Maximizing the Function y (1 - cos2x)/2: An SEO Optimized Guide
Understanding the maximum value of a mathematical function like y (1 - cos2x)/2 is crucial for students and professionals in various fields, including engineering, physics, and data analysis. This guide will provide a detailed explanation of the steps involved in finding the maximum value, along with the relevant trigonometric and calculus principles.
Introduction to Trigonometric Functions
Trigonometric functions, such as sine, cosine, and their variations, are fundamental in mathematics and have broad applications in various scientific and practical scenarios. The function we are focusing on, y (1 - cos2x)/2, is a specific application of these functions. To find the maximum value of this function, we need to apply principles from calculus, particularly differential calculus.
Step-by-Step Solution
Step 1: Identify the Function
The function to maximize is given by:
y (1 - cos2x)/2
Step 2: Find the First Derivative
To find the critical points of the function, we need to calculate the first derivative, y'.
y' 1/2 * sin(2x) * 2 sin(2x)
The steps and calculations to find the first derivative are critical in calculus and are necessary for identifying the critical points.
Step 3: Solve for y' 0
Setting the first derivative to zero will help us find the critical points where the function may attain its maximum or minimum values.
sin(2x) 0
Solving for x, we get:
2x 0 2kπ, 2x π 2kπ, where k ∈ Z
x kπ, x (π/2 kπ), where k ∈ Z
Step 4: Second Derivative Test
To determine the nature of these critical points (whether they are maxima, minima, or points of inflection), we need to calculate the second derivative, y''.
y'' 2cos(2x)
Evaluating the second derivative at the critical points will help us decide if it is a maximum, minimum, or point of inflection.
For x (π/2 kπ), we find:
y' sin(2x) sin(π 2kπ) 0
y'' 2cos(2x) 2cos(π 2kπ) -2 lt 0
Hence, x (π/2 kπ) are the maxima.
Step 5: Evaluate the Maximum Value
The maximum value of y can be found by substituting the value of x into the original function.
y_{max} (1 - cos2(x))/2
At x (π/2 kπ), we find:
y_{max} (1 - cos(π 2kπ))/2 (1 - (-1))/2 2/2 1
Conclusion
In summary, the function y (1 - cos2x)/2 reaches its maximum value of 1 at points where x (π/2 kπ), where k is an integer. These steps demonstrate the importance of applying calculus principles, particularly the first and second derivatives, in solving optimization problems.