We're asked to use Newton's method
to approximate the zero of the equation
5 x cubed plus 2x plus 2 equals zero as follows.
Let x1 equal negative 2 be the initial approximation.
Determine x2 and x3.
Round all answers to four decimal places.
Newton's method uses tangent lines
to approximate zeros of a function.
And the formulate for Newton's method is x sub n plus 1
equals x sub n minus f of sub n
divided by f prime of x sub n.
So first, because we're trying to find the zero
of the equation 5 x cubed plus 2x plus 2 equals zero,
the function f of x is
5 x cubed plus 2x plus 2.
If we were doing this by hand,
we would also need the derivative function,
which in this case, f prime of x
is equal to 15 x squared plus 2.
Now in this lesson, we are going to be using Desmos
to complete the table below
and make the approximations for x2 and x3.
But before we do, let's write the formulas for x2 and x3.
Notice for x2, n is equal to 1 since 1 plus 1 is 2.
X2 is equal to x1
minus f of x1
divided by f prime of x1.
In this case, we know x1 is negative 2,
and therefore this is equal to
negative 2 minus f of negative 2
divided by f prime of negative 2.
Let's also write the formula for x3.
For x3, notice n is equal to 2 since 2 plus 1 is 3.
X3 is equal to x2 minus f of x2
divided by f prime of x2.
We don't know x2
until we calculate this first approximation.
So now we'll create a similar table to the one below
using desmos.com, where when n is equal to 1,
we know x1 is given as negative 2.
So this first row is going to be f of negative 2,
f prime of negative 2,
and then finally, the last entry is going to be x2,
which again is equal to x1 minus f of x1
divided by f prime of x1.
Let's go to desmos.com and create this table.
First, let's make the calculation screen larger
by moving the cursor to the right
until we see the double arrow.
Click hold and drag to the right.
Next, click into cell 1
and enter the function f of x equals 5 x the power of 3,
right arrow, plus 2x plus 2.
We don't have to enter the derivative function in Desmos,
because Desmos will find derivative function values.
Next click in cell 2.
Click the Add Item button and select table.
And now we'll replicate the table on the previous slide.
Delete x1 and enter n, tab, tab.
Delete y1 and enter x, tab.
Next f of x, tab, f prime of x, tab.
And then we have the formula x minus f of x
divided by f prime of x.
And now let's go back to the n column
and enter 1, 2,
and 3.
And again, we know x1 is equal to negative 2,
and therefore we enter negative 2 in the x column
when n is 1.
And notice how the entire row automatically populates,
where f of negative 2 is negative 42,
f prime of negative 2 is 62,
and negative 2 minus f of negative 2
divided by f prime of negative 2
to four decimal places
is approximately negative 1.3226.
This value is x2.
Let's go back and record this in our table.
And now going to the next row, where n is equal to 2,
we know x2 is negative 1.3226.
Let's also record x2 above.
Again, we now know this is approximately negative 1.3226.
And now let's find x3 using the table in Desmos.
So going back to Desmos,
we simply go down to the row where n is equal to 2
and enter x2 to four decimal places,
which again, is negative 1.3226.
And again, the entire row populates,
where f of negative 1.3226
is approximately negative 12.2131.
F prime of negative 1.3226
is approximately negative 28.2391.
And finally, the last entry is our approximation for x3,
which is approximately negative 0.8901.
Once this table is set up,
we can keep going and going as long as we need to.
Notice x3 would be approximately negative 0.8901,
x4 would be approximately negative 0.6520,
from the last entry, and so on.
Let's go back and complete our table
and finalize the answers for x2 and x3.
Again, we now know x3 is approximately negative 0.8901.
I hope you found this helpful.