# Calculus II

Material to remember

There is a certain amount of mathematics that you have to remember, and have at
you finger-

tips, in order to understand the lectures and be perform well on the exams. The
are many

techniques for remembering information, for example using flash cards or
association. Personally, I

think it is best to remember a few fundamental things, and be able to work out
the less fundamental

things from these. Also, the more cross-links you have, the better you remember.
If you forget

one bit of information but remember a cross-link, then you can generally recover
the missing

information. Students who have a poor understanding of the subject matter, and
have not sorted

the fundamental points from the less fundamental ones, attempt to remember
everything. This

technique frequently fails in stressful situations (i.e. exams), especially when
you have poorly

developed cross-links. Even if your short term memory is particularly good, and
you can get

through the exam with a good grade, then you will have troubles in the next
course that has the

previous course as a prerequisite (because then you need long term memory, and
that does not

work with huge amounts of poorly sorted information). Much of the information
below is from

precalculus. I encourage you to create your own summary (with cross-links) for
all your courses

in mathematics.

(1) cos^{2} θ + sin^{2} θ = 1

(2) 1 + tan^{2} θ = sec^{2} θ

(3) cot^{2} θ + 1 = csc^{2} θ

I like to define cos θ and sin θ by the rule that (cos θ,
sin θ) is the point on the circle x ^{2}+ y^{2}= 1

inclined at an angle of θ to the positive x-axis. Then (1) is merely Pythagoras’
Theorem. Equations

(2) and (3) follow from (1). Dividing both sides of (1) by cos^{2}θ gives (2), and
dividing both sides

of (1) by sin^{2}θ gives (3).

**Euler’s Formula**
= cos θ + i sin θ where i^{2}= −1.

**Addition Formulas** Expanding
and equating real and imaginary
parts gives

(4) cos(α + β) = cos α cos β − sin α sin β,

(5) sin(α + β) = sin α cos β + cos α sin β.

The formula for tan (α + β) is obtained by dividing (5) by (4) and then dividing
the numerator

and denominator by cos α cos β to get

(6)

I do not remember the formulas for cos(α−β), sin(α−β) and tan(α−β). These can be
deduced

from (4)–(6) by replacing β by −β and recalling that cos(−β) = cos β, sin(−β) =
−sin β and

tan(−β) = −tan β (this last equation follows, of course, from the other two ). We
tend not to use

addition formulas for sec, csc and cot.

**Double-Angle Formulas** Expanding
and equating real and imaginary parts of

gives

(7)
cos(2θ) = cos^{2}θ − sin^{2}θ = 2 cos^{2}θ − 1 = 1 − 2 sin^{2}θ,

(8) sin(2θ) = 2 sin θ cos θ.

The second and third expressions for cos(2θ) were obtained from the first by
using sin^{2} θ = 1−cos^{2} θ

and cos^{2} θ = 1 − sin^{2} θ. Alternatively, put α = β = θ into (4) and (5). Dividing
(8) by (7) and

then dividing numerator and denominator by cos^{2} θ gives

(9)

In Math 172.2 we shall use the half-angle formulas to convert a trigonometric
integral to one

involving a rational function of t:

where t = tan(θ/2).

Properties of a ^{x} and log_{a} x Let a be a positive base, a
≠ 1. Remember that log_{a}
y = x means

a^{x} = y. Remember the graphs of y = a^{x}, and the inverse function y = log_{a} x, and
how to change

base from a to e using the formulas
and log_{a} x = ln x/ ln a. Also,
(ab)^{x} = a^{x}b^{x}.

You need not remember (2) as it follows from (1) by replacing y by −y, and d by
d^{-1}.

**Derivatives** You must remember the product, quotient and chain rules. You should
be able to

deduce the quotient rule from the product and chain (or composite) rules. A
common mistake is

to use the wrong numerator: uv′ − u′v instead of vu′ − uv′.

You have to remember the derivatives of cos x and sin x. You should be able to
work out the

derivative of tan x using the quotient (or product) rule, and the derivatives of
sec x, csc x, cot x

by differentiating (cos x)^{-1}, (sin x)^{-1}, (tan x)^{-1} using the chain rule.

**Completing the square **This technique is used to writing a quadratic ax^{2} + bx + c
in the

form a(x − h)^{2} + k. It is used in this course to evaluate integrals, and is used
in precalculus to

find the vertex of a parabola , or the center and radius of a circle which is not
in standard form

(x − h)^{2} + (y − k)^{2} = r^{2}. Completing the square is done as follows:

For example, 2x^{2} − 12x + 17 = 2(x^{2} − 6x + 9) − 2 × 9 + 17 = 2(x − 3)^{2} − 1.

** Polynomial division **You should be able to divide one polynomial into another and
find the

quotient and remainder. For example, dividing 2x^{2} + 1 into 2x^{3} − 6x^{2} + 5x − 4
gives a quotient

of x − 3 and a remainder of 4x − 1. These four polynomials are connected via the
equation

2x^{3} − 6x^{2} + 5x − 4 = (x − 3)(2x^{2} + 1) + (4x − 1).

**Definitions/Theorems **You should remember every definition. The derivative is
defined to be:

Note that f′(c) is the slope of the tangent line to y = f(x) at x = c.

**DMVT **If f(x) is continuous on the closed interval [a, b] and differentiable on
the open interval

(a, b), then there exists a c in (a, b) such that

Arithmetic sums An arithmetic progression has first term a, and subsequent terms
are obtained

by adding a fixed term d. Thus a, a + d, a + 2d, a + 3d, . . . , a + (n − 1)d is
a typical arithmetic

progression. An arithmetic sum can always be computed by adding the sum forwards
and

backwards. For example, to evaluate 1 + 3 + 5 + · · · + 101:

(sum is independent of order of addition)

Therefore, S = 2550. In general, an arithmetic sum is the half the number of
terms times the sum

of the first and last terms, e.g.

**Formulas/Theorems** You should remember the formulas for: the volume of a solid of
revolution

(using the disc and shell methods ), the length of a curve, the surface area of a
surface of revolution,

and the centre of mass of a rod with density ρ(x):

(disc method; about x-axis)

(shell method; about y-axis)

(length)

(surface area)

(center of mass = moment/mass)

You should be able to state the major theorems of the course. Stating a theorem
is more than

writing down a formula, you must specify the hypotheses and the conclusion
clearly using words.

For example,

FTC1 Let y = f(t) be continuous on the interval [a, b] and let x be an element
of [a, b]. The

derivative of the area function
or equivalently

FTC2 If f(x) is continuous on [a, b] and F′(x) = f(x), then

IMVT If f(x) is continuous on [a, b], then there exists a c in [a, b] such that

Integrals Of course, equations (15)–(18) can be written backwards as integral
formulas, so you

need not remember these. You should remember

(n
≠ −1),

(compare Eqn (15)),

(compare Eqn (15)),

**Techniques** Four important techniques for antidifferentiation are (1) the
substitution method (the

chain rule backwards), (2) integration by parts (the product rule backwards),
(3) partial fractions

(adding fractions backwards), and (4) trigonometric substitution. Know how to
compute improper

integrals (integrals with infinities in limits or integrand). In some years you
may also be expected

to know how to integrate in polar co-ordinates. (I do not expect you to remember
the formulas

for computing areas and arc lengths using polar coordinates .) It is good to try
the above methods

in order, i.e. (1) first and (4) last. The following expressions
suggest

the trigonometric substitutions x = sin θ, x = sec θ and x = tan θ respectively.
The motivation

behind these substitutions is Eqns (1) and (2).

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