Let's break down the problem step by step. --- ## (a) **Find the cos series for \( f(x) \) with period \( 2L \):** This is asking for the **Fourier cosine series** for a function \( f(x) \) that is defined on \( [0, L] \) and extended as an even function to \( [-L, L] \), with periodicity \( 2L \). ### **Step 1: General Form of the Cosine Series** The Fourier cosine series for a function \( f(x) \) defined on \( [0, L] \) is: \[ f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right) \] where the coefficients are: \[ a_0 = \frac{2}{L} \int_0^L f(x)\,dx \] \[ a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n\pi x}{L}\right) dx, \quad n \geq 1 \] This series represents the **even extension** of \( f(x) \) on \( [-L, L] \), and then it is repeated every \( 2L \) (thus, period \( 2L \)). --- ### **Step 2: Write the Final Answer (General Cos Series)** **The cosine series for \( f(x) \) with period \( 2L \) is:** \[ \boxed{ f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right) } \] where \[ a_0 = \frac{2}{L} \int_0^L f(x)\,dx, \qquad a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n\pi x}{L}\right) dx \] --- ## (b) **Sketch the graph of the function to which the series converges over 3 periods** ### **Step 1: Description of the Extended Function** - The cosine series converges to the even periodic extension of \( f(x) \) with period \( 2L \). - Over one period \( [0, 2L] \), \( f(x) \) is defined as: - On \( [0, L] \): \( f(x) \) (the original function) - On \( [L, 2L] \): \( f(2L - x) \) (mirror image) - Extend this pattern to the left and right for additional periods. ### **Step 2: Sketch (Describe since text-only)** - On \( [0, L] \): Draw the original \( f(x) \). - On \( [-L, 0] \): Draw the mirror image of \( f(x) \) (since it's even). - Repeat this \( 2L \)-long shape to the left and right for a total of 3 periods (\( -2L \) to \( 4L \)). **(You can sketch this as a "wavy" shape that repeats every \( 2L \), symmetric about \( x = 0, 2L, 4L, \ldots \))** --- ## **Summary** ### **(a) Cosine Series:** \[ f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right) \] with \[ a_0 = \frac{2}{L} \int_0^L f(x)\,dx, \qquad a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n\pi x}{L}\right) dx \] ### **(b) Sketch:** The function is the **even periodic extension** of \( f(x) \) with period \( 2L \), repeated over three periods. --- **If you provide a specific \( f(x) \), I can calculate the coefficients and show a specific plot!**

Certainly! Since I can't physically draw here, I'll guide you through how to sketch the graph of the function to which the Fourier cosine series converges, based on the typical form of such problems. --- ### Step-by-step guide to sketch the graph of the periodic extension: #### 1. **Identify the original function \( f(x) \) on \( [0, L] \):** - For example, if \( f(x) \) is given in the problem (e.g., a specific function like a constant, linear, or piecewise function), note its shape on \( [0, L] \). *Note:* Without the explicit \( f(x) \) from the image, let's assume a generic case for illustration, say \( f(x) = x \) on \( [0, L] \). #### 2. **Construct the even extension on \( [-L, L] \):** - Extend \( f(x) \) to \( [-L, 0] \) as an even function: \[ f(-x) = f(x) \] For \( f(x) = x \), the reflection is \( f(-x) = -x \), but since the Fourier cosine series corresponds to an **even extension**, the function should be made even, i.e., \( f_{even}(x) = |x| \) for the example. - For general \( f(x) \), mirror the graph across the y-axis. #### 3. **Repeat periodically with period \( 2L \):** - After constructing the \( 2L \)-periodic even extension, replicate this pattern to the left and right for multiple periods. - For **three periods**, sketch the pattern over \( x \in [-L, 3L] \): - **Period 1:** \( [0, L] \) - **Period 2:** \( [L, 2L] \) - **Period 3:** \( [2L, 3L] \) - The pattern repeats every \( 2L \), so the graph looks like a wave that repeats, with symmetry about the y-axis (even extension). #### 4. **Sketch the combined shape:** - For each period: - On \( [0, L] \), plot the original \( f(x) \). - On \( [L, 2L] \), plot the mirror image of \( f(x) \) about \( x = L \). - Connect the points smoothly to visualize the convergence of the Fourier series. --- ### **Summary of the sketch:** - A "wave-like" pattern repeating every \( 2L \). - Symmetric about the y-axis within each period. - The function's shape within one period is the even extension of the original \( f(x) \). --- ### **Example (if \( f(x) = x \) on \( [0, L] \)):** - On \( [0, L] \): linearly increasing from 0 to \( L \). - On \( [-L, 0] \): mirror image, decreasing from 0 to \(-L\), making the extension even. - Repeat this pattern over three periods. --- **If you'd like, I can help generate a specific sketch if you tell me the form of \( f(x) \).**

Certainly! To provide a clear sketch, I will assume a specific example function \(f(x)\) on \([0, L]\). Let's choose: \[ f(x) = x, \quad \text{for } 0 \leq x \leq L \] This is a common example and will illustrate the process well. --- ## Step-by-step sketch guide for \(f(x) = x\): ### 1. **Original function on \([0, L]\):** - A straight line starting from \(0\) at \(x=0\) to \(L\) at \(x=L\). ### 2. **Construct the even extension:** - Since the Fourier cosine series corresponds to the **even extension** of \(f(x)\), define: \[ f_{even}(x) = \begin{cases} x, & x \in [0, L] \\ -x, & x \in [-L, 0] \end{cases} \] - For \(x \in [-L, 0]\), the graph is a straight line decreasing from \(0\) at \(x=0\) to \(-L\) at \(x=-L\). - The overall extended function over \([-L, L]\) is symmetric about the y-axis, forming a "V" shape with vertex at the origin. ### 3. **Periodicity:** - The function repeats every \(2L\), so: \[ f_{periodic}(x) = f_{even}(x \bmod 2L) \] - Over the interval \([-L, L]\), it looks like a symmetric "V". - Over three periods, the graph will extend from \(-L\) to \(3L\): - \(\boxed{-L \text{ to } L}\): first period - \(\boxed{L \text{ to } 3L}\): second period, repeating the pattern ### 4. **Sketch the graph:** - **Center at \(x=0\):** \(f(0) = 0\). - **On \([0, L]\):** linearly increasing from 0 to \(L\). - **On \([-L, 0]\):** decreasing linearly from 0 at \(x=0\) to \(-L\) at \(x=-L\). - **Repeat for subsequent periods:** - From \([L, 3L]\), the pattern repeats: - On \([L, 2L]\): \(f(x) = x - 2L\), decreasing from \(f(L)=L\) at \(x=L\) to \(f(2L)=0\) at \(x=2L\). - On \([2L, 3L]\): \(f(x) = x - 2L\), increasing from \(f(2L)=0\) to \(f(3L)=L\). - The overall graph looks like a repeated wave of straight lines forming "V" shapes, symmetric about the y-axis, with each "V" spanning length \(2L\). --- ## **Summary of the sketch:** - **Shape:** Repeating "V" shapes with the vertex at each multiple of \(L\). - **Symmetry:** About the y-axis. - **Period:** \(2L\). --- ## **Final note:** If you wish, I can help generate a simple ASCII art sketch or guide you through plotting this with graphing software. Let me know!