Tests and Solutions

1. Test 1 Solutions

2. Test 2 Solutions

3. Test 3 - Test 3 Solutions

Final Exam Responsibilities

Self-Scheduled – May 11-15

1. Limit of a Sequence: Know how to use the limit “laws” and theorems in section 1, determine the limit of a sequence and when a sequence diverges – Know how to use the Monotone (Bounded) Sequence (Convergence) Theorem, p 709 – Practice problems: Ch 11 Rev TRUE-FALSE p 786: 1,3,11,16 p 787: 1,2,3,6,9; 11.2: 37

2. Types of Infinite Series: Know how to identify a geometric series, determine when a geometric series diverges, and use the formula for its sum when it converges, how to use the Divergence Theorem p 718, and how to identify and classify (according to convergence or divergence) a p-series (p 725) – Know how to use algebra to reduce an infinite series to infinite series involving geometric series – Practice Problems: Ch 11 Rev p 786-787: 1,12,27; 11.2: 16,22,27,35; 11.3: 10,29

3. Compare Series: Know how to apply the procedure if you know both the series for which you want to determine convergence and a “known” series with which to compare – know how to choose a series with which to compare a given series – Practice Problems: 11.4: 10,11,14,16,19 (Compare the series in problem 11 to the Harmonic series, and compare the series in problem 19 to a geometric series whose common ratio is 2/3.)

4. Alternating Series Test: Know how to identify an alternating series, how to determine whether the Alternating Series Test applies, and how to apply that test – know how to determine conditional convergence and absolute convergence and know how to estimate the sum of a convergent series to a desired degree of accuracy – Practice problems: 11.5: p 740: 11,13,14,27

5. Ratio and Root Tests: Know how to apply each of the two tests to given series of numbers and to power series – Practice Problems: 11.6: pp 745-746: 9,10,15,25

6. Direct Substitution Method of Integration: Know how to apply the procedure if you know the required substitution – Know how to make reasoned guesses for a substitution and apply the procedure for your substitution – Practice Problems: 5.5: p 420-421: 2,34,36,43,57,62

7. Integration by Parts Method of Integration: Know how to apply the procedure if you know the required identification of u and dv – Practice Problems: 7.1: p 480: 2,21,23,35

8. Folding 1/7 Sequence: Know how to apply the 3-step folding procedure and represent the result as a sequence of estimates for 1/7 starting with an arbitrary guess – Practice Problem: Start with an arbitrary estimate a[1] and show that the sequence of estimates generated by repeating the 3-step process on the result of each preceding 3-step process converges to 1/7. See the diagram for the process and Test 2 solutions.

9. Improper Integrals of Type I: know how to evaluate an improper integral of Type I as the limit of a definite integral – Practice Problems: 7.8: 20,24

10. Integral Test of Convergence of an Infinite Series: Know how to apply the Integral test and draw conclusions about the convergence of a related infinite series - Practice Problems: Ch 11 Rev p 787: 15

11. Center of Mass: Know how to evaluate moments and centers of mass by applying integrals and symmetry - Practice Problems: Ch 8 Rev p 583: 12

12. Probability Density Functions: Know how to use information about probability density and a mean to determine probabilities (and information about probability to determine the mean) - Practice Problems: Ch 8 Rev p 583: 21

13. Power Series: Know how todetermine the interval of convergence of a power series and how to use algebra, differentiation, and integration to determine power series representations for functions based on the representations of "well-known" functions (the functions listed at the bottom of page 767) - Practice Problems: 11.10 p 770: 26,27,28; 11.9 p 759: 9,14 (14a was done in class)

14. Taylor Series: Know how to generate the Taylor Series for a function using the Taylor formulas, pp 761-761; know how to evaluate an integral by replacing one representation of a function by its Taylor series representation - Practice Problems: Ch 11 Rev p 788: 5; 11.9 p 759: 27

15. Differential Equations: Be able to recognize a differential equation and to confirm that a proposed solution is valid, know how to apply the method of separation of variables to solve a differential equation, including ones with specified initial conditions, know how to apply the power series method for solving a differential equation - Practice problems: 9.3 p 607: 14,19; See examples 14-17 below for solutions of initial value problems by using power series.

Examples

1. Calculating the limit of a sequence whose nth term is in the form of a Rational Function

2. Divergence of the Harmonic Series

3. Balancing Cards (and boxes) Hints

4. A few Series Practice Problems

12. Probability and Waiting: Left Bank - Holden Call

17. Calculating the Taylor Series (MacLaurin Series) by using the Taylor Formulas

14. Solving an Initial Value Problem, y' + y = 2 cos(t); y(0) = 1, with Power Series

15. Solving Another Initial Value Problem, y'' + y = x^2 + 2; y(0)=0, y'(0)=1, with Power Series

16. Yet Another Power Series solution: Now for the IVP: (1+x^2)y'=xy; y(0)=1.

17. Final (!) Example of using a Power Series to Solve an IVP

Using the software Package Maple

1. Maple: Working with sequences

2. Maple: Working with infinite series

3. Maple: Finding integrals

4. Maple: Learning to use techniques of integration (Nov 2, 2006, class)

5. Maple: Power Series Radius of Convergence

6. Maple: Taylor Series & Taylor Polynoms.pdf

7. Maple: Fourier Series