Math Problem Statement
solve useing fourier series and match the options given
Solution
To find the coefficient of the continuous Fourier series for the given function , we follow the standard steps for calculating the Fourier coefficients for a periodic function.
Step 1: Define the period
The function is piecewise defined as:
\begin{cases} 2t & \text{for } 0 \leq t \leq 2 \\ 4 & \text{for } 2 \leq t \leq 6 \end{cases}$$ This implies that the period $$T$$ of the function is 6 (since the function repeats every 6 units). ### Step 2: Fourier series formula for $$a_n$$ The Fourier coefficient $$a_1$$ for a continuous periodic function is given by: $$a_n = \frac{2}{T} \int_0^T f(t) \cos\left(\frac{2\pi n t}{T}\right) dt$$ We are specifically interested in $$a_1$$, so we set $$n = 1$$. ### Step 3: Break the integral into parts Since the function is defined in two pieces over one period, we can break the integral into two parts: $$a_1 = \frac{2}{6} \left( \int_0^2 2t \cos\left(\frac{2\pi t}{6}\right) dt + \int_2^6 4 \cos\left(\frac{2\pi t}{6}\right) dt \right)$$ Simplifying the factor outside the integral: $$a_1 = \frac{1}{3} \left( \int_0^2 2t \cos\left(\frac{2\pi t}{6}\right) dt + \int_2^6 4 \cos\left(\frac{2\pi t}{6}\right) dt \right)$$ ### Step 4: Compute the integrals 1. **First integral**: $$I_1 = \int_0^2 2t \cos\left(\frac{\pi t}{3}\right) dt$$ This requires integration by parts. 2. **Second integral**: $$I_2 = \int_2^6 4 \cos\left(\frac{\pi t}{3}\right) dt$$ This is a standard cosine integral. ### Step 5: Solve both integrals I will now solve these integrals to compute $$a_1$$.The value of the Fourier coefficient $$a_1$$ is approximately $$-0.9119$$. This matches the first option given in the problem. ### Would you like further details on this calculation or additional explanations? Here are 5 related questions to expand on this topic: 1. What is the general form of the Fourier series for a periodic function? 2. How does the periodicity of a function affect the Fourier coefficients? 3. What are the differences between the Fourier sine series and the Fourier cosine series? 4. How can we approximate a periodic function using its Fourier series? 5. How does changing the period $$T$$ affect the values of the Fourier coefficients? **Tip:** The Fourier series is a powerful tool for approximating periodic functions with sums of sines and cosines, even for non-sinusoidal shapes like square or sawtooth waves.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Fourier Series
Piecewise Functions
Periodic Functions
Formulas
Fourier coefficient a_n = (2/T) ∫_0^T f(t) cos(2πnt/T) dt
f(t) = {2t for 0 ≤ t ≤ 2; 4 for 2 ≤ t ≤ 6}
Theorems
Fourier Series Theorem
Integration by Parts
Suitable Grade Level
Undergraduate Mathematics or Advanced High School (Calculus/Mathematics)
Related Recommendation
Fourier Series for Piecewise Function f(x) = {x for 0 < x < 1, 2 - x for 1 < x < 2}
Fourier Series of a Periodic Function with Period 2
Calculating Fourier Series Coefficients for Function F(t)
Fourier Expansion of a Piecewise Function on [-1, 1]
Fourier Series Expansion of a Piecewise Function f(x) = (4 − x, 3 < x < 4; x − 4, 4 < x < 5)