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3 years ago

F(x)=x^3+3

Mathematics

1 answer:

lawyer [7]3 years ago

5 0

Answer:

(a) -3/4

(b) -0.75

Step-by-step explanation:

It's a bit hard to tell what constitutes an "iteration" when using the bisection method to approximate a polynomial root. For the purpose here, we'll say one iteration consists of ...

evaluating the function at the midpoint of the bracketing interval
choosing a smaller bracketing interval
identifying the x-value known to be closest to the solution

Thus, the result of the iteration consists of a bracketing interval and the choice of one of the interval's ends as the solution approximation.

(a) We observe that the graphs intersect in the interval (-1, 0). For the first iteration, we evaluate f(x)-g(x) at x=-1/2. This tells us the solution is in the interval (-1, -1/2). The x-value closest to the root is x=-1/2.

For the second iteration, we evaluate the function f(x)-g(x) at x=-3/4. This tells us the solution is in the interval (-1, -3/4). The x-value closest to the root is x=-3/4.

For the third iteration, we evaluate the function f(x)-g(x) at x=-7/8. This tells us the solution is in the interval (-7/8, -3/4). The x-value closest to the root is x=-3/4.

(b) The graph tells us the solution is approximately 0.7549. Rounded to 2 decimal places, the solution is approximately 0.75.

(c) The above solution found after 3 iterations rounded to 2 decimal places is exactly 0.75.

See the attached table for function values.

_____

Comment on bisection iteration

Since you cut the interval containing the root in half with each iteration, you gain approximately one decimal place for each 3 iterations. When the function value is very nearly zero at one of the interval endpoints, it can take many more iterations to achieve a better result.

Here, it takes 4 more iterations before an x-value becomes closer to the solution (x≈-97/128). And it takes one more iteration to move the end of the interval away from -3/4. After these 5 more iterations (8 total), the solution is known to lie in the interval (-97/128, -193/256). The corresponding solution approximation is -193/256. It is still only correct to 2 decimal places.

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A bag of marbles contains 5 red, 3 blue, 2 green, and 2 yellow marbles. What is the probability that you choose a blue marble an

ira [324]

PART ANSWER:

Hi. I don't know the entire answer but I do know the probability of chosing a blue marble once is 25%.

5 + 3 + 2 + 2 = 12

3/12 or 1/4 = 25% or .25

Sorry I could not fully help.

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3 years ago

Read 2 more answers

Two landscapers must mow a rectangular lawn that measures 100 feet by 200 feet. Each wants to mow no more than half of the lawn.

Citrus2011 [14]

The total area of the complete lawn is (100-ft x 200-ft) = 20,000 ft².
One half of the lawn is 10,000 ft². That's the limit that the first man
must be careful not to exceed, lest he blindly mow a couple of blades
more than his partner does, and become the laughing stock of the whole
company when the word gets around. 10,000 ft² ... no mas !

When you think about it ... massage it and roll it around in your
mind's eye, and then soon give up and make yourself a sketch ...
you realize that if he starts along the length of the field, then with
a 2-ft cut, the lengths of the strips he cuts will line up like this:

First lap:
   (200 - 0) = 200
   (100 - 2) = 98
   (200 - 2) = 198
   (100 - 4) = 96

Second lap:
   (200 - 4) = 196
   (100 - 6) = 94
   (200 - 6) = 194
   (100 - 8) = 92

Third lap:
   (200 - 8) = 192
   (100 - 10) = 90
   (200 - 10) = 190
   (100 - 12) = 88

These are the lengths of each strip. They're 2-ft wide, so the area
of each one is (2 x the length).

I expected to be able to see a pattern developing, but my brain cells
are too fatigued and I don't see it. So I'll just keep going for another
lap, then add up all the areas and see how close he is:

Fourth lap:
   (200 - 12) = 188
   (100 - 14) = 86
   (200 - 14) = 186
   (100 - 16) = 84

So far, after four laps around the yard, the 16 lengths add up to
2,272-ft, for a total area of 4,544-ft². If I kept this up, I'd need to do
at least four more laps ... probably more, because they're getting smaller
all the time, so each lap contributes less area than the last one did.

Hey ! Maybe that's the key to the approximate pattern !

Each lap around the yard mows a 2-ft strip along the length ... twice ...
and a 2-ft strip along the width ... twice. (Approximately.) So the area
that gets mowed around each lap is (2-ft) x (the perimeter of the rectangle),
(approximately), and then the NEXT lap is a rectangle with 4-ft less length
and 4-ft less width.

So now we have rectangles measuring

   (200 x 100), (196 x 96), (192 x 92), (188 x 88), (184 x 84) ... etc.

and the areas of their rectangular strips are
   1200-ft², 1168-ft², 1136-ft², 1104-ft², 1072-ft² ... etc.

==> I see that the areas are decreasing by 32-ft² each lap.
   So the next few laps are
   1040-ft², 1008-ft², 976-ft², 944-ft², 912-ft² ... etc.

How much area do we have now:

   After 9 laps,    Area =   9,648-ft²
   After 10 laps, Area = 10,560-ft².

And there you are ... Somewhere during the 10th lap, he'll need to
stop and call the company surveyor, to come out, measure up, walk
in front of the mower, and put down a yellow chalk-line exactly where
the total becomes 10,000-ft².

There must still be an easier way to do it. For now, however, I'll leave it
there, and go with my answer of: During the 10th lap.

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3 years ago

What are the numbers of pi

tresset_1 [31]

3.14?

or do you want more digits?

3.141592653589793238462643383279502884197169399375105820974944592307816406286

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3 years ago

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aleksandrvk [35]

Answer:

66.5

Step-by-step explanation:

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Step-by-step explanation:

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3 years ago

F(x) = 2x2 - 5 and g(x) = 5x + 7 Find f(g(x)) = [ ? ]x2 + [ ]x +

lbvjy [14]

Answer:10x^2 -18

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3 years ago