\hline t & 16 & 18 & 20 & 22 & 24 & 26 & 28 & \\\\\hline g(t) & 17 & 15 & 13 & 11 & 14 & 19 & 17 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Period: 12, Amplitude: 4, \(g(34) = 11\), \(g(60) = 14\).

Step 1: Determine the Period

The period of a periodic function is the length of the interval over which the function repeats itself. In Table 1.43, observe the values of \(g(t)\). Note that at \(t = 0\), \(g(t) = 14\) and this value repeats at \(t = 12\). Therefore, the period \(T\) is 12 units.

Step 2: Determine the Amplitude

The amplitude of a periodic function is half the distance between its maximum and minimum values. From the table, the maximum value of \(g(t)\) is 19 and the minimum value is 11. The amplitude \(A\) is calculated as \(A = \frac{19 - 11}{2} = 4\).

Step 3: Estimate g(34)

Since the period is 12, we can use the equivalence \(g(34) = g(34 \mod 12) = g(10)\). From the table, \(g(10) = 11\). Thus, \(g(34) = 11\).

Step 4: Estimate g(60)

Similarly, using the period, \(g(60) = g(60 \mod 12) = g(0)\). From the table, \(g(0) = 14\). Therefore, \(g(60) = 14\).

Key Concepts

Period Estimation

A periodic function is one that repeats its values at regular intervals or periods. The period is one of the key characteristics of such functions. When estimating the period of a function given in tabular form, we look for the interval over which the function values repeat.

In our example:

• At time point \(t = 0\), the function \(g(t)\) equals 14.
• This value repeats itself at \(t = 12\).

Therefore, the full cycle occurs every 12 units of time, meaning the period \(T\) is 12.

The period is essential because it allows us to make predictions about the values of the function at points outside the given data range. We can extend the function infinitely in both directions using this known period, making periodic functions incredibly useful in practical scenarios, such as predicting tides or seasons.

Amplitude Calculation

The amplitude is a measure of how much a periodic function varies, essentially capturing the height of its peaks and valleys. For many functions, particularly trigonometric ones like sine and cosine, amplitude relates to the energy or intensity of the signal. It gives us an understanding of the function's variability from its average position.

To calculate amplitude:

• Identify the maximum value of the function; here, \(g(t)\) reaches 19.
• Find the minimum value, which is 11 in this case.

The amplitude \(A\) then becomes half of the distance between the max and min values: \(A = \frac{19 - 11}{2} = 4\).

This amplitude indicates how deeply the function dips and how high it peaks from its mean position, giving an insight into the range of the periodic signal.

Modular Arithmetic

Modular arithmetic, often thought of as "clock arithmetic," is a system which deals with integer division and finds remainder values. In periodic functions, it helps to find function values outside the immediate cycle by wrapping them around the period.

Here's how it works:

• If we need to find \(g(34)\): Calculate \(34 \mod 12\), which results in 10, so \(g(34) = g(10) = 11\).
• For \(g(60)\): Calculate \(60 \mod 12\), giving a remainder 0, thus \(g(60) = g(0) = 14\).

Using the modulo operation helps simplify calculations where the period of the function extends to values not immediately present in our data. It's especially useful in scenarios where periodic behaviors are repeated over time, such as studying cyclical patterns in data analysis or designing periodic signals in engineering.