Using the graph below select all statements that are true

Mathematics

1 answer:

Leokris [45]3 years ago

6 0

Answer:

A.This is the graph of the greatest integer function

E. f(-3.4)=-4

Step-by-step explanation:

The graph shown in the attachment is the parent greatest integer function.

f(x)=[x]

This takes any integer as an input and the output is the greatest integer less than the input.

For instance

f(4.9)=4 because 4 is the greatest integer less than 4.9

f(-3.4)=-4 because -4 is the greatest integer less than -3.4

This graph is not one-to-one because more x-values can correspond to one y-value.

For instance f(3.4)=3 and f(3.9)=3

The correct options are A and E

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The full range is $-\pi$ (length $2L=2\pi$ ), so the half range is $L=\pi$ . The half range sine series would then be given by

$f(x)=\displaystyle\sum_{n\ge1}b_n\sin\dfrac{n\pi x}L=\sum_{n\ge1}b_n\sin nx$

where

$b_n=\displaystyle\frac2L\int_0^Lf(x)\sin\dfrac{n\pi x}L\,\mathrm dx=\frac2\pi\int_0^\pi(\pi-x)\sin nx\,\mathrm dx$

Essentially, this is the same as finding the Fourier series for the function

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Integrating by parts yields

$b_n=\dfrac2\pi\left(\dfrac\pi n-\dfrac{\sin n\pi}{n^2}\right)=\dfrac2n$

So the half range sine series for this function is simply

$f(x)=\displaystyle\sum_{n\ge1}\frac{2\sin nx}n$