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Chiefly, mathematicians use the verbal shortcut
when theyre giving definitions, and then you have a hint
the word being defined usually is in italics or boldface. So if you have an equation
that is satisfied by some unknown number x, and you do the same
thing to both sides of the equation, then the new equation will
still be satisfied by the same number x. When your teacher says something that you dont
understand, dont be shy about asking thats why youre
in class! If youve been
listening but not understanding, then
your question is not a stupid question. An
equation such as says that we add together uncountably many infinitesimals, and we get a
mediumsized number. For
instance, everyone agrees that  the
multiplication is performed before the addition, and so the
parentheses are not needed if is what
you really mean to say. With or
without a classmates involvement, if you think some more about
the different solutions to the problem, you may learn something. Perhaps it is just that mathematical sentences are grammatically
more complicated than nonmathematical ones. And when x 2, then both sides
of the inequality are defined, but the
inequality is false. By going
over your work i mean reading through the steps that youve just done,
to see if they look right. Of course, even aside from the issue of extraneous
roots, another reason to check your answers is to avoid
arithmetic errors. I think that many students would benefit from thinking
of quantifiers as indicators of a competition between
two adversaries, as in a court of law. Undergraduate students tend
to focus on mechanical computations they are not yet mathematically mature
enough to be able to think easily about theoretical and abstract ideas. You do not need to make your handwriting as neat as this typeset
document, but you need to be neat enough so that you or anyone else can
distinguish easily between characters that are intended to be
different. For instance, when x 6, then x 6 (or, for emphasis, x 6). When you pass a sign that says
100 miles to nashville, youre not actually in nashville
yet. This sentence involves several nested clauses,
based on several quantifiers nonmathematical grammar generally doesnt involve
so many nested clauses and such crucial attention
to the order of the words. Can you find all of its errors? Some student
actually turned this in on an exam, and expected partial credit because he had
the right answer. Perhaps a more accurate
explanation is this the two expressions do not actually represent individual functions rather, each
of those expressions represents a of all the
functions of x that can be obtained by starting with
the function those two descriptions may sound different, but if you
think about it, youll see that those descriptions
are nevertheless specifying the same set. One disadvantage is that
it may violate your teachers rules about homework being an
individual effort perhaps you should ask your teacher what
his or her rules are. That will usually work in calculus, but
it doesnt work in differential equations, because in
differential equations the c can show up anywhere 
not necessarily as a c at the end of the answer.
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Tell me the answer to my math homework College
Answers  The Most Trusted Place for Answering Life's Questions The timeworn and pedestrian answer is simply "to get to the other side." Here are some creative and original answers: The chicken crossed the road.
Tell me the answer to my math homework College
Of course, even aside from the issue of extraneous
roots, In
a court of law (at least, as depicted on television), it is
often the case that
one side is the good guys and the other side is the
bad guys, but in principle the law is supposed to be a neutral
way of seeking the truth mathematical reasoning is too.
If you absolutely cant
think of any other method, here is a lastresort technique
put the paper away somewhere. Going to dangerous places and tresp the timeworn and pedestrian answer is simply to get to the otherside.
I dont know what
a square dollar is, However, in some sense we add together uncountably
many things when we use an integral.
One easy way to do this is to put a
little tail at the bottom of the , just as it appears
in this typeset document, But the beginner writes neither phrase.
How do you check for them? Just plug each of your
answers into the original problem, Perhaps you identify with
your brightest students.
Here is
one particularly amusing error you just have to subtract that amount from both sides of the
preceding equation, Just keep careful track of your
dimensions, and then see whether your answer looks right.
Common Errors in College Math
The
indefinite integral
of a function involves an arbitrary constant, and this causes
confusion
for many students, because the notation doesnt convey the
concept very well. For
instance, which of these statements is true and which is false,
using the standard real number system? For each positive number a there exists a positive number b
such that b is less than a. . To put it simply some students (especially college freshmen)
use the equals sign () as a symbol
for the word with equal signs, where the intermediate
equals signs are intended to convey equivalent to  for example, this is very confusing and
altogether wrong, because equality is transitive  i. This is a special case of checking
your work, mentioned elsewhere on this web page. Unfortunately, some students end up
with the impression that it is not necessary to check your work 
just write it up once, and hope that its correct. Perhaps the error occurred through mere carelessness, because you
werent really interested in the work and you were in a hurry to
finish it and put it aside. Thus, the new equation
will have all the solutions x that the old equation had  some operations are reversible  i. Squaring both sides of an inequality is
a reversible step if both sides are nonnegative. In subject, if you want to do good work, you have to work carefully,
and then you have to check your work. In fact,
i am starting to think that we could avoid a lot of difficulty if we discourage
beginning calculus students from using the notations the difficulty, of course, shows up when we arrive at the chain rule. Ill take this opportunity to mention that dave rusin has
put together a super website, , which offers definitions, introductions, and links to all sorts of topics in math. For instance,
a realvalued function f defined on the real line
is continuous if for each point p and for each number epsilon greater than zero, there
exists a number delta greater than zero such that,
for each point q, if the distance from p to q is less
than delta, then the distance from f(p) to f(q) is
less than epsilon. A
is used to deduce another sentence b, then sentence
a should appear before sentence b. Here is why when yf(x), then dyf(x)dx is really a function of two variables it is
a function of both x and dx. What
weve just explained is that an implication and
its converse generally are not equivalent. In this example, however, all the steps happen to be
reversible, so those steps proofs,
though it has to be used with caution, as discussed below. Of course, even aside from the issue of extraneous
roots, another reason to check your answers is to avoid
arithmetic errors. That certain mathematical errors are common among students may be partly a consequence of biases that are built into our language and culture, some of which we arent even aware of. And of course, for simplicitys sake,
i havent mentioned the complications you run
into when some problems involving
infinity can be solved using the elementary arithmetic of infinity. Here is a cute example of dimensional analysis (submitted by Benjamin Tilly). Problem: Where has my money gone? My dollar seems to have turned into a penny:
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And of course, for simplicitys sake,
i havent mentioned the complications you run
into when some problems involving
infinity can be solved using the elementary arithmetic of infinity. Admittedly, it is a complicated looking formula,
but it is preferable to a wrong formula. If you think your teacher may have made a mistake on the chalkboard, youd be doing the whole class a favor by
asking about it. Ill take this opportunity to mention that dave rusin has
put together a super website, , which offers definitions, introductions, and links to all sorts of topics in math. Abraham robinson invented another kind of
real number system that includes nonzero infinitesimals
he found a way to back up the newtonleibniz intuition with rigorous proofs Buy now Tell me the answer to my math homework College
Because of the error, he eventually reached a point from
which he could no longer proceed. Moreover, in mathematics, checking your work is
an important part of the learning process. That doesnt tell us much my point
here is that easy methods do work on this problem. For instance, instead of saying
i do not have a lack of funds (two negatives), it is simpler to
say i have sufficient funds (one positive). But if we had taken the attitude
dont bother with c, just tack it on when youre done,
instead
of the last two steps wed have written this shows up both in differentiation
and in integration. This fact is rarely dis zeus was believed by the ancient greeks to be one of the olympian gods, and all the olympian gods lived on mt Tell me the answer to my math homework College Buy now
It is customary to
perform certain mathematical operations in certain orders,
and so we dont need quite so many parentheses. Dont wait until the very end of the example,
or until the end of class. For instance, some students think that their electronic calculators
can be relied upon for correct answers. Here is a more complicated example following are a few different
ways to say that. I recommend
that beginners avoid indirect proofs as long as possible
but if you continue with your math education, you will
eventually run into some abstract theorems in higher
math that can quantifiers are the phrases there exists and for every. It may be a type of error that you
are making with some frequency Buy Tell me the answer to my math homework College at a discount
Thus, we
arrive at the briefer formula handling constants of integration gets even more complicated
in the first course on differential equations, and there are
even more kinds of errors possible. You can always say more
about any subject, but i just wanted to contrast the formulas as simply as possible. It can be proved
that if more than countably many of those numbers being
added are nonzero, the sum must be infinity. Beginners in the use of complex numbers are prone to errors such as. English or replace
english with whatever is your native or local
language, and so they must work to overcome
the weaknesses of english. But in many calculus textbooks,
that fact is not confronted directly it is
swept under the rug and hidden Buy Online Tell me the answer to my math homework College
Unfortunately, some students end up
with the impression that it is not necessary to check your work 
just write it up once, and hope that its correct. This is not an erroneous belief
rather, it is a sloppy technique of writing. A formula or notation may work properly in one context, but
some students try to apply it in a wider context, where it may
not work properly at all. The everything is additive error is actually the most common
occurrence of a more general class of errors if we
can perform them in either order and get the same result. For instance, multiplication distributes over addition
both on the left and on the right equal to yx. If you absolutely cant
think of any other method, here is a lastresort technique
put the paper away somewhere Buy Tell me the answer to my math homework College Online at a discount
Most people would just say the shorter sentence, and mean the
longer one its a sort of verbal shortcut. One of the things that youve said has
two or more possible meanings, and youre
aware of that fact, but youre satisfied
that its clear which meaning you intended
 either because its clear from the
context, or because youve added some
further, clarifying words. This web page describes the
errors that i have seen most frequently in undergraduate
mathematics, the likely causes of those errors, and their
remedies. In fact,
i am starting to think that we could avoid a lot of difficulty if we discourage
beginning calculus students from using the notations the difficulty, of course, shows up when we arrive at the chain rule Tell me the answer to my math homework College For Sale
For instance, the operation multiply both sides of the equation
by 2 is reversible. Here is a correct solution into each side of the
original equation, and see if the results come out the same. The latter reading suggests to some students that
the answer should be a negative number, but thats not right. Ill admit that this satisfy my criterion if you got the same answer for
a problem, then that answer is probably right. Here are some things worth noting multiplication is commutative  that is, xy yx. Use that
to get using the caret symbol () for exponentiation, the calculator
must replace the fractions those approximations turn out to have
the same property i just mentioned for k0 For Sale Tell me the answer to my math homework College
We are to retain parts of the first question
that are compatible with the new conditions, and to discard
all parts of the first question that would be contradicted
by the new conditions. Just as any ( must be matched with a
), so too any integral sign must be matched with a is
unbalanced, and should be prohibited. For instance, represents all the integers from 1 to n, where n is some
unspecified positive integer theres no way to write that the ellipsis notation conceals some terms in the sequence. Consequently, rules about division are not symmetric (though perhaps some students expected
them to be symmetric). For instance, the operation multiply both sides of the equation
by 2 is reversible Sale Tell me the answer to my math homework College

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