Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Graph \(f(x)=2^{\cos x}\). (a) Is the function periodic? If so, what is its period? (b) What is its maximum value? Its minimum value? Give exact answers.

Step-by-Step Solution

Answer

(a) The function is periodic, and its period is \(2\pi\). (b) The maximum value of the function is 2, and the minimum value is 0.5.

Step 1: Periodicity of the Function

To check if the function is periodic, we can use the standard periodic property of the trigonometric function cosine, which states that \(\cos(x + 2\pi) = \cos(x)\). This tells us that any function of the form \(f(x) = g(\cos x)\) is periodic, with a period of \(2\pi\). So, \(f(x)=2^{\cos x}\) is a periodic function, and its period is \(2\pi\).

Step 2: Maximum Value of the Function

The maximum value of the function can be obtained by finding the maximum value of the exponent of 2, which is \(\cos x\). Because cosine oscillates between -1 and 1, the maximum value of \(\cos x\) is 1. Therefore, the maximum value of the function \(f(x)\) is \(2^1 = 2\).

Step 3: Minimum Value of the Function

Similar to the maximum value, the minimum value of the function can be obtained by finding the minimum value of the exponent of 2. The minimum of \(\cos x\) is -1. Thus, the minimum value of the function \(f(x)\) is \(2^{-1} = 0.5\).

Key Concepts

Maximum Value

The concept of maximum value is crucial when analyzing functions. For a function, the maximum value is the highest point it reaches on its graph. In the context of trigonometric functions, specifically with the function \(f(x) = 2^{\cos x}\), the maximum value can be determined by focusing on the properties of the cosine function.

• The cosine function, \(\cos x\), oscillates between -1 and 1.
• Since we are interested in finding the maximum value of \(f(x)\), we must find the maximum value of \(\cos x\).
• The maximum value of \(\cos x\) is 1.
• Inserting this into \(f(x) = 2^{\cos x}\), we get \(2^1 = 2\).

These points illustrate that the maximum value of the function \(f(x) = 2^{\cos x}\) is 2. By understanding this process, the concept of identifying maximum values in functions, especially those involving trigonometric components, becomes more approachable.

Minimum Value

Just as we consider maximum values, the minimum value indicates the lowest point a function reaches on its graph. For the function \(f(x) = 2^{\cos x}\), identifying the minimum value involves examining the behavior of the cosine function.

• The cosine function, \(\cos x\), reaches its minimum at -1.
• To find the lowest value of \(f(x)\), we substitute the minimum value of \(\cos x\) into the function.
• Thus, \(f(x) = 2^{-1}\) simplifies to \(0.5\).

The function \(f(x) = 2^{\cos x}\) therefore has a minimum value of 0.5. This illustrates the importance of understanding the role of trigonometric values in determining the extrema of a function.

Trigonometric Functions

Trigonometric functions are foundational concepts in mathematics, dealing with the angles and sides of triangles. They are periodic, demonstrating repetitive patterns. The familiar trigonometric functions include sine, cosine, and tangent.In this instance, the focus is on the cosine function, because \(f(x) = 2^{\cos x}\) is built upon it. Here are a few key points:

• The function \(\cos x\) is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) interval.
• This periodicity impacts \(f(x)\), making it also a periodic function with a period of \(2\pi\).
• These periodic properties allow us to predict and understand the behavior of functions like \(f(x)\).

Grasping the concept of trigonometric functions helps in many areas of mathematics, physics, and engineering, particularly in modeling harmonic and oscillatory phenomena.