Math 142 Fall 2006 course log

MATH 142- FALL 2006- COURSE LOG

W 8/23 Syllabus, course policies
Antiderivatives: definition, linearity and linear substitutions,
derivatives of arc tan(x) and arc sin (x)
HW: p.332 3, 8, 21, 29, 47

F 8/25 Antiderivatives and mechanics: 3 examples
Area (intro)- sum of first n integers, first n squares
HW: p. 332 43, 54; p.352 1, 2, 17

M 8/28 Binomial theorem, Pascal's triangle
Area under a `parabola of degree p' (derivation)
HW- p. A49: 33, 41c, 45, 49

T 8/29 Problem session: 4.9 54, 56; 5.1 1; AppF 49

W 8/30 Left- and right-endpoint Riemann sums- example of exponentials
(upper/lower estimates for area); sum of n terms of a geometric sequence
Midpoint Rule, general Riemann sums and Riemann's theorem
HW: sect 5.2 - 5a,b, 7, 9, 11

F 9/1 Definition of definite integral/ properties: linearity and comparison
(use in estimating integrals). Indefinite integral = antiderivative.
(statement, examples) Mechanics: signed area under v(t)=net displacement
HW: sect 5.1 15, sect 5.2 35, 38, 49, 50 sect 5.3 11, 13, 19

M 9/4 Labor Day (no classes)

T 9/5 Quiz 1

W 9/6 2 examples involving areas (cp. 5.2-35,38)/ average value over an interval/
Fundamental thm of calculus- outline of proof based on the idea:
difference quotient of F=average value of f over a shrinking interval
examples of integration by substitution (rescaling the independent variable)
HW: (5.4) 3,9,11,15 (6.4) 1,2,5,12,13

F 9/8 definition of natural logarithm as the integral of 1/x: (i) L(x)=the integral satisfies
L(ab)=L(a)+L(b) (using substitution), hence is the log to some base; (ii) expansion
of L(1+1/b), integrated as an `infinite polynomial' (geometric series); (iii) use of
  e=lim (1+1/n)^n to show L(e)=1 (hence L(x)=log to base e).
HW: (5.5) 9, 12, 39, 40

Quiz 2 on 9/12 will be based on the HW from sections 5.4, 5.5, 6.4 (listed above)

M 9/11 integration by substitution
HW (5.5): 11,19,32, 33, 34, 45, 47, 48, 53, 54, 63

T 9/12 Quiz 2

W 9/13 integration by parts
HW(5.6) 7, 13, 17, 19, 23, 37

F 9/15 trigonometric integrals; reduction to rational functions by t=tan(x/2) substitution
rational functions: decomposition into partial fractions (start)
HW(5.7) 1, 3, 5, 6, 15, 17, 18, 19, 20 ; read: appendix G

Exam 1 on Tuesday 9/19: up to lecture on 9/15. The test will include problems (at the level of
the HW, or slightly harder) and theory questions (based on the lecture; see the entries in this
`course log' for the main points).

M 9/18 Discussion of HW problems (based on students' questions)

T 9/19 EXAM 1 (with answers)

W 9/20 partial fractions and integration of rational functions (end)
HW (5.7) 15, 16, 24/ App. G: 5, 6, 29, 31, 33

F 9/22 standard substitutions (involving square roots of quadratics); also: 1/(x^2+a^2)^n
Exam 1 returned
HW (5.7) 13, 14, 29, 30, 32
Correction to the original syllabus: the deadline to drop with a W is 10/3.

M 9/25 discussion of HW problems (integration, section 5.7)

T 9/26 discussion of problems in the integration worksheet
problem 3- complete solution due this Friday (optional)
     OPTIONAL READING: elliptic integrals- historical introduction

W 9/27 arc length for parametrized curves and graphs
HW (6.3) 5,7,14, 17, 23 App (H2) 36, 38
The quiz on Tuesday will be on integration (based on the HW from 5.7 and worksheet)

F 9/29 area of a surface of revolution (ex: paraboloid); arc length in polar coordinates
HW (app H2) 36, 37

M 10/2 arc length is independent of parametrization/ arc length function, use as a parameter
area enclosed by a parametrized simple closed curve
HW (6.1) 31,32,35,38 (app H2) 5, 7, 19

T 10/3 Quiz 3

W 10/4 area- examples (inc. polar coordinates)

F 10/6 conic sections: geometric definition (as a locus in the plane), equations in polar
and cartesian coordinates. Reflection property of the parabola: proof using mechanics
(parametrize by arc length, differentiate the geometric definition.)
HW: Review the material on conics in App. B, read the `discovery project' on p.A73
and do problems 5,8,9
PDF handout: Conic sections
(final version)

M 10/9 ellipses: polar and cartesian equations, geometric parameters, string and reflection properties

T 10/10 Kepler's laws; velocity and acceleration in polar coordinates. Newton's derivation
from an inverse-square law for gravity: Kepler's second law (=conservation of angular momentum),
the orbit equation (reduction to u''+u=const., u=1/r).
Exercises 1-4 in the online handout "conic sections" are to be turned in as HW on 10/17 (= 1 quiz).

W 10/11 Derivation of Kepler's first and third laws./Area of an ellipse/ Examples: earth/moon system
(estimate of distance), earth/sun system (Bode's mnemonic for distances to the sun in AU).
Newton's derivation of Kepler's laws
QUIZ 4 on 10/17: based on HW problems from Stewart listed above, from 9/27 to 10/6.

F 10/13 Fall break (no classes)

M 10/16 HW problems (arc length, area, conic sections)/ volume (start)
HW (6.2) 1, 7, 11, 31, 35

T 10/17 Quiz 4/ solution of quiz 4/ handout HW collected

W  10/18 Volume- examples (truncated cone, general cones, spherical segment- also surface area)
  center of mass- start (finite set of points)
   HW (6.2) 39, (6.5) 33

F 10/20 Centroid : formulas for a region bounded by two graphs, by a parametrized curve, or
by a closed curve given in polar coordinates. Symmetry principle /decomposition principle
HW (6.5): 37,38,39,40

Exam 2 on Tuesday 10/24 will consist of six problems: 1) arc length 2) area 3) volume or surface area
4) centroid 5) conics 6) Newton's derivation of Kepler's laws . (1) (2)(4) - includes regions bounded by graphs,
parametrized curves. or a curve given in polar coordinates. (3) surface area-surfaces of revolution only (5) review
online handout (6) review class notes and the online summary given above (class of 10/11)

M 10/23 Discussion of homework problems/ Pappus' theorem relating volumes and centroids.
Quiz 4 and handout HW returned

T 10/24 EXAM 2

W 10/25 Sequences: irrationality of sqrt(2)/ Newton's method/ bounded monotone sequences have limits/
rational approximations of sqrt(2) via Newton's method: proof that the limit exists
HW (4.8) 7,11,21,22
(although Newton's method is really part of Calc I, it is important- so the above problems
will give you a chance to review the topic, or learn it for the first time.)
HW( 8.1) 41 to 44, 46, 47, 48

F 10/27 limits of sequences: examples/ Squeeze principle, equivalence/ two limits defining e: proof
that both limits exist. HW (8.1) 11, 15, 17, 21, 23, 26, 27, 28

M 10/30 irrationality of sqrt(prime) [Coop] /two limits defining e (end)/
irrationality of e/ series with positive terms: geometric series, comparison,
divergence criterion, decimal representation of real numbers
HW(8.2) 5, 7, 13, 15, 35, 37

T 10/31 Problem session- problems on sequences and series, solution of exam 2 (start)

W 11/1 solution of exam 2 (end) harmonic series- divergence proof, existence of the
Euler-Mascheroni constant. Arithmetic-geometric means (start)

F 11/3 EXAM 2B

M 11/6 Arithmetic-geometric means (end)/ Improper integrals at a point (start)
Discussion of HW problems

T 11/7 Quiz 5- HW from 4.8, 8.1, 8.2
solution of quiz/ solution of exam 2B

W 11/8 Improper integrals at a point: p-th power, comparison, `limit comparison'
HW (5.10) 25, 27, 29, 30, 45
Improper integrals at infinity (start)- exponential examples, comparison

F 11/10    Improper integrals at infinity (end)pth power, comparison, examples
HW(5.10) 5, 13, 17, 19, 41, 43

M 11/13    series with positive terms and improper integrals/remainder estimate/
discussion of HW problems
HW(8.3) 15, 17, 21, 24, 25, 31, 33, 40

T 11/14 Quiz 6- HW from 5.10
Examples from Stewart: (8.3) 38, 39, Cantor middle-thirds set

W 11/15 Convergence tests: alternating series, absolute vs. conditional convergence,
(Riemann's rearrangment theorem), ratio test
HW(8.4) 5, 13, 17, 21, 25, 27

Exam 3 on Tuesday 11/21 will include the lectures from 10/25 through 11/17, including
`power series' (section 8.5 in Stewart), to be discussed on Friday 11/17.
main topics: convergence of sequences and series, improper integrals, power seires

F 11/17 power series: radius of convergence, open interval of conv./
   representation of rational functions by power series at 0: examples
HW (8.5) 7, 9, 11, 20, 25, 26
(remark: don't worry about determining convergence at the endpoints of
the interval of convergence)

M 11/19 review problems (driven by student questions)

T 11/20 EXAM 3

W 11/21 differentiation and integration of power series/applications
HW(8.6) 13 to 16 , 23, 24, 25, 27, 37

F 11/23 Thanksgiving break (no classes)

M 11/26 Taylor polynomials/ Taylor approximation with remainder estimate
   HW(8.9) 11, 13, 15, 17, 21, 23

T 11/27 Problem session/ solution of Exam 3 (returned)

W 11/28 Taylor series/ basic examples: exp, sin, cos, ln(1+x), arctan/
concept of analytic function/ example of differentiable function not
represented by its Taylor series/De Moivre's formula for exp(ix).
HW (8.7) 3, 6, 7, 11, 12, 23, 25, 26, 28, 29

F 12/1 Taylor series: examples / binomial series
(8.7) 33, 35, 39, 43, 49, 51
(8.8) 5, 9, 10, 13

M 12/4 Review session (problems from Stewart)

T 12/5 Euler's solution of Bernouilli's `Basel problem' via infinite products/
problems from Stewart/ course evaluations

FINAL EXAM: Friday 12/8, 10:15-12:15 (in the usual classroom)

The final will consist of 8 problems: 4 taken from Exams 1, 2, 2B or 3 (possibly
with some numbers changed), and 4 chosen from the HW problems listed above
for sections 8.6, 8.7, 8.8, 8.9 in Stewart (again, possibly with numbers changed).

Remember the final is worth 40% of the course average. This means that, regardless
of what your current estimated grade is, it could still change substantially- either way.
Monday/Tuesday will be review sessions, and I'll be available to answer questions
until Thursday.

FINAL GRADE DISTRIBUTION

A=5, B+=2, B=3, C+=1, C=3, D=2, F=2
(18 students took the final)