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Let f(x) = exp(-X?) (15 pts) Give the firsl ' three terms (i.e: Pz(x)) of the Taylor series for f (x) expanded about Xo-0. Explicitly write out the appropriate...

Question

Let f(x) = exp(-X?) (15 pts) Give the firsl ' three terms (i.e: Pz(x)) of the Taylor series for f (x) expanded about Xo-0. Explicitly write out the appropriate derivatives.(15 pts) Give the error term Ry(x) Explicitly write out the appropriate derivatives(15 pts) If x can be anywhere on the interval [-1,+1], find numerical upper bound on the magnitude of Ra(x):(15 pts) For x-0.3 , give an equation that could be solved for € the Greek letter in the formula for Ra(x): Do not solve the equat

Let f(x) = exp(-X?) (15 pts) Give the firsl ' three terms (i.e: Pz(x)) of the Taylor series for f (x) expanded about Xo-0. Explicitly write out the appropriate derivatives. (15 pts) Give the error term Ry(x) Explicitly write out the appropriate derivatives (15 pts) If x can be anywhere on the interval [-1,+1], find numerical upper bound on the magnitude of Ra(x): (15 pts) For x-0.3 , give an equation that could be solved for € the Greek letter in the formula for Ra(x): Do not solve the equation.



Answers

Compute the coefficients for the Taylor series for the following functions about the given point $a$, and then use the first four terms of the series to approximate the given number. $$f(x)=\sqrt[4]{x} \text { with } a=16 ; \text { approximate } \sqrt[4]{13}$$.

To start here. We're gonna take a few derivatives of the function one over the square root of X, which is X negative 1/2 power. If we insert a, which is four, into this function, we'll get one over the square root of four, which is 1/2 protect the first derivative of this function. Using the power rule, we're gonna get one over negative 1/2 X to the three halves power. You plug in four here, we're gonna get negative one over to times four to the three halves power which will just be negative one over to the fourth power. If we evaluate the second derivative, we're going to get using the power rule three over four times X to the five halves power. If we evaluate this, a four we're going to get three over two squared times four to the five halfs power 4 to 5 have powers to to the fifth power to the fifth power. Times two squared Mr to the seventh Power. Finally, the third derivative will be negative. 15 over eight X to the seven hands. If we insert in four into this expression, we're gonna get negative 15 over two cute times. Four to the seven halfs power for the seven house powers to to the seventh Power to to the seventh power times two Cubed is to to the 10th Power, and now we can take these four values. Plug him into the definition for the tailor. Siri's, which started with F of A, which we evaluated, was 1/2 minus one over to to the fourth times, X minus A, which is four plus three over to to the seventh times, one ever to factorial times X minus four. Squared minus 15 over to to the tent times 1/3 factorial times X minus for cubed and someone. And now we can take these this expression that we just got and try to evaluate what three to the negative 1/2 power is. And we do that by plugging in three for everywhere we see X in this approximation. So this will be approximately 1/2 minus three. Minus floor is negative. One negative. One times negative one is a positive one over to to the fourth plus three over two. Factorial is too. Two times two of the seventh is to to the eighth times negative one squared is a positive one, minus 15 over three factorial, which is six times two to the 10th Power times three minus four is negative, one cubed, which is a negative one, which makes this positive. Then, if you plug this into a calculator, you'll get approximately 0.5 77

To start here. We're gonna look at the first few derivatives of the cube root effects. We're gonna evaluated an equal 64. So if we start with the function Q. Bert of X is the same Miss X to the 1/3 power. We evaluate this at 64. We asked what Cube is 64 that's four. The first derivative of it. Ex, Using the power rule would then be won over three times X to the 2/3 power. Evaluating this at 64 will give 1/3 times the cube root of 64 squared, which will be or squared second derivative again using the power rule B negative too over nine x to the 5/3 this evaluated in 64. We'll then be negative, too, over nine times the cube root of 64 which is four to the fifth power and the third derivative will then be 10 over 27 x to the 8/3. This evaluated a 64 re 10 over 27 times the Cuba of 64 which is or to the eighth power. And now we can take these four terms and plug the minute the definition for the tailor Siri's that we have above to get that This is F of A with just four plus f prime of a, which to 1/3 times four squared times X minus a and a 64 plus second, Davide, which is negative two over nine times four to the fifth times, will not for two factorial times X minus 64 squared, plus the third Irvine today just 10 over 27 for two the eighth times 1/3 factorial times X minus 64 to the third power and so on. And we can simplify this a little bit to be four plus 1/3 times 16 times x minus 64 minus two. Factorial is to those two factors are gonna cancel out one over nine times forward to the fifth times X minus 64 All square plus 10 over 27 times for to the eighth times three Factorial, which is six x minus 64 cubed. And so I'm and these were the 1st 4 terms of our Taylor Siri's and every one of approximate the Cube root of 60. We just had to plug in 60 for everywhere we see X in the Taylor Siri's, which will then be four plus 1/3 times 16. I'm 16 minus 64 minus 1/9 times for the fifth. I'm 60 minus 64 squared, plus 10 over 27 times for the eighth time. Six time 16 1964 All Cube. And we know that 60 months 64 is negative for so we can write. This is four minus for over three times 16 minus foursquare to 16 divided by nine times for to the fifth. Power minus 10 times for huge all over 27 times for the eighth time. Six. And if you plug this into a calculator, you'll get that this is three 0.91 for nine.

We're going to start here by taking the first few derivatives of the square root of X. Evaluating it really equals 36. So we started with our function. The eagle to the square objects three. Evaluate this at 36. This will just be the square root of 36 which is your six. If you take the first derivative of spirit of X is extra the 1/2 power this be one over to spirit of X using the power rule. The first derivative evaluated 36 will therefore be won over two times the square root of 36 the 1/2 times six, which is 12. We take the second derivative again using the power rule. We're going to get negative one over four times X to the three halves power Plugging this in looking 36 into this mule debt Negative one over four times 36 to the three halves power which would be negative 1/4 time six Doomed. Finally, if we take the third derivative again using the power rule, we're going to get negative one over six times X to the five hands power plaguing 36 in the year we're going to get. We should be positive. One positive fraction, but one over six times 36 to the five halves power, which would just be 1/6 to the six power. And now we can use all of these terms to evaluate the 1st 4 terms in the Power or the Tailor Siri's, which starts with F of A, which was six plus f prime of A, which was 1/12 times X minus A where a is 36 plus the second derivative a day, which was negative. 1/4 times six cubed times, 1/2 factorial times X minus 36 squared class 1/6 to the sixth Power Tim's 1/3 sectorial times X minus 36 cubed and someone. Now we can simplify these terms to be six plus 1/12 tens X minus 36 minus two. Factorial is too. And if you bring it into the fraction we can read, this is 1/8 times six cube times X minus 36 squared plus three. Factorial is six. So if we multiply that with six to the sixth will get six to the seventh power Times X minus 36 Huge and someone and these are the 1st 4 terms in our Taylor Siri's, and now we can approximate with Square Root of 39 is by plugging in 39 for X in our 1st 4 terms. To get that spirit of 39 is approximately six plus 1/12 times 39 minus 36 minus 1/8 times six to you times 39 minus 36 squared, plus 1/6 to the seven times 39 minus 36 o cubed. Simplifying some of this will be six plus 39 minus 36 is three divided by 12 is 1/4 minus 39 minutes. 36 is again three squared nine over eight time six cubed plus three cubed is 27 over six to the seven. And if you plug this into a calculator, you're gonna get that This is equal to six point 24 for eight, which is approximately 6.245


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