7-24-2005
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
sweeps out
9-28-2014
Newtons Method
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.
10-13-2010
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
1-18-2006
Inverse Trig Functions
If you restrict f(x) = sin x to the interval
x , the function increases:
2
2
y = sin x
-p/2
p/2
This implies that the function is one-to-one, and hence it has an inverse. The inverse is called the
inverse sine or arcsine
1-9-2014
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
7-22-2013
Miscellaneous Substitutions
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
7-21-2005
LHpitals Rule
o
LHpitals Rule is a method for computing a limit of the form
o
lim
xc
f(x)
.
g(x)
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
8-5-2013
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 .
n=0
The number c is called the expansion point.
A power series may represent a function f (x), in the sense that wherever the se
7-24-2005
Improper Integrals
b
Roughly speaking, an integral
f(x) dx is improper if:
a
1. One of the limits is innite.
2. The integrand blows up somewhere on the interval of integration.
For example,
e3x dx and
0
x2
x
dx
+9
are improper because they have
9-19-2014
Implicit Dierentiation
Example. The Folium of Descrates is given by the equation x3 + y 3 = 3xy. Picture:
2
1
-3
-1
-2
1
2
-1
-2
-3
The graph consists of all points (x, y) which satisfy the equation. For example, (0, 0) is on the graph,
because
1242013
Integration by Parts
If u and v are functions of x, the Product Rule says that
d(uv)
dv
du
=u
+v .
dx
dx
dx
Integrate both sides:
d(uv)
dx =
dx
uv =
u
dv
dx +
dx
v
u dv +
v du,
u dv = uv
du
dx,
dx
v du.
This is the integration by parts formula. T
10-9-2005
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)
8-2-2005
Alternating Series
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
11-2-2005
Denite Integrals
The area under a curve can be approximated by adding up the areas of rectangles.
1
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
7-7-2013
Parametric Equations of Curves
A pair of equations
x = f (t),
y = g(t),
atb
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
5-24-2007
The Natural Logarithm
The Power Rule says
xn dx =
1
xn+1 + C
n+1
provided that n = 1. The formula does not apply to
1
dx.
x
An antiderivative F (x) of
1
would have to satisfy
x
d
1
F (x) = .
dx
x
But the Fundamental Theorem implies that if x > 0
1-23-2006
Trigonometric Integrals
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
7-27-2005
Work
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
10-31-2005
Substitution
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
du
du
for dx. To do this, co
7-18-2005
Trig Substitution
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
If
8-8-2005
Constructing Taylor Series
The Taylor series for f(x) at x = c is
f(c) + f (c)(x c) +
f (c)
f (n) (c)
f (c)
(x c)2 +
(x c)3 + =
(x c)n .
2!
3!
n!
n=0
(By convention, f (0) = f.) When c = 0, the series is called a Maclaurin series.
You can constru
10-31-2005
Summation Notation
Summation notation is used to denote a sum of terms. Usually, the terms follow a pattern or formula.
n
f(k)
is shorthand for f(0) + f(1) + + f(n).
k=1
In this case, 1 and n are the limits of summation and k is the summation v
7-27-2005
Sequences
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
8-2-2005
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
i
9-22-2008
Related Rates
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
4-12-2013
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
7-7-2013
Polar Coordinates
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
7-19-2005
Partial Fractions
Partial fractions is the opposite of adding fractions over a common denominator. It applies to integrals
of the form
P (x)
dx, whereP (x)
and
Q(x) are polynomials.
Q(x)
P (x)
into a sum of smaller terms which are easier to inte