Name:_____________________ Improper Integrals Done Properly



Math 2414 Activity 18 (Due by August 14)

1. Using first and second Taylor polynomials with remainder, show that for [pic],

[pic].

2. Using a second Taylor polynomial with remainder, find the best constant C so that for [pic],

[pic].

3. The nth Derivative Test : Suppose that f has n continuous derivatives and

[pic], but [pic].

Case I: If n is even and [pic], then from Taylor’s Theorem, we know that [pic] for some z between x and c. Or equivalently, [pic]. Since [pic] is continuous and [pic], for x close to c, [pic]. This means that[pic], or [pic] for x close to c. So f has a local minimum at c.

a) Investigate Case II: If n is even and [pic].

b) Investigate Case III: If n is odd and[pic].

4. Test [pic] as a local extremum for the following functions:

a) [pic] b) [pic]

c) [pic] d) [pic]

5. Let [pic]

a) Find the second Maclaurin polynomial for [pic].

b) Find the fourth Maclaurin polynomial for [pic].

c) Find the fourth Taylor polynomial centered at [pic] for [pic].

6. Explain why the polynomial [pic] cannot be the fourth Maclaurin polynomial for the function graphed:

7. Let [pic] be the second Maclaurin polynomial generated by the function f graphed below. Determine the signs of [pic], [pic], and [pic].

8. Given the graph of the differentiable function f, which has a minimum at [pic] and inflection point at [pic], determine the signs of the coefficients in the following Taylor polynomials for f:

a) [pic]

b) [pic]

c) [pic]

d) [pic]

9. How accurate are the following Maclaurin polynomial approximations if [pic]?

a) [pic] b) [pic]

c) [pic] d) [pic]

10. For [pic], the nth Maclaurin polynomial is

[pic].

And the remainder is [pic], where [pic]. So for a fixed value of [pic], [pic], but [pic], so [pic].

a) Perform the ratio test on the series [pic].

b) What does the result of part a) tell you about [pic] for any value of x?

c) What does the result of part b) tell you about [pic]?

11. Use the first Maclaurin polynomial with remainder for [pic], with [pic] and [pic] to get an inequality between [pic] and [pic].

12. Find the fourth Taylor polynomial centered at 2 for the function [pic], and show that it represents f exactly.

13. Find the third Taylor polynomial centered at 1 for the function [pic], and show that it represents f exactly.

14. The fourth Maclaurin polynomial for [pic], [pic], is really a third degree polynomial since the coefficient of [pic] is zero. So [pic].

a) Show that if [pic], then [pic].

b) Approximate [pic] with [pic], and give an upper bound on the error.

15. Find Maclaurin series for the following functions:

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

16. Express the following antiderivatives as infinite series:

a) [pic] b) [pic] c) [pic]

17. Use series to approximate the following definite integrals to within [pic] of their exact values:

a) [pic] b) [pic]

18. Use series to evaluate the following limits:

a) [pic] b) [pic]

c) [pic] d) [pic]

e) [pic] f) [pic]

19. Use multiplication, division , or a trig identity to find at least the first three nonzero terms in the Maclaurin series of the following functions:

a) [pic] b) [pic]

c) [pic] d) [pic]

20. Use Maclaurin series to find the sum of the following series:

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic]

f) [pic] g) [pic]

h) [pic] i) [pic]

21. Find the following derivatives of the given function at [pic]:

a) [pic], [pic] b) [pic], [pic]

c) [pic], [pic] d) [pic], [pic]

e) [pic], [pic] f) [pic], [pic]

22. Given the two Maclaurin series:

[pic] and [pic], find an equation relating [pic] and [pic] for [pic].

{Hint: Partial fractions.}

23. What is the coefficient of [pic] in the Maclaurin series for [pic]?

24. Consider the improper integral [pic]. For [pic] large, [pic], and [pic]. So the improper integral converges.

a) Make the substitution [pic] to convert the integral into a different improper integral.

b) Use the series [pic] and the fact that [pic] to find the value of the improper integral.

25. Consider the power series [pic], where [pic] and [pic] for [pic].

a) Find the first four terms and the general term of the series.

b) What function is represented by this power series?

c) Find the exact value of [pic]

26. Suppose that f has derivatives of all orders for all numbers and that [pic], [pic], [pic], and [pic].

a) Find the third Maclaurin Polynomial for f, and use it to approximate [pic].

b) Find the fourth Maclaurin Polynomial for the function g if [pic].

27. Show that the series

[pic]

converges to 0 if [pic], and converges to 1 if [pic].

{Hint: Use Mathematical Induction to show that [pic] for [pic].}

Everyone is familiar with the Quadratic Formula(Babylon, circa 1600 B. C.). For [pic] with [pic], the solution is given by [pic].

Most people are not familiar with the Cubic Formula(Italy, circa 1500 A.D.). For [pic], a solution is given by [pic]. For a general cubic, [pic] with [pic], divide through by a to get [pic]. Now substitute [pic] to get [pic], which expands into [pic]. The Cubic Formula can be applied to the last equation, and subtracting [pic] gets you back to a solution of the original equation.

There is also a Quartic Formula, but it was proven that there is no Quintic Formula or any higher power formula like the previous ones that express the solution using roots of the coefficients. Using substitutions, it was shown that the general quintic could be transformed into [pic]. The German mathematician Gotthold Eisenstein(1823-1852) at the age of 15 found a power series solution of the reduced quintic.

28. Determine the interval of convergence of Eisenstein’s Quintic Power Series Solution: [pic].

29. Find all functions which can be represented by a power series centered at 0 which solve the functional equation [pic], and determine their interval of convergence.

{Hint: If [pic], then the functional equation leads to

[pic]. Match the corresponding coefficients, and solve for them.}

30. [pic] and

[pic]. Use the previous series to find Maclaurin series for [pic] and [pic]. Use these series to find the exact value of [pic].

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