Repost from https://mathforums.com/threads/verify-given-function.356475/#post-644773 by same guy as nycmath, https://mathhelpboards.com/members/rtcntc.8471/, https://mathhelpboards.com/members/harpazo.8631/, https://mathhelpboards.com/members/mathland.19617/.nycmathdad said:Verify the given function has a zero in the indicated interval. Then use the Intermediate Value Theorem to approximate the zero correct to three decimal places by repeatedly subdividing the interval containing the zero into 10 subintervals.
f (x) = x3 − 4x + 2; interval: (1, 2)
I don't understand the instructions. How is this done?
nycmathdad said:Verify the given function has a zero in the indicated interval. Then use the Intermediate Value Theorem to approximate the zero correct to three decimal places by repeatedly subdividing the interval containing the zero into 10 subintervals.
f (x) = x^3 − 4x + 2; interval: (1, 2)
I don't understand the instructions. How is this done?
Do you know what the "Intermediate Value Theorem" is?
f(1)= 1- 4+ 3= -1 and f(2)= 8- 8+ 2= 2. The Intermediate Value Theorem says that since this polynomial is continuous and positive at one end of the interval and negative at the other end there must be some point in the interval where f(x)= 0.
Now, divide (1, 2) into 10 subintervals:
(1, 1.1), (1.1, 1.2), (1.2, 1.3), (1.3, 1.4), (1.4, 1.5), (1.5, 1.6), (1.6, 1.7), (1.7, 1.8), (1.8, 1.9), and (1.9, 2.0).
Evaluate f(x) at the endpoints of those. If there is an interval where f(x) has different signs at the endpoints then, by the "Intermediate Value Theorem", f(x)= 0 somewhere in that interval. Divide that interval into 10 more subintervals and repeat. That will get you to things like 1.a0, 1.a1, 1,a2, etc, two decimal places. Repeat one more time to get three decimal places.
Prove It said:I have a feeling that when it says to divide into subintervals, they are meaning to Bisect it ten times to close in on the point...
nycmathdad said:Yes but I am beyond this topic in my studies.
Prove It said:Are you saying you've already done the Bisection Method? Good, then attempt it.
Not if you have someone else do it for you!nycmathdad said:I have done several questions involving the Bisection Method. It is very time consuming.
Country Boy said:Not if you have someone else do it for you!
Translation: I posted actual examples on other math sites and inveigled others to do it for me.nycmathdad said:I did a few myself. Struggled but actually did it.
The purpose of verifying the function is to ensure that it is working correctly and producing the expected results. This helps to identify any errors or bugs in the code and allows for necessary adjustments to be made.
To verify the function, you can run test cases with different input values and compare the output to the expected results. You can also use debugging tools or step through the code to see how it is executing.
If the function fails to verify, it means that there is an error in the code or the function is not producing the expected results. This could be due to incorrect logic, syntax errors, or missing components.
Verifying the function is important because it ensures the accuracy and reliability of the code. It helps to catch any mistakes or issues before the code is implemented in a larger project or system.
Yes, the function can and should be verified multiple times throughout the development process. This helps to catch any new errors or changes that may have been introduced and ensures that the function continues to work as intended.