[Technology Exercise] Roots

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Question

[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

c. It can be shown that the exact value of the solution in part (b) is

(1/2 + √69/18)¹/³ + (1/2 ― √69/18)¹/³

Evaluate this exact answer and compare it with the value you found in part (b).

Accepted Answer

Welcome back, everyone. Consider the graph of the function F X equals 13 of X cubed minus 3 X + 4 as shown in the graph given. Approximate the root of the given function using Newton's method, rounding the answer to 6 decimal places. A says it's negative 3.452553, B 3.458333. C 3.522523, and D-3.522333. Now, are we going to approximate the root of this function using Newton's method? What do we know about Newton's method? But recall The Newton's method. Basically tells us OK. That The next approximation XN + 1 equals the current approximation XN minus F of XN, the function whose root is being approximated, divided by the derivative of F of XN. Once the derivative of F of XN is not equal to zero. No, in our case then. If we can have an initial guess for our root, which is gonna, we can call X not, and we can also find the derivative of our function F of X, then we can, we can apply Newton's method until the XN values converge to a root. So let's go ahead and do that. Now, from our graph, this can help us to get our initial guess. Notice that it crosses the it crosses the X axis, sorry, at a point where X looks to be approximately 3.5, OK? So we can let our initial guess X not be equal to 3.5. My apologies. That would be -3.5 here, OK? No, OK, this is approximately -3.5. If we differentiate F of X, OK. Then the derivative is going to be equal to X2 minus 3. So now that we have X0 and the derivative, then we can start to approximate with X1 and then continue until, like we said, the XN values converge to a root. So let's do that down here, OK? So no, OK. This would imply. That X1 is going to be equal to X minus F of X0 divided by the derivative of F of X0. So now that's gonna be -3.5. Minus F of X knot, so that's minus 3 of 3.5 cubed. -3 multiplied by -3.5. Plus 4. So anywhere we see X in F of X, we put -3.5 divided by the derivative of F of X where X equals -3.5. So that's gonna be -3.5 squared minus 3. And now we go ahead when we go and calculate, we should get X1 to be approximately equal to -3.522523. Let's use that to find the value X2, because X2 will not be equal to X1 minus F of X1 divided by the derivative of F of X1. Now, we're just gonna repeat the process. Wherever we see X, we're gonna put X1, OK? And when we do that, we should get X2 to be approximately equal to -3.522333, OK? 3.522333. No, we can go ahead and solve for X3. It's gonna be X2 minus F of X2 divided by the derivative of F of X2. And again, when we substitute negative 3.522333, then we get X3 to be approximately equal to -3.522333. So what is this telling us? Well, notice now that the 2nd and 3rd iterations show that the values of XN converge to negative 3.522333. Therefore, the root is approximately equal to this value which is going to be equal to answer choice D. Thanks a lot for watching everyone. I hope this video helped.

[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

c. It can be shown that the exact value of the solution in part (b) is

(1/2 + √69/18)¹/³ + (1/2 ― √69/18)¹/³

Evaluate this exact answer and compare it with the value you found in part (b).

Key Concepts

Roots of a Polynomial

Cubic Functions

Exact vs. Approximate Values

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