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Topics to be covered and recommended practice problems.

In the list of problems 1-30 means that there are 30 similar problems in the book, and that you should pick at least two of these, and more if you feel you need more practice.

METHODS OF INTEGRATION

We learn a few of the many existing tricks which help you find the antiderivative F(x) of a function f(x). The main methods are:
§7.1 Integration by parts, and reduction formulas.
Problems: 1-30, 37,38, 41, 42, 43, 57, 58, 60
§7.3 Trigonometric substitution. Certain types of integrals can be made easier by substituting u=sin(x), or something similar. (Review the method of substitution if you have forgotten this)
Problems: 1, 2, 3, 7, 16 , 27 , 35
§7.4 Partial Fractions. You can find the antiderivative of every function of the form P(x)/Q(x) where P(x) and Q(x) are polynomials. The recipe for finding the antiderivative is what this section is about. The method turns out to be useful in other situations as well (e.g. finding Taylor-Maclaurin series(this course) or Laplace transforms(math 319)).
Problems: 1-16, 17, 19, 22, 24, 45


IMPROPER INTEGRALS

§7.9 The integral of a function f(x) which is not bounded, or over an interval which is infinitely long. You must know and be able to use the definition, as well the comparison test which somtimes allows you to decide on convergence of an improper integral without actually having to compute it.
Problems: 1, 2, 3, 4, 7, 9, 10, 11, 13, 27-31, 32, 60abc, 66

1st ORDER DIFFERENTIAL EQUATIONS

Differential Equations occur everywhere in Science and Engineering. We go through some methods for solving a few of the many different kinds of equations which are out and about.
Read §8.1 for your information, as an introduction. This material is repeated in more detail in §15.1, where we begin.
§15.1 Separable and Homogeneous equations are two kinds of diffeqs which you can solve by integration. Direction Fields provided a graphical interpretation of what you’re really doing when you “solve a diffeq.”
Problems: 1-4, 5-8, 11-14, 15, 16, 21, 22
§15.2 First order linear equations. Remember HOW you get the solution rather than memorizing than formula (we will ask for the derivation with the integrating factor on the exam!)
Problems: 15-20, 28, 29, 33.

INFINITE SERIES, POWER SERIES, THE TAYLOR MACLAURIN FORMULA

Power series are like polynomials, except they have infinitely many terms. The Taylor-Maclaurin formula shows that many functions can be represented by such a “power series.” Because of this, and the fact that certain calculations (notably integration and differentiation) are easier with power series than with ordinary functions makes it worth while to study power series. The logical order in which we will do this is first to study infinite series (how do you add infinitely many numbers?), then
§10.2 Series (aka “sums with infinitely many terms.”) Definition of convergence of a series, and the sum of a series.
Problems: 1, 2, 3, 7, 9, 13, 19, 21, 39, 43
§10.6 The ratio and root tests for convergence of a series.
Problems: 1-12 (only apply the ratio/root tests; your answer may be “undecided” if that’s what the ratio/root tests give you) 27, 28
§10.8 Power series (definition, radius of convergence).
Problems: 3, 7, 9, 13, 15, 17, 28, 29, 31
§10.9 Representing functions as Power Series. Adding, multiplying, (long)division, differentiation and integration of power series.
Problems: 1-8, 9, 11, 12, 17(hint ln(a/b)=ln a - ln b), 21, 25, 36
§10.10 Taylor-Maclaurin series. The formula, and the formula for the remainder term.
Problems: 1, 3, 4, 7, 14, 17, 20, 22, 25, 41, 42
§10.11 Newton’s Binomial formula. The “Pascal triangle formula” for (1+x)ⁿ also works when n is not an integer: Newton figured out how to make sense of this.
Problems: 1, 3, 7, 13.

PARAMETRIC EQUATIONS

§9.1 Parametrized curves. We study the motion of a point in the plane, and the curve traced out by a point as it moves through the plane. Learn how to find the formula for the position (x(t), y(t)) of a point at time t from a verbal description of its motion.
Problems: 1, 2, 3, 7, 9, 18, 19, 20, 21, 30, 31, 35 (take a=1/2 if you wish)
§9.2 Tangents and areas.
Problems: 1-4, 15-18, 25, 27, 30, 31, 33
§9.3 Arclength of a parametrized curve.
Problems: 1, 2, 5, 6
§9.4 Polar coordinates. A different way of keeping track of where you are.
Problems: 1-6, 13-16, 17-22, 25-30, 37, 39, 45, 48, 63-68, 76, 77
§9.5 Areas and arclengths of curves given in polar coordinates.
Problems: 7-12, 13, 14, 15-2043-48, 49.


THREE DIMENSIONAL GEOMETRY AND VECTORS

§11.1 Coordinates in three dimensional space.
Problems: 1-4, 5-8, 9-10, 21, 27, 28-43
§11.2 Vectors.
Problems: 1-6, 7-10, 11-18, 19-24, 25(make a drawing), 27, 29, 42
§11.3 The “dot product” of two vectors.
Problems: 51, 1-8, 10, 11-16, 17, 19-24, 30, 37-42, 55, 56, 57
§11.4 The cross product and 3 by 3 determinants.
Problems: 39, 1-7, 8, 11, 44, 20, 21-24, 27, 29, 33, 35
§11.5 Equations of lines and planes.
Problems: 5-10, 11, 12, 13, 14, 15, 16, 17, 23-26, 27-30, 31-34, 35-38, 39, 40, 55, 57, 58, 61, 63, 72.


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