The difference between two natural numbers is 39. If the larger number is divided by the smaller one, then the quotient is 3 and
1 answer:
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Define x and y :
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Let one number be x and the other number will be x + 39
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Construct equation :
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When the big number divided by the smaller number, the quotient is 3 with 11 remaining
⇒
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Solve x :
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Find the numbers:
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small number = x = 14
big number = x + 39 = 14 + 39 = 53
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Answer: The smaller number is 14.
-------------------------------------------
Define x and y :
-------------------------------------------
Let one number be x and the other number will be x + 39
-------------------------------------------
Construct equation :
-------------------------------------------
When the big number divided by the smaller number, the quotient is 3 with 11 remaining
⇒
-------------------------------------------
Solve x :
-------------------------------------------
-----------------------------
Find the numbers:
-----------------------------
small number = x = 14
big number = x + 39 = 14 + 39 = 53
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Answer: The smaller number is 14.
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Answer:
(a) -3/4
(b) -0.75
(c) -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.
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(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.
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(b) The graph tells us the solution is approximately 0.7549. Rounded to 2 decimal places, the solution is approximately 0.75.
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(c) The above solution found after 3 iterations rounded to 2 decimal places is exactly 0.75.
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See the attached table for function values.
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<em>Comment on bisection iteration</em>
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.
