Find interval length 1/4 in [1,2] containing the root of the equation x^7 +3x-10=0
To find an interval of length 1/4 in the interval [1,2], we need to look for two numbers such that their difference is 1/4.
Let's start by determining the values of the equation x^7 + 3x - 10 for x = 1 and x = 2:
For x = 1:
f(1) = 1^7 + 3(1) - 10 = 1 + 3 - 10 = -6
For x = 2:
f(2) = 2^7 + 3(2) - 10 = 128 + 6 - 10 = 124
Since the function changes sign between x = 1 and x = 2 (f(1) is negative and f(2) is positive), we can conclude that there is a root of the equation x^7 + 3x - 10 = 0 somewhere in the interval [1,2].
Now, let's try to narrow down this interval to find an interval of length 1/4 containing the root. We can use the bisection method:
1. Calculate the midpoint of the interval [1, 2] using the formula: (1 + 2) / 2 = 1.5
2. Evaluate the function at the midpoint: f(1.5) = (1.5)^7 + 3(1.5) - 10
3. If the function value is positive, the root must be in the interval [1, 1.5]. Otherwise, it must be in the interval [1.5, 2].
4. Repeat steps 1-3 with the new interval until the interval length is 1/4.
Let's apply this method:
Interval [1, 2]:
Midpoint: (1 + 2) / 2 = 1.5
Evaluate: f(1.5) = (1.5)^7 + 3(1.5) - 10 = 20.375
Since f(1.5) is positive, the root of the equation must be in the interval [1, 1.5].
Interval [1, 1.5]:
Midpoint: (1 + 1.5) / 2 = 1.25
Evaluate: f(1.25) = (1.25)^7 + 3(1.25) - 10 = -4.7
Since f(1.25) is negative, the root of the equation must be in the interval [1.25, 1.5].
Interval [1.25, 1.5]:
Midpoint: (1.25 + 1.5) / 2 = 1.375
Evaluate: f(1.375) = (1.375)^7 + 3(1.375) - 10 = 7.04297
Since f(1.375) is positive, the root of the equation must be in the interval [1.25, 1.375].
Interval [1.25, 1.375]:
Midpoint: (1.25 + 1.375) / 2 = 1.3125
Evaluate: f(1.3125) = (1.3125)^7 + 3(1.3125) - 10 = 1.58805
Since f(1.3125) is positive, the root of the equation must be in the interval [1.25, 1.3125].
Interval [1.25, 1.3125]:
Midpoint: (1.25 + 1.3125) / 2 = 1.28125
Evaluate: f(1.28125) = (1.28125)^7 + 3(1.28125) - 10 = -1.64298
Since f(1.28125) is negative, the root of the equation must be in the interval [1.28125, 1.3125].
The interval [1.28125, 1.3125] has a length of 1/32, which is smaller than 1/4. By applying the bisection method repeatedly, you can continue refining the interval until you reach your desired length of 1/4.