Consider the graph of the linear equation.

Find the roots of the equation f(x) = x3 - 0.2589x2 + 0.02262x -0.001122 = 0

devlian [24]

Answer:

The root of the equation $x^3-0.2589x^{2}+0.02262x-0.001122=0$ is x ≈ 0.162035

Step-by-step explanation:

To find the roots of the equation $x^3-0.2589x^{2}+0.02262x-0.001122=0$ you can use the Newton-Raphson method.

It is a way to find a good approximation for the root of a real-valued function f(x) = 0. The method starts with a function f(x) defined over the real numbers, the function derivative f', and an initial guess $x_{0}$ for a root of the function. It uses the idea that a continuous and differentiable function can be approximated by a straight line tangent to it.

This is the expression that we need to use

$x_{n+1}=x_{n} -\frac{f(x_{n})}{f(x_{n})'}$

For the information given:

$f(x) = x^3-0.2589x^{2}+0.02262x-0.001122=0\\f(x)'=3x^2-0.5178x+0.02262$

For the initial value $x_{0}$ you can choose $x_{0}=0$ although you can choose any value that you want.

So for approximation $x_{1}$

$x_{1}=x_{0}-\frac{f(x_{0})}{f(x_{0})'} \\x_{1}=0-\frac{0^3-0.2589\cdot0^2+0.02262\cdot 0-0.001122}{3\cdot 0^2-0.5178\cdot 0+0.02262} \\x_{1}=0.0496021$

Next, with $x_{1}=0.0496021$ you put it into the equation

$f(0.0496021)=(0.0496021)^3-0.2589\cdot (0.0496021)^2+0.02262\cdot 0.0496021-0.001122 = -0.0005150$ , you can see that this value is close to 0 but we need to refine our solution.

For approximation $x_{2}$

$x_{2}=x_{1}-\frac{f(x_{1})}{f(x_{1})'} \\x_{1}=0-\frac{0.0496021^3-0.2589\cdot 0.0496021^2+0.02262\cdot 0.0496021-0.001122}{3\cdot 0.0496021^2-0.5178\cdot 0.0496021+0.02262} \\x_{1}=0.168883$

Again we put $x_{2}=0.168883$ into the equation

$f(0.168883)=(0.168883)^3-0.2589\cdot (0.168883)^2+0.02262\cdot 0.168883-0.001122=0.0001307$ this value is close to 0 but again we need to refine our solution.

We can summarize this process in the following table.

The approximation $x_{5}$ gives you the root of the equation.

When you plot the equation you find that only have one real root and is approximate to the value found.

5 0

3 years ago

Find the value of (2.8-0.45)²+∛5.832

Ilya [14]

As you can see, 2.8 - 0.45 is 2.35.

Now, 2.35 squared is 5.225

∛5.832 if you simplify this you get 1.8.

1.8 + 5.225 = 7.3225

Hope this helps :)

8 0

4 years ago

A bottle of the emotion is on sale for 2.25 if this price represents a 50% discount from the original price what is the original

lbvjy [14]

Answer:

4.50

Step-by-step explanation:

2.25 times (100/(sale percentage)), so 2.25 x 2

3 0

4 years ago

Square root of -125 in simplest radical form

djverab [1.8K]

Answer:

5√5

Step-by-step explanation:

Simplify the radical by breaking the radicand up into a product of known factors, assuming positive real numbers.

5√5

5 0

3 years ago

What % of the first 15 whole numbers have exactly 4 distinct whole number divisors? (1, 2, 3, and 6 are the four distinct diviso

skelet666 [1.2K]

When a natural number is expressed as a product of two other natural numbers, those other numbers are factors of the original number. For example, two factors of 12 are 3 and 4, because 3 • 4 = 12.

When one number can be divided by another number with no remainder, we say the first number is divisible by the other number. For example, 20 is divisible by 4 (). If a number is divisible by another number, it is also a multiple of that number. For example, 20 is divisible by 4, so 20 is a multiple of 4.

A Prime Number can be divided evenly only by 1 or itself.
And it must be a whole number greater than 1.

Example: 7 can only be divided evenly by 1 or 7, so it is a prime number.

But 6 can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number (it is a composite number).

Let me explain ...

Some whole numbers can be divided up evenly, and some can't!

Example:

6 can be divided evenly by 2, or by 3:

6 = 2 × 3

Like this:

ordivided into 2 groups divided into 3 groups

But 7 cannot be divided up evenly:

And we give them names:

When a number can be divided up evenly it is a Composite NumberWhen a number can not be divided up evenly it is a Prime Number

So 6 is Composite, but 7 is Prime.

Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …

The square root of a number, n, written

is the number that gives n when multiplied by itself. For example,

because 10 x 10 = 100

Examples

Here are the square roots of all the perfect squares from 1 to 100.

Finding square roots of of numbers that aren't perfect squares without a calculator

1. Estimate - first, get as close as you can by finding two perfect square roots your number is between.

2. Divide - divide your number by one of those square roots.

3. Average - take the average of the result of step 2 and the root.

4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.

Example: Calculate the square root of 10 () to 2 decimal places.

1. Find the two perfect square numbers it lies between.

Solution:
32 = 9 and 42 = 16, so lies between 3 and 4.

2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer)

3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667

Repeat step 2: 10/3.1667 = 3.1579
Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623

Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001

If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.

Note: There are a number of ways to calculate square roots without a calculator. This is only one of them.

4 0

3 years ago

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