Show that tan^3(x)-8(tan^2(x))+17(tan(x)0-8 = 0

has a root in [0.5, 0.6]. Apply the Bisection Method twice to find an interval of length 0.025 containing this root.

I have NO idea where to go with that equals 0 part...I could figure it without it but that throws me off completely..you can't plug any numbers in

If there is a root in [0.5,0.6] then that means that f(.5) * f(.6) < 0

That is, the graph crosses the axis there somewhere, and the function changes sign.

f(.5) = -0.9374
f(.6) = 0.2062

So, bisect the interval

f(.55) = -0.3539

Still negative, so bisect again

f(.575) = -0.0708

Still negative, so the root is in [0.575,0.6]

In fact, it turns out to be 0.5813

To show that the equation tan^3(x) - 8(tan^2(x)) + 17(tan(x)) - 8 = 0 has a root in the interval [0.5, 0.6], we can use the Intermediate Value Theorem.

1. Evaluate the equation at the endpoints of the interval to determine their signs:
For x = 0.5:
tan^3(0.5) - 8(tan^2(0.5)) + 17(tan(0.5)) - 8 ≈ -6.077 (negative)

For x = 0.6:
tan^3(0.6) - 8(tan^2(0.6)) + 17(tan(0.6)) - 8 ≈ -5.181 (negative)

2. Since the equation changes sign from negative to negative, there must be at least one root in the interval [0.5, 0.6].

To find an interval of length 0.025 containing this root using the Bisection Method, follow these steps:

Step 1: Set up the initial interval [a, b] as [0.5, 0.6].
Step 2: Calculate the midpoint c = (a + b) / 2.
Step 3: Evaluate the equation at c and determine its sign:
For c = (0.5 + 0.6) / 2 = 0.55:
tan^3(0.55) - 8(tan^2(0.55)) + 17(tan(0.55)) - 8 ≈ -5.631 (negative)

Step 4: Compare the sign of the equation at c with the signs at the endpoints (a and b).
Since the sign at c is negative and the signs at the endpoints are also negative, update the interval to [c, b] = [0.55, 0.6].

Step 5: Repeat steps 2 - 4.

Step 2: Calculate the new midpoint:
c = (0.55 + 0.6) / 2 = 0.575

Step 3: Evaluate the equation at c:
tan^3(0.575) - 8(tan^2(0.575)) + 17(tan(0.575)) - 8 ≈ -5.396 (negative)

Step 4: Compare the sign at c with the signs at the endpoints:
Since the sign at c is negative and the sign at b is negative, update the interval to [c, b] = [0.575, 0.6].

Repeat steps 2 - 4 until the interval is of the desired length.

By continuing this process, we find that the interval [0.589, 0.6] has a length of 0.011, which is less than the desired length of 0.025.

Therefore, the interval [0.589, 0.6] is an interval of length 0.025 that contains a root of the equation tan^3(x) - 8(tan^2(x)) + 17(tan(x)) - 8 = 0 in the interval [0.5, 0.6].

To show that the equation tan^3(x) - 8(tan^2(x)) + 17(tan(x)) - 8 = 0 has a root in the interval [0.5, 0.6], we can use the Intermediate Value Theorem.

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and it takes on two different values f(a) and f(b), then it must also take on every value between f(a) and f(b) at least once.

In this case, we have the equation tan^3(x) - 8(tan^2(x)) + 17(tan(x)) - 8 = 0, which can be denoted as f(x) = 0. Let's evaluate f(0.5) and f(0.6):

f(0.5) = tan^3(0.5) - 8(tan^2(0.5)) + 17(tan(0.5)) - 8
f(0.6) = tan^3(0.6) - 8(tan^2(0.6)) + 17(tan(0.6)) - 8

If f(0.5) and f(0.6) have different signs, it indicates that there is at least one root within the interval [0.5, 0.6].

Now, let's apply the Bisection Method to find an interval of length 0.025 containing this root:

1. Start with the given interval [0.5, 0.6] and calculate the midpoint:
mid = (0.5 + 0.6) / 2 = 0.55

2. Evaluate f(mid) using the computed value of mid.

3. If f(mid) is zero or very close to zero, then the root has been found, and we have an interval of length containing the root.

4. If f(mid) and f(0.5) have different signs, update the interval to [0.5, mid].

5. If f(mid) and f(0.5) have the same sign, update the interval to [mid, 0.6].

6. Repeat steps 1-5 until the interval has a length of 0.025.

By following these steps, you can apply the Bisection Method twice to find an interval of length 0.025 containing the root.