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Chapter 7: Problem 19

State whether the function \(y=\sin x\) is increasing or decreasing on the given interval. (The terms increasing and decreasing are explained on page \(159 .\) ) $$3 \pi / 2

Short Answer

Expert verified
The function is decreasing on the interval \( \frac{3\pi}{2} < x < 2\pi \).

Step by step solution

01

Understand the problem

We need to determine if the function \( y = \sin x \) is increasing or decreasing within the interval \( \frac{3\pi}{2} < x < 2\pi \). This requires analyzing the behavior of the sine function on this specific interval.
02

Analyze the properties of sine function

The sine function, \( y = \sin x \), reaches its minimum value of \(-1\) at \( x = \frac{3\pi}{2} \) and returns to \( 0 \) at \( x = 2\pi \). We can use this information to explore how the function behaves between these two points.
03

Use derivatives to check increasing/decreasing behavior

The derivative of \( y = \sin x \) is \( y' = \cos x \). A function is increasing when its derivative is positive and decreasing when its derivative is negative. Let's evaluate \( \cos x \) in the interval \( \frac{3\pi}{2} < x < 2\pi \).
04

Evaluate the derivative on the interval

On the interval \( \frac{3\pi}{2} < x < 2\pi \), the cosine function \( \cos x \) is negative. This is because it ranges from \( 0 \) (at \( x = \frac{3\pi}{2} \)) to \( -1 \) (as it approaches \( x = 2\pi \)).
05

Conclude based on the derivative

Since the derivative \( \cos x \) is negative on the interval \( \frac{3\pi}{2} < x < 2\pi \), the function \( y = \sin x \) is decreasing on this interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Function
The sine function, commonly written as \( y = \sin x \), is one of the most frequent trigonometric functions. It represents the y-coordinate of a point on the unit circle as the angle \( x \) in radians is rotated from the positive x-axis.

Key features of the sine function include:
  • Its range is between -1 and 1.
  • It is periodic with a period of \( 2\pi \), meaning it repeats its pattern every \( 2\pi \) units of \( x \).
  • It has a maximum value of 1 and a minimum value of -1.
  • Its graph is smooth and continuous, resembling a wave-like pattern.
The sine function is fundamental in many areas of mathematics, especially in geometry and calculus. It helps describe natural phenomena like sound and light waves.
Function Behavior
Understanding the behavior of functions is crucial in calculus and mathematics in general. "Behavior" refers to how the function's output, \( y \), changes as the input, \( x \), varies. Functions can exhibit patterns such as increasing, decreasing, or being constant over intervals.

For the sine function, \( y = \sin x \), its behavior can be described by:
  • Oscillating: The function increases and decreases periodically.
  • Smooth transitions: Changes in the output are gradual and continuous.
In the context of the interval \( \frac{3\pi}{2} < x < 2\pi \), the sine function shows a decreasing behavior, beginning at its minimum point and rising back to zero. Recognizing function behavior within specific intervals is useful for predicting and understanding systems modeled by mathematical equations.
Derivatives
Derivatives are a core concept in calculus that help us understand how a function changes. They represent the rate of change, or the slope of the function, at any given point. A derivative provides powerful insights into the function's behavior over its domain.

For the sine function, the derivative is \( y' = \cos x \). This means:
  • If \( y' > 0 \), the function is increasing at that point.
  • If \( y' < 0 \), the function is decreasing at that point.
  • If \( y' = 0 \), the function could be at a peak or trough鈥攅ssentially a flat point.
In our analysis of \( y = \sin x \), we look at the derivative \( \cos x \) within a given interval to see if it is positive or negative, providing evidence for the function's increasing or decreasing nature on that interval.
Increasing and Decreasing Functions
Determining whether a function is increasing or decreasing over an interval is a fundamental task in calculus. A function is said to be increasing if, as \( x \) grows, \( y \) also grows. Conversely, it is decreasing if \( y \) decreases as \( x \) increases.

To determine this for any function, you can:
  • Calculate the derivative \( y' \) of the function.
  • Examine the sign of \( y' \) over the desired interval.
For the sine function \( y = \sin x \), we explored the interval \( \frac{3\pi}{2} < x < 2\pi \). In this range, the derivative \( \cos x \) was found to be negative, confirming the function is decreasing. Understanding these changes is invaluable in fields like physics and engineering where such calculations predict behavior in dynamic systems.

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Most popular questions from this chapter

Use the Pythagorean identities to simplify the given expressions. $$\frac{\csc ^{4} \theta-\cot ^{4} \theta}{\csc ^{2} \theta+\cot ^{2} \theta}$$

(a) Graph the equations \(y=\tan x\) and \(y=2\) in the standard viewing rectangle. (b) Use the graph to give a rough estimate for the smallest positive root of the equation \(\tan x=2\) Answer:\(\quad x \approx 1\) (c) Use the graphing utility to determine the root more accurately, say, through the first four decimal places. (d) Let \(r\) denote the root that you determined in part (c). Is the number \(r+\pi\) also a root of the equation \(\tan x=2 ?\)

In the expression \(\sqrt{9-x^{2}},\) make the substitution \(x=3 \sin \theta\left(0<\theta<\frac{\pi}{2}\right),\) and show that the result is \(3 \cos \theta\).

Graph each function for one period, and show (or specify) the intercepts and asymptotes. $$y=\cot 2 \pi x$$

Four functions are defined as follows: $$\begin{array}{ll}f(x)=\csc x & T(x)=\tan x \\\g(x)=\sec x & A(x)=|x|\end{array}$$ In each case, graph the indicated function over the interval \([-2 \pi, 2 \pi]\). $$f \circ A$$
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