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.)

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

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:

so now we can find f'(0)

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:

so:

we can rewrite this derivative as:

and use the chain rule to get:

which simplifies to:

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

which yields:

f''(0)=0

so now we can complete the second order Maclaurin polynomial:

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:

so:

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:

in this case:

p=2x

p'=2

So when using the quotient rule we get:

which simplifies to:

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

which yields:

f'''(0)=-2

so now we can complete the third order Maclaurin polynomial:

which simplifies to:

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

so:

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

in this case:

p'=12x

So when using the quotient rule we get:

which simplifies to:

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

which yields:

so now we can complete the fourth order Maclaurin polynomial:

which simplifies to:

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:

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

well if he bought them at 5 cedis each sold them at 7 cedis each then he would make a profit of 2 cedis of each book. if in total he made 150 cedis profit that means he sold 75 books. he lost 12 books at the start so that means he originally bought a total of 82 books.

step-by-step explanation: