Volumes of Revolution by Slicing
Start with an area a planar region which you can imagine as a piece of cardboard. The cardboard
is attached by one edge to a stick (the axis of revolution). As you spin the stick, the area revolves and
The work required to raise a weight of P pounds a distance of y feet is P y foot-pounds. (In m-k-s
units, one would say that a force of k newtons exerted over a distance of y feet does k y newton-meters, or
joules, of work.)
Example. If a 1
Vector spaces and linear transformations are the primary objects of study in linear algebra. A
vector space (which Ill dene below) consists of two sets: A set of objects called vectors and a eld (the
Denition. A vector spa
Unitary Matrices and Hermitian Matrices
Recall that the conjugate of a complex number a + bi is a bi. The conjugate of a + bi is denoted
a + bi or (a + bi) .
In this section, Ill use ( ) for complex conjugation of numbers of matrices. I want to
You can use substitution to convert a complicated integral into a simpler one. In these problems, Ill
let u equal some convenient x-stu say u = f(x). To complete the substitution, I must also substitute
for dx. To do this, co
Trig substitution reduces certain integrals to integrals of trig functions. The idea is to match the
given integral against one of the following trig identities:
1 (sin )2 = (cos )2
1 + (tan )2 = (sec )2
(sec )2 1 = (tan )2
For trig integrals involving powers of sines and cosines, there are two important cases:
1. The integral contains an odd power of sine or cosine.
2. The integral contains only even powers of sines and cosines.
I will look
A coordinate transformation of the plane is a function f : R2 R2 . Ill usually assume that f has
continuous partial derivatives, and that f is essentially one-to-one in the region of interest. (A function is
Limits and Derivatives of Trig Functions
If you graph y = sin x and y = x, you see that the graphs become almost indistinguishable near x = 0:
That is, as x 0, x sin x. This approximation is often used in appl
Linear Systems with Constant Coecients
Here is a system of n dierential equations in n unknowns:
x = a11 x1 + + a1n xn ,
x = a21 x1 + + a2n xn ,
xn = an1 x1 + + ann xn .
This is a constant coecient linear homogeneous system. Thus, the
Constructing Taylor Series
The Taylor series for f(x) at x = c is
f(c) + f (c)(x c) +
f (n) (c)
(x c)2 +
(x c)3 + =
(x c)n .
(By convention, f (0) = f.) When c = 0, the series is called a Maclaurin series.
You can constru
Tangent Lines and Rates of Change
Given a function y = f (x), how do you nd the slope of the tangent line to the graph at the point
P (a, f (a)?
(Im thinking of the tangent line as a line that just skims the graph at (a, f (a), without going thr
Review: Convergence Tests for Innite Series
When you are testing a series for convergence or divergence, its helpful to run through your list of
convergence tests if you dont see what to do immediately just as you might run through your list of
Summation notation is used to denote a sum of terms. Usually, the terms follow a pattern or formula.
is shorthand for f(0) + f(1) + + f(n).
In this case, 1 and n are the limits of summation and k is the summation v
The Spectral Theorem
Theorem. (Schur) If A is an n n matrix, then there is a unitary matrix U such that U AU 1 is upper
triangular. (Recall that a matrix is upper triangular is the entries below the main diagonal are 0.)
Proof. Use induction on
An innite sequence is a list of numbers. The following examples should make the idea clear.
Example. Here is a familiar sequence:
1, 2, 4, 8, 16, . . . , 2n , . . .
Sequences are often written using subscript notation. This one might b
Row reduction (or Gauss-Jordan elimination) is the process of using row operations to reduce a
matrix to row reduced echelon form. This procedure is used to solve systems of linear equations, invert
matrices, compute determinants,
The Remainder Term and Error Estimation
If the Taylor series for a function f (x) is truncated at the nth term, what is the dierence between f (x)
and the value given by the nth Taylor polynomial? That is, what is the error involved in using the
Related rates problems deal with situations in which several things are changing at rates which are
related. The way in which the rates are related often arises from geometry, for example.
Example. The radius of a circle increases
Polar coordinates are an alternative to rectangular coordinates for referring to points in the plane.
A point in the plane has polar coordinates (r, ). r is (roughly) the distance from the origin to the point;
is the angle betw
The Mean Value Theorem
A secant line is a line drawn through two points on a curve.
The Mean Value Theorem relates the slope of a secant line to the slope of a tangent line.
The Mean Value Theorem. If f is continuous on a x b and dierentiable on
The Ratio Test and the Root Test
It is a fact that an increasing sequence of real numbers that is bounded above must converge.
The picture shows an increasing sequence that is bounded above by some number M . It seems reasonable
that the terms
Partial fractions is the opposite of adding fractions over a common denominator. It applies to integrals
of the form
dx, whereP (x)
Q(x) are polynomials.
into a sum of smaller terms which are easier to inte
Parametric Equations of Curves
A pair of equations
x = f (t),
y = g(t),
are parametric equations for a curve. You graph the curve by plugging values of t into x and y, then
plotting the points as usual.
Example. The parametric equations
x = c
The Natural Logarithm
The Power Rule says
xn dx =
xn+1 + C
provided that n = 1. The formula does not apply to
An antiderivative F (x) of
would have to satisfy
F (x) = .
But the Fundamental Theorem implies that if x > 0
Row Space, Column Space, and Null Space
Denition. Let A be an m n matrix.
(a) The row vectors of A are the vectors in F n corresponding to the rows of A. The row space of A
is the subspace of F n spanned by the row vectors of A.
(b) The column ve
When an integral contains a quadratic expression ax2 + bx + c, you can sometimes simplify the integrand
by completing the square. This eliminates the middle term of the quadratic; the resulting integral can
then be co
Absolute Maxima and Minima
Ill begin with a couple of examples to illustrate the kinds of problems I want to solve.
Example. A string 6 light years in length is cut into two pieces. One piece is used to make a circle, while
the other piece is u