Use Newton’s Method to approximate 3^(√7) to four decimal places.

Use x1 = 2 as your seed. Round off intermediate iterates to five decimal place

let f(x) = x^3 - 7

so f(∛7) = 0

See
http://keisan.casio.com/exec/system/1244946907

just enter your function, its derivative, and initial guess.

the answer I should get is 18.2955

let x = 3^√7
then x^(1/√7) = 3
x^(1/√7) - 3 = 0

let y = x^(1/√7) - 3 = x^.3779645 - 3
dy/dx = (1/√7)^(1/√7 - 1) = .3779645 x^-.62204

iteration expression:

x - (x^.3779645 - 3)/(.3779645x^-.62204)

x = 2 -----> 15.06815
x = 15.06815 --->18.10237..
x = 18.10237.. ---> 18.294858..
x = 18.29456 ----> 8.2955 , which was my calculator answer.

I find this a very strange question.
There is no practical way to do the above calculations without using a scientific calculator.
So why not find that scientific calculator to find the answer in the first place ?

Good job, Reiny. I keep seeing 3^√ being used as cube root, so maybe I was way off.

In any case, the problem is solved.

To approximate 3^(√7) using Newton's Method, we can start with an initial guess, also known as a seed, and then iterate until we reach a desired level of accuracy. Here's how you can do it:

Step 1: Determine the function and its derivative
The function we want to approximate is f(x) = 3^(√7). To apply Newton's Method, we also need to find the derivative of this function. Let's call the derivative as f'(x).

Step 2: Set up the iteration formula
Newton's Method uses the following iteration formula to refine the initial guess:
x(i+1) = x(i) - (f(x(i)) / f'(x(i)))

Step 3: Choose an initial guess or seed
In this case, the given initial guess or seed is x1 = 2.

Step 4: Calculate the intermediate iterates
Using the iteration formula, we can calculate the intermediate iterates to refine our approximation. We will round off these values to five decimal places as instructed.

Iterate 1: x(1) = x1 = 2
Iterate 2: x(2) = x(1) - (f(x(1)) / f'(x(1)))

To find f(x(1)), substitute x(1) = 2 into the function f(x):
f(x(1)) = 3^(√7) = 3^(√7)

To find f'(x(1)), substitute x(1) = 2 into the derivative f'(x):
f'(x(1)) = df(x) / dx = derivative of 3^(√7)

Then use the formula:
x(2) = x(1) - (f(x(1)) / f'(x(1)))

Iterate 3: x(3) = x(2) - (f(x(2)) / f'(x(2)))

Repeat this iteration until you reach the desired level of accuracy or until the difference between two consecutive iterates is small enough.

After several iterations, you should obtain an approximation for 3^(√7) accurate to four decimal places.