Question

0<x<2π aralığındaki f(x)=x÷2 fonksiyonunun 0 < x < 4π için grafiğini çiziniz ve 0<x<2n için Fourier seri dönüşümünü gerçekleştiriniz.

280

likes
1402 views

Answer to a math question 0<x<2π aralığındaki f(x)=x÷2 fonksiyonunun 0 < x < 4π için grafiğini çiziniz ve 0<x<2n için Fourier seri dönüşümünü gerçekleştiriniz.

Expert avatar
Bud
4.6
77 Answers
Öncelikle, f(x) = x/2 fonksiyonunun grafiğini çizelim.

0 < x < 2π aralığı için f(x) = x/2 fonksiyonunun grafiği aşağıdaki gibi olacaktır:

\begin{align*} \text{Grafiğin üzerindeki noktalar:} \ (0, 0), (\pi/2, \pi/4), (\pi, \pi/2), (3\pi/2, 3\pi/4), (2\pi, \pi) \ \text{Ve bu noktaları birleştiren bir doğru elde edeceğiz.}\end{align*}

Grafiği çizdikten sonra, Fourier serisi dönüşümünü gerçekleştirelim.

Fourier Serisi Dönüşümü için a0, an ve bn katsayılarını bulmamız gerekiyor. Bunları hesaplayalım:

\begin{align*} a0 = \frac{1}{\pi} \int_{0}^{2\pi} f(x) dx = \frac{1}{\pi} \int_{0}^{2\pi} \frac{x}{2} dx = \frac{1}{2\pi} \left[ \frac{1}{2} x^2 \right]_{0}^{2\pi} = \frac{1}{2\pi} \left[ \frac{1}{2} (2\pi)^2 \right] = \frac{\pi}{2}\end{align*}

\begin{align*} an = \frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos(nx) dx = \frac{1}{\pi} \int_{0}^{2\pi} \frac{x}{2} \cos(nx) dx\end{align*}

Burada x/2'nin çift veya tek olduğunu kontrol etmeliyiz. Eğer x/2 çift ise, an = 0 olacaktır. Eğer x/2 tek ise, an kısmını hesaplamalıyız:

\begin{align*} an = \frac{1}{\pi} \int_{0}^{2\pi} \frac{x}{2} \cos(nx) dx = \frac{1}{2\pi} \left[ x \sin(nx) + \frac{1}{n} \cos(nx) \right]_{0}^{2\pi} = \frac{1}{2n\pi} (0 + \frac{1}{n} - (0 - \frac{1}{n})) = \frac{2}{n^2\pi}\end{align*}

\begin{align*} bn = \frac{1}{\pi} \int_{0}^{2\pi} f(x) \sin(nx) dx = \frac{1}{\pi} \int_{0}^{2\pi} \frac{x}{2} \sin(nx) dx\end{align*}

Burada x/2'nin çift veya tek olduğunu kontrol etmeliyiz. Eğer x/2 çift ise, bn kısmını hesaplamalıyız:

\begin{align*} bn = \frac{1}{\pi} \int_{0}^{2\pi} \frac{x}{2} \sin(nx) dx = \frac{1}{2\pi} \left[ - x \cos(nx) + \frac{1}{n} \sin(nx) \right]_{0}^{2\pi} = \frac{1}{2n\pi} (0 - (\pi\cos(2n\pi) - 0)) = \frac{1}{2n\pi} \pi = \frac{1}{2n}\end{align*}

Bu şekilde Fourier serisi dönüşümünü elde ettik.

Sonuç olarak, 0 < x < 4π aralığı için f(x) = x/2 fonksiyonunun grafiği aşağıdaki gibi olacaktır:

\begin{align*} \text{Grafiğin üzerindeki noktalar:} \ (0, 0), (\pi/2, \pi/4), (\pi, \pi/2), (3\pi/2, 3\pi/4), (2\pi, \pi), (5\pi/2, 5\pi/4), (3\pi, 3\pi/2), (7\pi/2, 7\pi/4), (4\pi, \pi) \ \text{Ve bu noktaları birleştiren bir doğru elde edeceğiz.}\end{align*}

Fourier serisi dönüşümü için:

\begin{align*} f(x) = \frac{\pi}{2} + \sum_{n=1}^{\infty} \frac{2}{n^2\pi} \cos(nx) + \frac{1}{2n} \sin(nx)\end{align*}

Yukarıdaki seriyi 0 < x < 2n için gösterdik.

Frequently asked questions (FAQs)
Math Question: What is the 4th derivative of f(x) = 3x^5 - 2x^3 + x^2 - 10x + 4?
+
Math question: In circle O, if angle BOC measures 110°, what is the measure of the arc BC?
+
What is the length of the hypotenuse in a right triangle if the other two sides are 8 cm and 15 cm?
+
New questions in Mathematics
Simplify the expression sin³(x)+cos³(x), using trigonometric functions
A college believes that 22% of applicants to that school have parents who have remarried. How large a sample is needed to estimate the true proportion of students who have parents who have remarried to within 5 percentage points?
-6(3x-4)=-6
String x = 5 Int y=2 System.out.println(x+y)
-11+29-18
A juice shop prepares assorted juices, for their juices they have 5 different types of fruit. How many types of assortments can be prepared in total, if it is considered an assortment to a juice made with two or more fruits?
A pair of die is thrown and the absolute difference of the two scores is recorded. What is the probability of the absolute difference being 4 or more?
∫ √9x + 1 dx
You mix a powder drug with a 4.5ml of liquid to get a reconstituted solution with a concentration of 250mg/ml. The prescribers order is for 500 mg . You will give what ml of the reconstituted solution
Find all real numbers x that satisfy the equation \sqrt{x^2-2}=\sqrt{3-x}
Two business partners have a bank balance of $17,942.00. After the first year their interest brings their balance to $18,928.91. What rate of interest is earned?
9 x² + 2x + 1 = 0
(a) List the set of possible rational zeros of the polynomial function F(x) = 2x3 - 11x2 + 13x - 4. (b) Find all rational zeros of F(x). Only do part B
5x+13+7x-10=99
Write the detailed definition of a supply chain/logistics related maximization problem with 8 variables and 6 constraints. Each constraint should have at least 6 variables. Each constraint should have At least 5 variables will have a value greater than zero in the resulting solution. Variables may have decimal values. Type of equations is less than equal. Numbers and types of variables and constraints are important and strict. Model the problem and verify that is feasible, bounded and have at least 5 variables are nonzero.
Calculate the area of the parallelogram with adjacent vertices (1,4, −2), (−3,1,6) 𝑦 (1, −2,3)
a coffee shop has 9 types of creamer and 11 types of sweetener. In how any ways can a person make their coffee?
Sally’s sales for last Sunday were $1,278. That was an increase of 6.5% over her sales for the previous Saturday. What were her sales for the previous Saturday?
Two trains leave stations 294 miles apart at the same time and travel toward each other. One train travels at 95 miles per hour while the other travels at 115 miles per hourHow long will it take for the two trains to meet?
To apply a diagnostic test, in how many ways can 14 students be chosen out of 25? if the order does not matter