Kemonotonan Fungsi Trigonometri

Summary

In this video, the presenter explains how to determine intervals where trigonometric functions are increasing or decreasing using derivatives. By analyzing the derivative of functions such as cos(x) and sin(2x), they demonstrate the step-by-step process to find the intervals where the function's slope is positive (increasing) or negative (decreasing). Through detailed examples, the presenter explains the method for finding critical points, testing intervals, and ultimately identifying the behavior of the functions within given ranges. The video aims to help students understand and apply the concept of monotonicity in trigonometric functions.

Takeaways

• 😀 The concept of monotonicity is introduced, which refers to whether a function is increasing or decreasing based on its first derivative.
• 😀 A function is said to be increasing if its first derivative (f'(x)) is greater than 0 and decreasing if f'(x) is less than 0.
• 😀 To determine whether a trigonometric function is increasing or decreasing, we start by calculating its first derivative.
• 😀 In the example, the first derivative of cos(x) is -sin(x), which is used to analyze the intervals where the function is increasing or decreasing.
• 😀 For the function f(x) = cos(x), we determine the points where f'(x) = 0, which occurs at x = 180° and x = 360°.
• 😀 The function f(x) = cos(x) is decreasing in the interval [0°, 180°] and increasing in the interval [180°, 360°].
• 😀 For the second example, the function f(x) = sin(2x) is analyzed, with its first derivative being 2cos(2x).
• 😀 For the function f(x) = sin(2x), the critical points where the first derivative equals 0 occur at x = 45° and x = 135° within the given interval.
• 😀 The function f(x) = sin(2x) is increasing in the intervals [0°, 45°] and [135°, 180°] and decreasing in the interval [45°, 135°].
• 😀 By testing values within the intervals, we can confirm where the function is increasing or decreasing, based on the sign of the first derivative.

Q & A

What is the concept of monotonicity in trigonometric functions?

Monotonicity in trigonometric functions refers to whether a function is increasing (rising) or decreasing (falling). A function is increasing if its first derivative is greater than zero, and it is decreasing if its first derivative is less than zero.

How can we determine if a function is increasing or decreasing?

To determine if a function is increasing or decreasing, we look at its first derivative. If the first derivative (f'(x)) is positive, the function is increasing. If it is negative, the function is decreasing.

In Example 1, what is the function given, and what interval is being analyzed?

In Example 1, the function given is f(x) = cos(x), and the interval being analyzed is from 0° to 360°.

What is the first derivative of f(x) = cos(x)?

The first derivative of f(x) = cos(x) is f'(x) = -sin(x).

How do we find the intervals where the function is increasing or decreasing?

To find the intervals where the function is increasing or decreasing, we first find the first derivative, set it equal to zero, solve for x, and then test intervals around these critical points to see if the derivative is positive or negative.

What are the critical points for f(x) = cos(x) on the interval 0° to 360°?

The critical points for f(x) = cos(x) are at x = 180° and x = 360° since sin(x) = 0 at these points.

What is the result when testing the intervals for f(x) = cos(x)?

Testing the intervals for f(x) = cos(x) results in the function being decreasing on the interval (0°, 180°) and increasing on the interval (180°, 360°).

In Example 2, what is the function given, and what interval is being analyzed?

In Example 2, the function given is f(x) = sin(2x), and the interval being analyzed is from 0° to 180°.

What is the first derivative of f(x) = sin(2x)?

The first derivative of f(x) = sin(2x) is f'(x) = 2cos(2x).

How do we solve for the critical points of f(x) = sin(2x)?

To solve for the critical points of f(x) = sin(2x), we set the first derivative equal to zero: 2cos(2x) = 0, which gives cos(2x) = 0. This results in the critical points x = 45°, 135°.

What are the intervals where the function f(x) = sin(2x) is increasing or decreasing?

For f(x) = sin(2x), the function is increasing on the intervals (0°, 45°) and (135°, 180°), and it is decreasing on the interval (45°, 135°).

What role does testing values within intervals play in determining monotonicity?

Testing values within intervals helps determine whether the function is increasing or decreasing by checking if the first derivative is positive (increasing) or negative (decreasing) in those intervals.