if you have Mathematica; 20 if you must write your own code.] Calculate the approximate values f

If you have Mathematica; 20 if you must write your own...

Key Concepts

Numerical Analysis

This is the study of algorithms that use numerical approximation for problems where an exact solution is difficult or impossible to obtain. It includes methods for approximating integrals, solving equations, and simulating systems, and often involves iterative techniques that converge on a solution within a desired level of accuracy.

Iterative Algorithms

Iterative algorithms repeatedly perform a set of computations to gradually approach a final result. Each iteration refines the approximation based on the previous value, and the process continues until changes between iterations become negligibly small as determined by a convergence criterion.

Error Tolerance

Error tolerance, often designated by a symbol like ?, is a predetermined threshold that defines an acceptable margin of error in numerical computations. It guides iterative algorithms by specifying when the approximation is sufficiently close to the true value, thereby stopping further iterations.

Computational Implementation

This concept involves using software tools or writing custom code to implement numerical algorithms. Such implementations allow for efficient calculation of approximate solutions and provide flexibility to adapt algorithms to specific problems or error tolerances.

Transcript

00:02 Okay, so we want to use the idea of differentials to estimate the value of this expression, 4th root of 624.

00:13 So since we have a 4th root, so our starting point will be, we'll start by taking the function to be 4th root of x, which is actually x to the part of 1 4th.

00:29 And also we will find the derivative of this function, f -dash of x, using the power rule.

00:38 So we have one -fourth, x to the power of one -fourth minus one.

00:45 And this gives us one -fourth, x to the part of negative 3 over 4.

00:55 Now, by the idea of differentials, we have this, we know this following relation, that f of a plus d x is approximately equal to f of a plus f dash evaluated at the point a multiplied by d.

01:24 So we want to find a such that a should be a number as close as possible.

01:30 To the number in question 624 so that a has a perfect 4th root so we see that in this case our choice of a is very nicely it is 625 because if you calculate f of a which is f of 625 this 4th root of 6 to 5, this turns out to be 5 by very straight calculation in the calculator.

02:11 And also we need this left side expression inside the function is value.

02:19 A plus d x, we need that to be 624, the number of those 4th root we are trying to estimate.

02:31 And since a is already 625, so my obvious value of d of x, turns out to be negative 1.

02:42 So now taking all these things together in this given expression, so we have f of 625 plus d of x which is negative 1 approximately equal to f of 625 plus f dash of a.

03:10 So if that we have found out on the top 1 4th now this has to be evaluated at a...

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