Math 222 Lecture Notes -- Fall 1999

These are VERY brief lecture notes that quickly recap what we actually did in the lecture.

Lecture 1 (09/02/99) 7.1 Integration by parts
We calculated the integral from 0 to x of 2 s cos s² (i.e. the antiderivative that vanishes at 0). The answer was sin x² . Check the answer. Sketch the integrand. Show why the integral is always between -1 and 1 eventhough the integrand becomes larger and larger as s increases. (Hint: show that the distance between two consecutive zeroes of the integrand, s_n and s_n+1 say, decreases like n^-1/2 while the size of the integrand in that interval grows like n^1/2, as n increases).

Lecture 2 (09/07/99) 7.2 Some Trig integrals
As I mentioned in class, Sect 7.2 is a bit of overkill. Don't study the formulas by heart!! especially not those involving tangents and secants (what's a sec anyway?) on page 425. Examples 8 and 9 are preposterous, what human could come up with that substitution?! A: One who found the answer another way then used "reverse engineering"!
The main idea of 7.2 is: if you have an odd power of sine or cosine think SUBSTITUTION + PYTHAGORAS and you can reduce the trigonometric integral to a simpler algebraic integral.
If all the powers are even, think HALF-ANGLE IDENTITY (or double angle: cos 2 x = cos² x -sin² x =2 cos² x - 1=1-2 sin² x).
The tangent and sec stuff is similar in spirit if you remember (tan x)' = 1/cos ²x (and of course there is also a (cot x)' = -1/sin ²x), so if you have Integral of STUFF/(cos ²x) dx, think w=tan x SUBSTITUTION and PYTHAGORAS which tells you that 1/cos² x = (cos² x + sin² x)/cos² x = 1+ tan² x, so you may be able to rewrite everything in terms of tan x and end up with an algebraic integral for w = tan x.
Example 10 and exercises 59-62 are important. That stuff comes up in approximation of functions, signal processing, heat transfer, .... (Fourier is Mister Heat transfer, he thought of "Fourier Series" to solve the Heat equation).

Lecture 3 (09/09/99) 7.4 Partial Fractions
we covered 7.4 integration of rational functions, starting from Example 8 and 9 in Stewart 7.2. We did it the way a human does it, first with a substitution ( w= sin x) because the power of cosine is odd (see 7.2) to end up with the integral of a rational function. We do not know complex variables, unfortunately, so instead of TWO cases of (proper) rational functions, we have to deal with FOUR cases (see 7.4). We gave up after the first 3 cases: (1) distinct linear factors (i.e. distinct roots), (2) repeated linear factors (i.e repeated roots), (3) distinct quadratic factors (i.e. distinct COMPLEX roots). We will not struggle with (4) repeated quadratic factors (i.e. repeated complex roots).

Lecture 4 (09/14/99) 7.3 Trig substitutions
We "digested" the basic formula for the antiderivatives of 1/sqrt(1-x ²) and 1/(1+x²), arcsin x and arctan x respectively, which suggest the substitutions

Lecture 5 (09/16/99) Maple excursion (7.7 + some of 7.8 and 7.9).

Lecture 6 (09/21/99) Maple illustration of 7.8 . Also finish 7.9. We talked about improper integral, i.e. integrals for which one (or both) limit is infinity or integrals where the integrand blows up (singularity) in the domain of integration. In both cases we make sense of these integrals by using limits. In lecture however, it was the electric power that blew up and things became rather obscure... So read the red boxes (1) and (3) in Sect 7.9. If you understand the basic idea (using limits), those definitions will seem very natural, it's not a matter of studying those big red boxes by heart. The key integral is int of x^-|p|. We'll use it as the main integral to compare to when deciding whether an improper integral makes sense or not (i.e. "converges" or "diverges").

Lecture 7 (09/23/99) Finished up 7.9 and started 8.2: arclength.
Looked at several examples for 7.9. Key skill is to be able to decide whether an improper integral converges without necessarily calculating it through the Fundamental theorem. For arclength, key ideas: Pythagoras + approximate curve by collection of straight segments + take limit i.e. integral!. Arclength formula not trivial but very intuitive. You must understand where that formula comes from. If you do, it will take you 3 secs to rederive it from Pythagoras. We'll see an even quicker and intuitive approach through the "infinitesimal" Pythagoras:

Lecture 8 (09/28/99) Diff Eq linear first order y'=ky (population growth, interest compounding, radioactive decay,...), 2nd order linear a y'' + b y' + cy =0 (e.g. Newton's law F=ma, harmonic oscillator y''= - k² y, LRC electric circuits,...) , 2nd order nonlinear (pendulum y''= - sin y .)

Lecture 9 (09/30/99) (15.5) linear constant coefficients -> exponential solution. Complex numbers, complex exponentials -> Euler's formula.

Lecture 10 (10/05/99) More morsels from 15.5 and 15.7. Damped harmonic oscillator [15.7 Eq. (3)] as canonical example for 2nd order, linear, constant coefficient Diff Eq. Discuss c² < 4 k m and c² = 4 k m cases, with numerical examples.

Lecture 11 (10/07/99) 15.2, linear, first order Diff Eqs including non-constant coefficients and non-homogeneous (i.e. forcing terms): --> use an integrating factor . Discussed both indefinite integral and definite integral formulation. Also qualitative description and understanding of solutions, terminal velocity.

Lecture 12 (10/12/99) 15.1 Population growth with competition dP/dt = a P - b P² (a.k.a. Logistic equation). Qualitative understanding. Solution by separation. Separable equations. Homogeneous equations (i.e. scale-invariant equations, do not change if replace x and y by K x and K y respectively, for any K not zero.

Lecture 13 (10/12/99) 15.1 continued. Homogeneous (i.e. scale-invariant) equation (one example). General first order Diff Eq : y'= f(x,y) with y(a) = b (initial condition). Direction fields (quickly) and Euler's method (graphically and algebraically). Test of Euler for the simple y'= k y . Prove convergence in the limit of step size h going to zero (i.e. number of steps n going to infinity) for that particular case. Discussed how to pick the step size h . For simple example must have k h small compared to 1. Example of what can happen if k h not small enough ( y'=-1000 y with h =0.1 gives wild growing oscillations while exact solution decays exponentially). For general y'= f(x,y) , this generalizes to |df/dy| h small compared to 1 (Note that for the simple f(x,y) = ky, indeed df/dy =k ).

Chapter 15 summary: you are supposed to be able to solve the following classes of ODEs:

Lecture 14 (10/19/99) 9.1 Parametric representation of a circle, a straight line and the cycloid.

Lecture 15 (10/21/99) 9.2, 9.3 Tangent, area, arclength. Tangent, area, arclength of the cycloid. TH2 problem 6 -> conservation of energy. The Tautochrone [see p. 530, Fig 8] property of the cycloid (a factor of 2 was missing in my rapid fire substitutions at the end of class, the final answer for the time to go from the original position parametrized by theta_0 to the bottom of the cycloid, parametrized by theta = Pi is T = Pi (R/g)^(1/2), independent of theta_0 !!).

Lecture 16 (10/26/99) 9.4 Polar coordinates. Slope. Radial line, circle, algebraic spiral r = theta (called "spiral of Archimedes"), equiangular spiral r=exp(theta) , cardioid r = 1 - sin (theta) . Area in polar coordinates (Sect. 9.5). (see also Famous curves (collection of Java applets) ).

Lecture 17 (10/28/99) 9.5 Area and arclength in polar coords, application to the cardioid. 9.6, brief reminder of what conics are: ellipse, parabola and hyperbola together with their geometric properties.

Lecture 18 (11/02/99) 10. Introduction to Sequences and Series. Local polynomial approximation to a function (see chap 2.9). Sine and cosine near x=0 . Solution of diff eq by series y'=y with y(0)=1 . TH2 #1 by series approximation. Geometric series. Partial sum. Sequence. Precise definition of the limit of a sequence.

Lecture 19 (11/04/99) 10.1, 10.2, 10.3 Monotonic sequence, bounded sequence. Geometric series, convergence. Algebraic series. Theorem 6 in 10.2. Comparison to integrals (Riemann sums, see Figs 1 and 2 in 10.3).

Lecture 20 (11/9/99) (10.2, 10.3) Geometric series, algebraic series, rate of convergence. Comparison to integral. Approximation of algebraic sums by integrals. Fundamental theorem of Calculus, telescoping sums. S_n=1 +2 + 3 + ...+ n= n(n+1)/2. Sum of squares, sum of cubes, ... Approximation of geometric series by an integral, when is it a good approximation?

Lecture 21 (11/11/99) We surfed through 10.4, 10.5, 10.6, 10.7, 10.8, 10.9 and 10.10 today!!! 10.4 Comparison of series to decide on convergence or divergence (e.g. comparison to algebraic or geometric series). We are basically skipping 10.5, alternating series, however you should know what an alternating series is and why it typically converges better than its non-alternating counterpart (see example in class). 10.6 Absolute vs. conditional convergence. 10.6 the RATIO TEST (i.e. comparison to a geometric series). Local polynomial approximation of a function --> power series (2.9 and 10.8). TAYLOR SERIES (10.10), Taylor series for the exponential, convergence for all x by the ratio test.

Lecture 22 (11/16/99) 10.8-10.10. Taylor Series (TS) for e^x, sin x , cos x, TS for 1/(1-x). Variations on the geometric series ==> TS for 1/(1+x), 1/(1+x²), 1/(1+x⁴), ... . Derivative and integral of series expansions ==> TS for ln (1+x), 1/(1-x)², ... . Concept of Radius of convergence R. Radius of convergence for e^x, sin x , cos x, 1/(1-x), 1/(1+x), 1/(1+x²), 1/(1+x⁴), ... . TS about x=0 for 1/(1-x) has R = 1 but TS about x=a for the same 1/(1-x) has R = |1-a| , picture. Applications of Taylor Series. 1/0.998= 1.002004008016032064128256513.....

Lecture 23 (11/18/99) Finish up chapter 10. Radius of convergence = distance to nearest singularity. Singularity may be complex. Examples (continued from previous lecture). Taylor's formula (extension of mean value theorem), applications. Binomial formula.

Lecture 24 (11/23/99) Begin chapter 11. Vectors from a geometric point of view, addition and subtraction of vectors. Decomposition into components. examples: plane in wind, forces, weight on an inclined plane. Frame of reference, Cartesian coordinates, right-handed vs. left-handed frame. Coordinates of a point, distance between two points. Equation of a sphere. Components of a vector, addition, subtraction, multiplication by a scalar.

Lecture 25 (11/30/99) Review problems. Problem 1 p. 573, dog chasing a rabbit.

Lecture 26 (12/02/99) Back to Chap. 11. Unit vectors. Dot product, projection, orthogonality. Geometric and algebraic points of view. Dot product is invariant with respect to changes in the frame of reference. Longitude and Latitude.

Lecture 27 (12/07/99) Longitude, latitude (spherical coordinates). Distance between two points on a sphere. 11.4, Physical motivation: Torque => Cross product of two vectors, geometric point of view. Basic properties. Cross-product in component form. Cyclic symmetry.

Lecture 28 (12/09/99) 3 ways to compute cross-product in component form. Determinants. Area of parallelogram. Mixed product, volume of parallelepiped, cyclic symmetry of mixed product; Calculation of mixed product (3-by-3 determinant). Equations of lines and planes: parametric and Cartesian.

Lecture 29 (12/14/99) Various Review problems, 11.5 #23, #51, 11.1 #26. 10.2 # 31, quick reasoning: when n is big, sin(1/n) ~ 1/n , and sin(1/n) - sin(1/(n+1)) ~ 1/(n(n+1)) . So because of cancellation, the sum of 2 consecutive terms will decrease like 1/n² which is plenty fast enough for convergence. [Rigorous treatment: sin(1/n) - sin(1/(n+1)) = 2 cos ((2n+1)/(2n(n+1))) sin(1/(2n(n+1))) < 1/(n(n+1)) because sin a - sin b = 2 cos((a+b)/2) sin ((a-b)/2) and sin x < x for x > 0 .]