/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Draw the graphs of $$ f(x)=\... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 53

Draw the graphs of $$ f(x)=\sin x+\cos x $$

Short Answer

Expert verified
The graph of \(f(x)=\sin(x) + \cos(x)\) is continuous and periodic. It crosses the x-axis at \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\), reaches a peak at \(x=\frac{\pi}{4}\), and a minimum at \(x=\frac{5\pi}{4}\).

Step by step solution

01

Understand the individual functions

First, consider the function \(\sin(x)\) and \(\cos(x)\) separately. Both are periodic functions with a period of \(2\pi\). They both oscillate between -1 and 1. However, \(\sin(x)\) starts at 0 when \(x=0\), while \(\cos(x)\) starts at 1.
02

Add the functions together

When combined into the function \(f(x)=\sin(x) + \cos(x)\), the new function takes the sum of the y-values of the two functions at each x-value. This changes the amplitude and phase of the wave, but it is still a periodic function.
03

Graph the function

Plot the function \(f(x) = \sin(x) + \cos(x)\) on a graph. For each x-value, calculate the y-value as the sum of the y-values of \(\sin(x)\) and \(\cos(x)\) at that x-value and plot the point. Do this for a complete cycle from \(x = 0\) to \(x = 2\pi\) to see one complete cycle of the wave. Due to the periodicity, the graph repeats this pattern for larger or smaller x-values. It crosses the x-axis at \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\), reaches a peak at \(x=\frac{\pi}{4}\), and reaches a minimum at \(x=\frac{5\pi}{4}\).

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