Use the Bisection Method to find the following values to the nearest tenth. The negative root of

step 1 negative numbers that create an interval where the root is. So I'm going to start with negative

Use the Bisection Method to find the following values to the...

Key Concepts

Bisection Method

The Bisection Method is a numerical technique for finding roots of continuous functions by repeatedly narrowing an interval in which the function changes sign. At each step, the midpoint of the current interval is computed, and the interval is then halved based on the sign change, ensuring that the root remains within the new subinterval.

Intermediate Value Theorem

The Intermediate Value Theorem guarantees that if a continuous function takes on values of opposite sign at the endpoints of an interval, then there is at least one root within that interval. This theorem underpins the Bisection Method by ensuring that a sign change implies the existence of a root within the interval.

Root Finding

Root finding refers to the numerical problem of locating the value(s) at which a function equals zero. Techniques such as the Bisection Method provide systematic ways to isolate and approximate these values, especially when analytical solutions are difficult or impossible to obtain.

Termination Criteria

Termination criteria in numerical algorithms, such as achieving a specified accuracy or error tolerance, dictate when the iterative process can be stopped. In the context of the Bisection Method, stopping once the interval is sufficiently small ensures that the approximated root is within an acceptable range, such as being rounded to the nearest tenth.

Transcript

00:01 Negative numbers that create an interval where the root is.

00:03 So i'm going to start with negative 3 and negative 2.

00:09 So we're going to start by plugging those numbers in.

00:13 So f of negative 3, it's negative 3 squared minus 5, which is 4.

00:20 And then we'll plug in negative 2, get negative 2 squared minus 5, which is negative 1.

00:28 Now this is the correct interval since one of them is negative and one of them is positive.

00:34 So using the bisection method, we're going to find the midpoint between negative 3 and negative 2, which is negative 2 .5.

00:45 Now we have two separate intervals from negative 3 to negative 2 .5 and from negative 2 .5 to negative 2 .5 to negative 2 .2.

00:54 So now we need to figure out within there which interval has the root.

00:59 So we'll find again the function for negative 2 .5, which is going to be 1 .25.

01:16 So since that one's positive, we want to find the other one where it's negative.

01:21 So the root will be between negative 2 and negative 2 .5.

01:27 So again, we'll take the midpoint of negative 2 .5 and negative 2, which is negative 2 .25.

01:42 And then we'll create our two intervals again, so negative 2 .5 to negative 2 .25 and then to negative 2 .25.

01:53 And then to negative 2...

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