Approximating f(x)=e^x Refer to Problem 89 . (a) Approximate f(-2.4) with n=3. (b) Approximate f

Approximating f(x)=e^x Refer to Problem 89 . (a) Approximate...

Key Concepts

Taylor Series

The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This concept allows us to approximate complex functions with simpler polynomial expressions, which is particularly useful for both analytical and numerical calculations.

Maclaurin Series

A Maclaurin series is a special case of the Taylor series that is centered at zero. Many elementary functions, such as the exponential function, have well-known Maclaurin series that enable straightforward computation of approximations by summing a finite number of terms.

Error Analysis and Remainder Estimation

When using a finite number of terms to approximate an infinite series, it is important to estimate the approximation error. Techniques such as the Lagrange remainder provide a way to bound the error, ensuring that the approximation meets a desired level of accuracy, such as reaching a certain number of correct decimal places.

Numerical Precision in Series Approximations

Numerical precision in series approximations involves determining how many terms are required to achieve a given accuracy. This involves balancing computational effort with the desired error tolerance, often using numerical methods or trial and error to verify that the approximation is correct to the specified number of decimal places.

Use of Graphing Utilities and Numerical Methods

Graphing utilities and numerical methods can facilitate the process of checking and refining approximations. These tools support techniques like trial and error to visually inspect the convergence and error behavior of the series, thus aiding in determining the minimum number of terms needed for the desired accuracy.

Transcript

00:01 For this problem, we're asked to approximate what e to the negative 2 .4 is using this summation rule right here.

00:13 So the first one, we're asked to part a is asking us to approximate what f of negative 2 .4 is with n equals to 3.

00:28 So therefore, this is going to be equal to, so using this summation rule, so k equals zero, where n is 3, x is going to be negative 2 .4 raised k over k factorial.

00:44 So now i'm going to expand this summation.

00:47 So this is equal to negative 2 .4, k starts at 0 over 0 factorial, plus negative 2 .4.

00:58 To the first over 1st factorial plus negative 2 .4 to the second power over 2 factorial plus negative 2 .4 to the third factorial over 3 fact i mean negative 2 .4 to the third power over 3 factorial and if you use your calculator to simplify this you will get the sum of approximately negative point 824.

01:27 So now the next thing is asking us to approximate f of negative 2 .4 with n equals to 6.

01:40 So therefore my summation rule is going to be k equals to 0 to 6 of negative 2 .4 is my x to the k power over k factorial.

01:52 So basically i'm adding when k equals 0 and k equals 1, k equals 2, and k equals 3, which i have already done from part a.

02:02 So i'm going to separate this summation.

02:06 So this is equivalent to the sum of k to the 0 to 3, which we did from part a plus the summation of k equals to 4 to 6.

02:26 So we want to end at 6 at the end.

02:29 So negative 2 .4 to the k over k factorial.

02:33 So we already know what the answer for this one is.

02:38 It's negative point, you know, 8 to 4.

02:43 So now i just need to figure out what the sum of this part is, which is going to be right here.

02:49 So this sum is going to be negative 2 .4 where k starts at 4 over 4 factorial plus negative 2 .4 to the fifth power over 5 factorial plus negative 2 .4 to the 6 power over 6 factorial.

03:09 And if you use your calculator to simplify this, you will get approximately 0 .9842688.

03:21 So therefore, my final answer, i'm going to add negative, negative point.

03:29 A24 plus 0 .9 .82688.

03:38 So my final answer equals to for part b is 0 .160 -2688.

03:49 So and part c is asking us to approximate f of negative 4 using our calculator.

03:58 So f of negative 4, using our calculator.

03:58 So f of negative 2 .4.

04:01 So if we use our calculator, our calculator has the e.

04:04 So i can just type in what's e to negative 2 .4 is.

04:08 So the exact answer for this one should be, so find the e in your calculator, raise that to parentheses negative 2 .4, close of parentheses.

04:22 So the answer should be approximately 09071795.

04:29 Okay, so you can see that when we did end to the six, our previous answer is still not very close to the exact answer.

04:39 And that's what d is asking us to do.

04:42 Okay, d is asking us to figure out, you know, using our calculator in the sequence mode, figure out, you know, the value of n required to obtain up to eight correct decimal places.

05:00 So to do that, the first thing you're going to do is turn your calculator into a sequence mode.

05:09 So click on mode and go down to function and go over to sequence mode...

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