Given the line below.

Mathematics

1 answer:

Assoli18 [71]3 years ago

8 0

Answer:

y + 1 = $\frac{1}{2}$ (x + 2)

Step-by-step explanation:

The equation of a line in point- slope form is

y - y₁ = m(x - x₁)

where m is the slope and (x₁, y₁ ) a point on the line

Calculate m using the slope formula

m = $\frac{y_{2}-y_{1} }{x_{2}-x_{1} }$

with (x₁, y₁ ) = (- 2, - 1 ) and (x₂, y₂ ) = (2, 1 )

m = $\frac{1-(-1)}{2-(-2)}$ = $\frac{1+1}{2+2}$ = $\frac{2}{4}$ = $\frac{1}{2}$ and (x₁, y₁ ) = (- 2, - 1 ) , then

y - (- 1) = $\frac{1}{2}$ (x - (- 2) ) , that is

y + 1 = $\frac{1}{2}$ (x + 2)

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F(x)=x^3+3

lawyer [7]

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.

_____

<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.