Use the method of bisection to find the root of the equation x^5 + 3x − 7 = 0
accurate to two decimal places.
since f(1) < 0 and f(2) > 0,start with the interval (1,2):
step left right middle f(middle)
1: 1.000 2.000 1.500 5.094
2: 1.000 1.500 1.250 -0.198
3: 1.250 1.500 1.375 2.040
4: 1.250 1.375 1.312 0.832
5: 1.250 1.312 1.281 0.297
6: 1.250 1.281 1.266 0.044
7: 1.250 1.266 1.258 -0.078
8: 1.258 1.266 1.262 -0.017
actual root: 1.26282
To find the root of the equation x^5 + 3x - 7 = 0 using the method of bisection, we need to follow these steps:
Step 1: Determine an interval [a, b] where the root lies. Here, we can choose any interval where the function changes sign. Let's try the interval [-2, 2].
Step 2: Calculate the midpoint c of the interval [a, b] using the formula c = (a + b) / 2.
Step 3: Evaluate the function f(c) at the midpoint c. If f(c) = 0, then c is the root. Otherwise, proceed to the next step.
Step 4: Determine a new interval [a, b] where the root lies. If f(c) and f(a) have opposite signs, the root lies in the interval [a, c]. Otherwise, the root lies in the interval [c, b].
Step 5: Repeat steps 2 to 4 until the desired accuracy is achieved. If the interval [a, b] is smaller than the desired accuracy, then the midpoint c is the approximate root.
Let's apply these steps to find the root of the equation x^5 + 3x - 7 = 0:
Step 1: We start with the interval [-2, 2].
Step 2: The midpoint is c = (-2 + 2) / 2 = 0.
Step 3: Evaluate f(0) = 0^5 + 3(0) - 7 = -7. Since f(c) is not equal to 0, we move to the next step.
Step 4: Since f(c) = -7 and f(a) = -7, which have the same sign, the new interval is [0, 2].
Step 2 (repeated): The midpoint is c = (0 + 2) / 2 = 1.
Step 3 (repeated): Evaluate f(1) = 1^5 + 3(1) - 7 = -3. Since f(c) is not equal to 0, we move to the next step.
Step 4 (repeated): Since f(c) = -3 and f(a) = -7, which have different signs, the new interval is [0, 1].
Step 2 (repeated): The midpoint is c = (0 + 1) / 2 = 0.5.
Step 3 (repeated): Evaluate f(0.5) = 0.5^5 + 3(0.5) - 7 = -2.734375. Since f(c) is not equal to 0, we move to the next step.
Step 4 (repeated): Since f(c) = -2.734375 and f(a) = -7, which have different signs, the new interval is [0.5, 1].
Step 2 (repeated): The midpoint is c = (0.5 + 1) / 2 = 0.75.
Step 3 (repeated): Evaluate f(0.75) = 0.75^5 + 3(0.75) - 7 = -1.83447265625. Since f(c) is not equal to 0, we move to the next step.
Step 4 (repeated): Since f(c) = -1.83447265625 and f(a) = -7, which have different signs, the new interval is [0.75, 1].
Step 2 (repeated): The midpoint is c = (0.75 + 1) / 2 = 0.875.
Step 3 (repeated): Evaluate f(0.875) = 0.875^5 + 3(0.875) - 7 = -0.8974838256835938. Since f(c) is not equal to 0, we move to the next step.
Step 4 (repeated): Since f(c) = -0.8974838256835938 and f(a) = -7, which have different signs, the new interval is [0.875, 1].
We continue with this process until we achieve the desired accuracy of two decimal places.