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We will use the degree 4 Taylor polynomial for e at z = 0 to approxmate First enter in the degree 4 Taylor polynomial (hint: you don't need to calculate it with derivatives, you can use the Taylor series for e) P₁(z)= Now evaluate 0.27 P4 (2) dz to get the approximate value for the original integral, giving your answer accurate to 4 decimal places 数字 Note that the answer will most likely give a different answer to a calculator for the original integral, but it won't be too far off (the calculator is also using an approximation like this, but a better one). Let's try a bigger value. Use the same method to approximate 1.27 -2² dz 数字 You should find now that it is is quite different to a calculator value. It turns out that this approximation starts to significantly diverge from the function above about = 0.6 and then goes badly wrong above z= 1. You can see this for yourself if you plot them together on a graph. To get an approximation that works for larger z you need more terms from the Taylor Series. If you include enough terms you can get a good approximation of any z1

We will use the degree 4 Taylor polynomial for e at z = 0 to approxmate First enter in the degree 4 Taylor polynomial (hint: you don't need to calculate it with derivatives, you can use the Taylor series for e) P₁(z)= Now evaluate 0.27 P4 (2) dz to get the approximate value for the original integral, giving your answer accurate to 4 decimal places 数字 Note that the answer will most likely give a different answer to a calculator for the original integral, but it won't be too far off (the calculator is also using an approximation like this, but a better one). Let's try a bigger value. Use the same method to approximate 1.27 -2² dz 数字 You should find now that it is is quite different to a calculator value. It turns out that this approximation starts to significantly diverge from the function above about = 0.6 and then goes badly wrong above z= 1. You can see this for yourself if you plot them together on a graph. To get an approximation that works for larger z you need more terms from the Taylor Series. If you include enough terms you can get a good approximation of any z1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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We will use the degree 4 Taylor polynomial for e 0.276 e-² da First enter in the degree 4 Taylor polynomial (hint: you don't need to calculate it with derivatives, you can use the Taylor series for e) at z = 0 to approxmate Now evaluate 0.27 P₁(x) da to get the approximate value for the original integral, giving your answer accurate to 4 decimal places 数字 Note that the answer will most likely give a different answer to a calculator for the original integral, but it won't be too far off (the calculator is also using an approximation like this, but a better one). Let's try a bigger value. Use the same method to approximate 1.27 -2² dz |数字 You should find now that it is is quite different to a calculator value. It turns out that this approximation starts to significantly diverge from the function above about = 0.6 and then goes badly wrong above z=1. You can see this for yourself if you plot them together on a graph. To get an approximation that works for larger a you need more terms from the Taylor Series. If you include enough terms you can get a good approximation of any a
Transcribed Image Text:We will use the degree 4 Taylor polynomial for e 0.276 e-² da First enter in the degree 4 Taylor polynomial (hint: you don't need to calculate it with derivatives, you can use the Taylor series for e) at z = 0 to approxmate Now evaluate 0.27 P₁(x) da to get the approximate value for the original integral, giving your answer accurate to 4 decimal places 数字 Note that the answer will most likely give a different answer to a calculator for the original integral, but it won't be too far off (the calculator is also using an approximation like this, but a better one). Let's try a bigger value. Use the same method to approximate 1.27 -2² dz |数字 You should find now that it is is quite different to a calculator value. It turns out that this approximation starts to significantly diverge from the function above about = 0.6 and then goes badly wrong above z=1. You can see this for yourself if you plot them together on a graph. To get an approximation that works for larger a you need more terms from the Taylor Series. If you include enough terms you can get a good approximation of any a
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