Mathematics, 08.08.2021 07:10, egreer9287

Find the first, second, third and fourth order Maclaurin polynomials of f(x) = arctan(x). Draw the graph of f(x) and the four polynomials on the same
diagram. (Sketch by hand or use software.)

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answered: carlo123

The first, second, third and fourth order Maclaurin polynomials of f(x)=arctan(x) are:

The first order Maclaurin polynomial is f(x)=xThe second order Maclaurin polynomial is also f(x)=xThe third order Maclaurin polynomial is $f(x)=x-\frac{1}{3}x^{3}$ The fourth order Maclaurin polynomial is also $f(x)=x-\frac{1}{3}x^{3}$ You can see the graph on the attached picture.

So let's start by finding the first order maclaurin polynomial:

f(x)=f(0)+f'(0)x

so let's find each part of the function:

f(0)=arctan(0)

f(0)=0

now, let's find the first derivative of f(x)

f(x)=arctan(x)

This is a usual derivative so there is a rule we can use here:

$f'(x)=\frac{1}{x^{2}+1}$

so now we can find f'(0)

$f'(0)=\frac{1}{(0)^{2}+1}$

f'(0)=1

So we can now complete the first order Maclaurin Polynomial:

f(x)=0+1x

which simplifies to:

f(x)=x

Now let's find the second order polynomial, for which we will need to get the second derivative of the function:

$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}$

so:

$f'(x)=\frac{1}{x^{2}+1}$

we can rewrite this derivative as:

$f'(x)=(x^{2}+1)^{-1}$

and use the chain rule to get:

$f''(x)=-1(x^{2}+1)^{-2}(2x)$

which simplifies to:

$f''(x)=-\frac{2x}{(x^{2}+1)^{2}}$

now, we can find f''(0):

$f''(0)=-\frac{2(0)}{((0)^{2}+1)^{2}}$

which yields:

f''(0)=0

so now we can complete the second order Maclaurin polynomial:

$f(x)=0+1x+\frac{0}{2!}x^{2}$

which simplifies to:

f(x)=x

Now let's find the third order polynomial, for which we will need to get the third derivative of the function:

$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+\frac{f'''(0)}{3!}x^{3}$

so:

$f''(x)=-\frac{2x}{(x^{2}+1)^{2}}$

In this case we can use the quotient rule to solve this:

Quotient rule: Whenever you have a function in the form , then it's derivative is:

$f'(x)=\frac{p'q-pq'}{q^{2}}$

in this case:

p=2x

p'=2

$q=(x^{2}+1)^{2}$

$q'=2(x^{2}+1)(2x)$

$q'=4x(x^{2}+1)$

So when using the quotient rule we get:

$f'(x)=\frac{p'q-pq'}{q^{2}}$

$f'''(x)=\frac{(2)(x^{2}+1)^{2}-(2x)(4x)(x^{2}+1)}{((x^{2}+1)^{2})^{2}}$

which simplifies to:

$f'''(x)=\frac{-2x^{2}-2+8x^{2}}{(x^{2}+1)^{3}}$

$f'''(x)=\frac{6x^{2}-2}{(x^{2}+1)^{3}}$

now, we can find f'''(0):

$f'''(0)=\frac{6(0)^{2}-2}{((0)^{2}+1)^{3}}$

which yields:

f'''(0)=-2

so now we can complete the third order Maclaurin polynomial:

$f(x)=0+1x+\frac{0}{2!}x^{2}-\frac{2}{3!}x^{3}$

which simplifies to:

$f(x)=x-\frac{1}{3}x^{3}$

Now let's find the fourth order polynomial, for which we will need to get the fourth derivative of the function:

$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+\frac{f'''(0)}{3!}x^{3}+\frac{f^{(4)}(0)}{4!}x^{4}$

so:

$f'''(x)=\frac{6x^{2}-2}{(x^{2}+1)^{3}}$

In this case we can use the quotient rule to solve this:

$f'(x)=\frac{p'q-pq'}{q^{2}}$

in this case:

$p=6x^{2}-2$

p'=12x

$q=(x^{2}+1)^{3}$

$q'=3(x^{2}+1)^{2}(2x)$

$q'=6x(x^{2}+1)^{2}$

So when using the quotient rule we get:

$f'(x)=\frac{p'q-pq'}{q^{2}}$

$f^{4}(x)=\frac{(12x)(x^{2}+1)^{3}-(6x^{2}-2)(6x)(x^{2}+1)^{2}}{((x^{2}+1)^{3})^{2}}$

which simplifies to:

$f^{4}(x)=\frac{12x^{3}+12x-6x^{3}+12x}{(x^{2}+1)^{4}}$

$f^{4}(x)=\frac{6x^{3}+24x}{(x^{2}+1)^{4}}$

now, we can find f^{4}(0):

$f^{4}(x)=\frac{6(0)^{3}+24(0)}{((0)^{2}+1)^{4}}$

which yields:

$f^{4}(0)=0$

so now we can complete the fourth order Maclaurin polynomial:

$f(x)=0+1x+\frac{0}{2!}x^{2}-\frac{2}{3!}x^{3}+\frac{0}{4!}x^{4}$

which simplifies to:

$f(x)=x-\frac{1}{3}x^{3}$

you can find the graph of the four polynomials in the attached picture.

So the first, second, third and fourth order Maclaurin polynomials of f(x)=arctan(x) are:

You can find further information on the following link:

Find the first, second, third and fourth order Maclaurin polynomials of f(x) =

arctan(x). Draw the

answered: Guest

your answer would be 10x - 1

answered: Guest

d) 48

step-by-step explanation:

16,32,48,64,80,96,112,128,144,160,176,192,208,224,240,256,272,288,304,320,336

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Find the first, second, third and fourth order Maclaurin polynomials of f(x) = arctan(x). Draw the graph of f(x) and the four polynomials on the same diagram. (Sketch by hand or use software.)

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Find the first, second, third and fourth order Maclaurin polynomials of f(x) = arctan(x). Draw the graph of f(x) and the four polynomials on the same
diagram. (Sketch by hand or use software.)