Answer:
2
Step-by-step explanation:
In figure 2, find x if the perimeter is 4. Does each answer make sense? Why or why not?
1 answer:
The value of x is equal to 1.5.
It is required to find x.
<h3>What is quadratic equation?</h3>Any equation containing one term in which the unknown is squared and no term in which it is raised to a higher power. The quadratic formula to solve a quadratic equation ax2 + bx + c = 0 is x = [-b ± √(b2 - 4ac)]/2a.
Given that:
So for the triangle, the perimeter would be:
By solving in above equation, we get
So, we take x= 1.5 and neglect -4 because we can't take negative roots of the quadratic equation.
Hence, the value of x is equal to 1.5.
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Answer:
The Fourier series of f(x) converges to 3 at the points x= π+2kπ, where k is an integer.
Step-by-step explanation:
First, recall that the function f(x) is extended 2π periodic to the whole real line, in order to obtain a valid Fourier expansion. Remember that a Fourier series is formed by a sines and cosines, which are 2π-periodic.
So, the 2π-periodic expansion of f(x) is discontinuous at the points π+2kπ, in particular π and -π. Check the attached figure to a better understanding.
Now, the Dirichlet theorem on the convergence of a Fourier series tells us that the series converges to the function at the points of continuity, and at points of discontinuity the sum of the series is
.
Here we understand the notation and
as
and
.
In this particular case
.
For the limit , with
recall that our function is 2π-periodic, so the values of f near π, with x>π are the same when x is near -π and x>-π. Again, check the attached figure. So,
.
Thus,
.
Note: In the attached figure we only have drawn three repetitions of the 2π-periodic extension of , recall that the extension is <em>ad infinitum</em>. Also, the points drawn in the dotted lines are the sum of the series at the points of discontinuity.
