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
Newtons method is simple to describe pictorially. To nd a root of an equation f (x) = 0, start at a
point x0 . Go up to the curve. From there, slide down the tangent line till you hit the x-axis. Thats x1 .
Now repeat the process.
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
Inverse Trig Functions
If you restrict f(x) = sin x to the interval
x , the function increases:
y = sin x
This implies that the function is one-to-one, and hence it has an inverse. The inverse is called the
inverse sine or arcsine
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
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
LHpitals Rule is a method for computing a limit of the form
c can be a number, +, or . The conditions for applying it are:
1. The functions f and g are dierentiable in an open interval containing c. (c may al
Intervals of Convergence of Power Series
A power series is an innite series
a0 + a1 (x c) + a2 (x c)2 + =
an (x c)n .
The number c is called the expansion point.
A power series may represent a function f (x), in the sense that wherever the se
Roughly speaking, an integral
f(x) dx is improper if:
1. One of the limits is innite.
2. The integrand blows up somewhere on the interval of integration.
e3x dx and
are improper because they have
Example. The Folium of Descrates is given by the equation x3 + y 3 = 3xy. Picture:
The graph consists of all points (x, y) which satisfy the equation. For example, (0, 0) is on the graph,
Integration by Parts
If u and v are functions of x, the Product Rule says that
Integrate both sides:
u dv +
u dv = uv
This is the integration by parts formula. T
Increasing and Decreasing Functions
A function f increases on a interval if f (a) < f (b) whenever a < b and a and b are points in the
interval. This means that the graph goes up from left to right.
A function f decreases on a interval if f (a)
If a series has only positive terms, the partial sums get larger and larger. If they get large too rapidly,
the series will diverge.
However, if some of the terms are negative, the negative terms may cancel with the positive te
The area under a curve can be approximated by adding up the areas of rectangles.
from x = 0 to x = 1 using 20 equal subintervals and
1 + x3
evaluating the function at the left-hand endpoints.
Example. Approximate the area unde
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
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
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
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
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
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
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
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
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
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
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
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