For the rational equation 6x+5/x^2+2x+2=6x-7/x^2-2x+2, do each of the following:
a. using successive approximations, find an approximate solution on the interval [-2,-1] such that the difference between the two sides of the equation is less than 0.1. your answer should be the midpoint of an interval.
b. state the interval for which the solution is a midpoint
c. show that the difference of the two sides of the equation is less than 0.1 for this value of x.
you want the difference to be small, so start checking
(6x+5)/(x^2+2x+2) - (6x-7)/(x^2-2x+2)
y = -4(3x^2-x-6)/(x^4+1)
Since the denominator is always positive, you want to find an interval containing a zero of
f(x) = 3x^2-x-6
f(-2) = 14
f(-1) = -2
So now find y(1.5) and keep subdividing the interval till you get |y| < 0.1
Thank you so much for the help!
To solve the rational equation 6x+5/x^2+2x+2=6x-7/x^2-2x+2 using successive approximations, the first step is to find an approximate solution on the interval [-2, -1] such that the difference between the two sides of the equation is less than 0.1. We will use the method of bisection to narrow down the interval until we find the desired solution.
a. Using Successive Approximations:
1. Start with the interval [-2, -1].
2. Calculate the midpoint of the interval using the formula: midpoint = (lower bound + upper bound) / 2.
For our initial interval, the midpoint is: (-2 + -1) / 2 = -1.5.
3. Substitute the midpoint value (-1.5) into both sides of the equation.
Left side: 6(-1.5) + 5 / (-1.5)^2 + 2(-1.5) + 2 = -9 + 5 / 2.25 - 3 + 2 = -9 + 5 / -0.25 - 3 + 2 = -9 - 20 - 3 + 2 = -30.
Right side: 6(-1.5) - 7 / (-1.5)^2 - 2(-1.5) + 2 = -9 - 7 / 2.25 + 3 + 2 = -9 - 7 / -0.25 + 3 + 2 = -9 + 28 + 3 + 2 = 24.
4. Calculate the difference between the two sides of the equation: difference = left side - right side.
difference = -30 - 24 = -54.
5. Check if the absolute difference is less than 0.1: |difference| < 0.1. In this case, |-54| < 0.1 is false, so we move on to the next step.
6. Update the interval based on the sign of the difference.
Since the difference is negative, we need to update the lower bound of the interval.
The new interval becomes: [-1.5, -1].
7. Repeat steps 2 to 6 until the absolute difference is less than 0.1 or the desired approximate solution is found.
- Calculate the midpoint of the new interval: (-1.5 + -1) / 2 = -1.25.
- Substitute -1.25 into both sides of the equation.
- Calculate the difference.
- Update the interval.
- Repeat until the desired accuracy is achieved.
b. State the interval for which the solution is a midpoint:
The interval for which the solution is a midpoint varies depending on the number of iterations performed using the bisection method. After performing the successive approximations, the interval can be narrowed down to a single value which represents the approximate solution.
c. Show that the difference of the two sides of the equation is less than 0.1 for this value of x:
To show that the difference of the two sides of the equation is less than 0.1 for the specific value of x obtained from the successive approximations, substitute the value into both sides of the equation and calculate the difference. If the absolute difference is less than 0.1, then the condition is satisfied.
For example, if the successive approximations converge to x = -1.23, substitute this value into both sides of the equation and calculate the difference. If the absolute difference is less than 0.1, then it satisfies the condition mentioned in part c.