Find an interval of length

0.1 in [1, 2] containing a root of the equation. (Enter your answer using interval notation.)

x7 + 7x − 10 = 0

f(1) = -2

f(2) = 134

so there is indeed at least one root in [1,2]

f'(x) = 7x^6 + 7
f'(x) > 0 everywhere, so f(x) is strictly increasing.

Since f(2) is so much greater than f(1), I'd expect to find the root near 1

f(1) = -2
f(1.05) = -1.243
f(1.10) = -0.352
f(1.15) = +0.710

so, there is a root in [1.05,1.15]

Well, let's start searching for a root in the given interval, [1, 2].

First, let's check if the equation is satisfied at the endpoints:
For x = 1: (1)^7 + 7(1) - 10 = 1 + 7 - 10 = -2
For x = 2: (2)^7 + 7(2) - 10 = 128 + 14 - 10 = 132

Since the equation evaluates to a negative value at x = 1 and a positive value at x = 2, we know that there is a root somewhere in between.

Now, let's narrow down the interval.

We'll try the midpoint: x = (1 + 2)/2 = 1.5
For x = 1.5: (1.5)^7 + 7(1.5) - 10 ≈ 1.5 + 10.5 - 10 ≈ 2

Great! The equation evaluates to a positive value at x = 1.5, which means our root is still within the interval.

So, we can confidently say that there is a root of the equation x^7 + 7x - 10 = 0 in the interval [1, 1.5].

Using interval notation: [1, 1.5]

To find an interval containing a root of the equation x^7 + 7x - 10 = 0, we will use the Intermediate Value Theorem.

Let's evaluate the equation at the endpoints of the interval [1, 2] to determine if there is a sign change.

f(1) = (1)^7 + 7(1) - 10 = -1
f(2) = (2)^7 + 7(2) - 10 = 114

Since there is a sign change from negative to positive between f(1) and f(2), by the Intermediate Value Theorem, there exists at least one root of the equation in the interval [1, 2].

To find a specific interval of length 0.1 containing the root, we can narrow down the interval by applying the Bisection Method.

Let's set the midpoint of the interval [1, 2] as our first approximation for the root:

midpoint = (1 + 2) / 2 = 1.5

Now, we evaluate the equation at the midpoint:

f(1.5) = (1.5)^7 + 7(1.5) - 10 ≈ -0.898

Since f(1.5) is negative, we know that the root is somewhere in the interval [1.5, 2].

Next, we calculate the midpoint of the new interval:

midpoint = (1.5 + 2) / 2 = 1.75

Evaluating the equation at the new midpoint:

f(1.75) = (1.75)^7 + 7(1.75) - 10 ≈ 0.633

Since f(1.75) is positive, we update our interval to be [1.5, 1.75].

We repeat the process one more time:

midpoint = (1.5 + 1.75) / 2 = 1.625

Evaluating the equation at the new midpoint:

f(1.625) = (1.625)^7 + 7(1.625) - 10 ≈ -0.136

Since f(1.625) is negative, we update our interval to be [1.625, 1.75].

Thus, an interval of length 0.1 in [1, 2] containing a root of the equation x^7 + 7x - 10 = 0 is [1.625, 1.75].

To find an interval of length 0.1 in [1, 2] containing a root of the equation x^7 + 7x - 10 = 0, we can use the Intermediate Value Theorem. According to the theorem, if a function is continuous on a closed interval [a, b], and if the function takes on values f(a) and f(b) with opposite signs, then there exists at least one root in the interval (a, b).

1. First, let's find the values of the function at the endpoints of the interval:
f(1) = 1^7 + 7(1) - 10 = -1
f(2) = 2^7 + 7(2) - 10 = 128 + 14 - 10 = 132

2. Since f(1) = -1 and f(2) = 132 have opposite signs, we know that there is at least one root in the interval (1, 2).

3. Now, we need to find a more specific interval of length 0.1 within (1, 2) that contains the root. We can use the "bisection method" to do this.

4. Start by dividing the interval (1, 2) into two equal halves: (1, 1.5) and (1.5, 2).

5. Evaluate the function at the midpoints of these two intervals:
f(1.5) = 1.5^7 + 7(1.5) - 10 ≈ -0.890625

6. Since f(1.5) = -0.890625 is negative, we know that the root lies in the interval (1.5, 2).

7. Now, repeat steps 4-6 with the interval (1.5, 2), dividing it into two equal halves again:
- Evaluate the function at the midpoint of the new interval.
- Determine in which subinterval the root lies (based on the sign of the function value).

8. Continue this process of dividing the interval in half and evaluating the function at the midpoints until you find an interval with a length of 0.1 that contains the root. This may require several iterations.

Using this method, you can find an interval of length 0.1 in [1, 2] containing a root of the equation x^7 + 7x - 10 = 0.