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Chapter 3: Problem 7

Find (by hand) the intervals where the function is increasing and decreasing. Use this information to sketch a graph. $$y=\sin x+\cos x$$

Short Answer

Expert verified
The function increases on the intervals \(x \in (0 + n\pi, \frac{\pi}{4}+n\pi)\) and decreases on the intervals \(x \in (\frac{\pi}{4}+n\pi, \pi+n\pi)\) where \(n\) is an integer.

Step by step solution

01

Derivative of the function

Firstly, differentiate the function \(y=\sin x+\cos x\). The derivative of \(\sin x\) is \(\cos x\) and the derivative of \(\cos x\) is \(-\sin x\). So, \(y'=\cos x-\sin x\).
02

Find the Critical Points

The function is increasing or decreasing, if the derivative of the function is positive or negative, respectively. In this case, we need to look for the value of 'x' that makes the derivative equal to zero (these points are known as critical points). Equate \(y'=0\), so \(\cos(x)-\sin(x)=0\). Solving it further we get \(x=\frac{\pi}{4} + n\pi \) where \(n\) is an integer.
03

Determine increasing and decreasing intervals

Test the intervals from the critical points to determine where the function's rate of change is positive (increasing) or negative (decreasing). Pick a test point in each interval and evaluate the derivative at these test points. If the value is positive, it means the function is increasing and if it鈥檚 negative then the function is decreasing. Based on the value of x, the function would be increasing for \(x \in (0 + n\pi, \frac{\pi}{4}+n\pi)\) and decreasing for \(x \in (\frac{\pi}{4}+n\pi, \pi+n\pi)\) where \(n\) is an integer.
04

Graph the function

The graph of the function \(\sin x + \cos x\) will have a similar nature to sinusoidal curves but with a different frequency and amplitude. The function would be increasing and decreasing for the intervals obtained in the previous step. Peaks occur at \(x=\frac{\pi}{4}+n\pi\) and troughs occur when \(x = \pi+n\pi\). The graph will be plotted according to these peaks and troughs.

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Key Concepts

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

Derivative
The derivative of a function allows us to determine how that function changes at any given point. It essentially acts as a tool to calculate the "slope" of the function wherever we need it. For the function \(y = \sin x + \cos x\), the derivative is found by differentiating both component functions.
  • The derivative of \(\sin x\) is \(\cos x\).
  • The derivative of \(\cos x\) is \(-\sin x\).
Therefore, the derivative of the function is \(y' = \cos x - \sin x\). This expression tells us how the function \(y\) behaves as \(x\) changes.
Critical Points
Critical points are values of \(x\) at which the derivative of a function is zero or undefined. These points are important because they can indicate potential maxima, minima, or points of inflection in the function. For \(y = \sin x + \cos x\), we determined the critical points by setting the derivative equal to zero:
  • \(\cos(x) - \sin(x) = 0\)
This simplifies to \(x = \frac{\pi}{4} + n\pi\), where \(n\) is any integer. These points are periodic, repeating every \(\pi\) interval, due to the periodic nature of sine and cosine functions.
Increasing and Decreasing Intervals
One of the primary uses of derivatives is to find intervals where a function is increasing or decreasing. To determine this, we test the sign of the derivative at various intervals defined by the critical points. In the case of \(y = \sin x + \cos x\), we discovered the following:
  • The function is increasing when \(y' = \cos x - \sin x > 0\) within the intervals \((0 + n\pi, \frac{\pi}{4} + n\pi)\).
  • It is decreasing when \(y' = \cos x - \sin x < 0\) within intervals \((\frac{\pi}{4} + n\pi, \pi + n\pi)\).
These intervals act as guides for understanding how the function behaves over its domain.
Graph Sketching
Graphing functions becomes easier and more insightful with knowledge of derivatives and interval behavior. For \(y = \sin x + \cos x\), the graph follows a periodic pattern. Understanding critical points and intervals of increase/decrease helps in accurately plotting distinctive characteristics, such as:
  • Peaks occurring at \(x = \frac{\pi}{4} + n\pi\).
  • Troughs at \(x = \pi + n\pi\).
These insights allow us to sketch a graph that provides a visual representation of the function's periodic upward and downward movements. Always remember, understanding the nature of the function not only helps in sketching but also in predicting behavior in applied scenarios.

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Most popular questions from this chapter

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