Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=3 \cos x+2 \sin x\) (a) What is the period of \(f ?\) (b) What are the maximum and minimum values of \(f ?\)

Step-by-Step Solution

Answer

The period of \(f\) is \(2\pi\), The maximum value of \(f\) is \(\sqrt{13}\), and the minimum value of \(f\) is \(-\sqrt{13}\).

Step 1: Determining the period

The period of any individual trigonometric function, whether it is \(sin\) or \(cos\), is \(2\pi\). Although these two functions are added together in the function \(f(x)=3 \cos x+2 \sin x\), this does not affect the period. So, as the period of both \(sin x\) and \(\cos x\) are \(2\pi\), the period of \(f(x)=3 \cos x+2 \sin x\) is also \(2\pi\).

Step 2: Converting to amplitude-phase form

In order to find the maximum and minimum values of the function, it is helpful to convert it into a form \(R \cos(x-\alpha)\) or \(R \sin(x-\alpha)\) using the R-formula - \(R \cos(x-\alpha) = A \cos x + B \sin x\), where \(R = \sqrt {A^2+B^2}\) and \(\alpha = arctan(B/A)\). Here, \(A = 3\) and \(B = 2\). Doing these calculations, we find that \(R = \sqrt {3^2+2^2} = \sqrt {13}\) and \(\alpha = arctan(2/3)\). So, the function \(f(x)\) can be rewritten as \(f(x) = \sqrt {13} \cos(x-\alpha)\).

Step 3: Finding the maximum and minimum values

The maximum and the minimum values of the function are given by the amplitude \(R\). A trigonometric function \(R \cos(x-\alpha)\) has a maximum value of \(R\) and a minimum value of \(-R\). So for the function \(f(x) = \sqrt {13} \cos(x-\alpha)\), the maximum value is \(\sqrt {13}\) and the minimum value is \(-\sqrt {13}\)

Key Concepts

Periodicity

Periodicity refers to the repeating nature of trigonometric functions over specific intervals. For functions like \(\sin x\) and \(\cos x\), their graphs repeat every \(2\pi\). This means they have a period of \(2\pi\), indicating the length of one complete cycle. In the function \(f(x) = 3 \cos x + 2 \sin x\), each individual sine and cosine component retains this periodic characteristic.
Since both components have a period of \(2\pi\), the combined function also has a period of \(2\pi\). This is because the addition of two trigonometric functions sharing the same period does not alter their periodicity.

• The period gives insights into how the function behaves over one complete cycle.
• Knowing the period helps in graphing the function accurately.
• It's an essential aspect for understanding oscillatory behavior such as waves.

Amplitude

Amplitude measures the height or strength of a wave represented by a trigonometric function. It defines the maximum distance from the middle of the wave to the peak or the trough. For the standard cosine function \(\cos(x)\), the amplitude is 1, but it changes when coefficients are added in front, like in \(3 \cos x\).
In the exercise, to find the amplitude of \(f(x) = 3 \cos x + 2 \sin x\), you can use the formula \(R = \sqrt{A^2 + B^2}\). This formula comes from converting the expression into the form \(R \cos(x - \alpha)\), helping to find the amplitude as a single value:
\[ R = \sqrt{3^2 + 2^2} = \sqrt{13} \]

• The amplitude \(\sqrt{13}\) represents the maximum value that the function can reach.
• The minimum value of the function is \(-\sqrt{13}\).
• Amplitude affects how "tall" the wave appears on a graph, which is vital for visual representation.

Phase Shift

Phase shift describes how a wave is shifted horizontally from its standard position. It indicates how the graph of the function moves left or right. In the expression \(R \cos(x - \alpha)\), the value \(\alpha\) represents the phase shift and is calculated using \(\alpha = \arctan(B/A)\). This helps adjust the starting point of the cosine wave on a graph.
In the function \(f(x) = 3 \cos x + 2 \sin x\), converting to amplitude-phase form aids in identifying the phase shift:
\[ \alpha = \arctan(\frac{2}{3}) \]

• The phase shift \(\alpha\) changes where the function's cycle begins.
• A positive \(\alpha\) shifts the function to the right, while a negative shifts it to the left.
• Understanding phase shift is crucial for aligning the function with real-world phenomena, such as signals and waveforms.

Recognizing phase shifts helps in comparing and analyzing similar functions effectively.