Tests and 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
5. Factorial Exercises & Factorial Solutions
9. Integral Practice Spring 07
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