Question

Number 9 requires number 8 so please can you answer both? Thanks. Here's more context:

There are also approximations of higher order derivatives that can be computed using only values of the original function. Consider the approximation: u(a + 2h)-2u(a + h) + u (a) h2 8. Using your knowledge of Taylor series, what derivative is approximated by Equa Many different combinations of terms can be used to create approximations to deriva- tion??? What is the order of the approximation? tives. 9. Create the highest order approximation to u'(a) that you can using the Taylor series and the expressions u(a), u(a + h), ua +3h) You can determine what the order of error is for a numerical approximation of this type by using Taylor series expansions. 1. Write down the 4th degree Taylor polynomial for u() expanded around the point a 2. Evaluate the 4th degree Taylor polynomial for u(x) at the point a +h 3. Use your results from 2) to calculate the order of the error in approximating u'(a) with the difference quotient a th-u(a) 4. Check your result from 3) with u(x)n(x and u'(x cos(x) by calculating the error of this numerical approximation with a 1 and h .1, h .01, h .001. h0001 and h00001. Compare the results that you have computed. Is your error getting smaller at a rate comparable to h" for the n that you found in 3)? Explain why your results support your computation in 3) It is possible to use different combinations of terms to get a numerical approximation to u' (a) that has a higher order of error. Consider the rule that 2h 5. Evaluate the Taylor polynomial for u(x) at the point a - h 6. Use your results from 1), 2) and 5) to calculate the order of the error in approximating u'(a) with the centered difference approximation u'(a) = u(a + h)-u(a-h) 2h 7. Check your result from 6) with a(z) = sin(x) and u,(x) = cos(z) by calculating the error with a 1 and h-1, h-.01, h-.001, h0001 andh00001. Does your answer make sense? Why or why not? There are also approximations of higher order derivatives that can be computed using only values of the original function. Consider the approximation u(a 2h) - 2u(a + h) + u(a) h2 8. Using your knowledge of Taylor series, what derivative is approximated by Equ Many different combinations of terms can be used to create approximations to deriva- tion ??? What is the order of the approximation? tives 9. Create the highest order approximation to u'(a) that you can using the Taylor series and the expressions u(a), u(a + h), u(a + 3h) 10. What is the order of error of your approximation from 9)? 11. Check your results from 10) with u(x)-sin(x) and u'()co( by calculating the error you find with a = 1, and h = .1, h = .01, h = .001, h = .0001 and h = .00001 Does your answer make sense? Why or why not?

Answer 1

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