(Perhaps Not So) FREQUENTLY ASKED QUESTIONS
From Students Spring 2007
Derivatives and Integrals
1. In finding the anti-derivative of a series, do we integrate the series with
respect to 'x' or with respect to 'n'?
If you take a look at the expanded form of a power series like
∞
1
∑n x
n =1
n
= x+
1 2 1 3 1 4
x + x + x + ... , you can see that it doesn’t actually depend on
2
3
2
any variable n. It is only if you consider the function that tells you the nth
1
n
term of the series, f (x, n) = x n that you might consider differentiating or
integrating with respect to either ‘x’ or ‘n’.
2. I have a question regarding question number seven in the review sheet
you gave us today. Using the fact provided ( Loga x =
ln x
), I found that
ln a
d
1
by using the quotient rule (or the rule for the derivative of a
[Loga x] =
dx
x ln a
constant multiple of a function). Assuming I am correct in my derivative,
then I would get:
1
∫ x ln a dx =Log x + c
a
However, I would have no idea how to antidifferentiate
1
without simply
x ln a
d
1
. Is this some rule of derivatives that I don't
[Loga x] =
dx
x ln a
1
know? Or is there an easier way to antidifferentiate
?
x ln a
knowing that
It is the point of this set of exercises that whenever you differentiate a
function, the result tells you an integral formula. So, it is entirely reasonable
that you wouldn’t have any idea how to do this particular antidifferentiation
without having already done the corresponding differentiation problem
unless you had already done the differentiation of lnx. Had you done that
differentiation, you could get the result
1
1
1
ln x
∫ x ln a dx = ln a ∫ x dx = ln a + c .
3. Why is the derivitive of a x equal to a x ln a instead of it being xa x −1 ?
Yet
d n
(x ) = nx n −1 , which is the Power Rule. (This is the same as
dx
d a
(x ) = ax a −1 .) They are almost the same, yet different. What is the
dx
difference between "a" and "n"?
Watch for: what symbol represents a constant and what symbol represents
the variable with respect to which the function is being differentiated (or
integrated). When you say, “the derivitive of a x equal to a x ln a ” you are
assuming that the symbol ‘a’ is constant and the differentiation is relative to
the variable x.
There just isn’t much of any reason to expect that the derivative of an
exponential function would have a form similar to that of a power function.
To convince yourself of the distinction, you should go back to the definition
of derivative and write the difference quotients that approximate the
derivative for each of those two types of functions. Compare the how the
derivatives of the functions 2 x and x 2 are formed.
4. In the proof for the fundamental theorem of calculus, how can you
differentiate in respect to x when the integration is respect to t? How is it in
the end that you get the original function?
2
Consider the example ∫ t sin(π t)dt . This definite integral is the number you
1
get when you integrate a specific function. The function that is integrated is
expressed in terms of a variable t, but the result is the same, regardless of
2
what variable is used: You get the same number for
∫1 ysin(π y)dy . Now
x
suppose that the upper limit is chaged from 2 to a variable x: ∫ t sin(π t)dt .
1
The result is that the integral is now a function of x, and, as such, we could
think about determining its derivative with respect to x. And, not only could
we think about such a differentiation, but one form of the Fundamental
x
d
Theorem of Calculus says what the answer is: ( t sin(π t)dt ) =
dx 1
∫
xsin(π x) .
I.e., we get back the original function, except that it is expressed as a
function of x instead of t.
b
5. In the integral
∫a f (x)dx , why must x be greater than or equal to the
number a and less than or equal to the number b?
The function f itself need not be restricted to values of x between the
numbers a and b. However, when you consider the integral with limits of
integration a and b, you are in effect saying, “I’m interested in this function
and some aspect of that function for numbers in the domain of f between
the numbers a and b. The aspect of interest might be an area, a force, an
average value, a probability, … depending on what the function represents.
6. What are uses of derivatives in the real world?
Many of the “laws of nature” are written as differential equations, equations
involving derivatives (first, second, and higher order). Such equations
become the foundation for theories (models) – in chemistry, the MichaelisMenten equations that describe reaction rates; electrical circuits that have
resistors and capacitors; in finance, option pricing methods; in psychology,
equations that describe rates of learning. (Look at section 9.3 and 9.4 for
examples.)
7. I was wondering if you could go more into depth about what "C" really is
and what it really means when you find the integral and how it will apply
when we actually find the integral of a function. I know that there is some
type of constant involved but why?
The addition of a constant to any function F whose derivative is f stems
from the fact that the rate of change of a constant function is zero. So, if
F’=f, then (F+C)’=f, too. What is not so obvious, but is true, is that once
you’ve found one antiderivative of a function, you know all of them – all
antiderivatives of a given function differ by just some constant.
So, when we write
∫ f (x)dx = F(x) + c , what we mean is the set of all
functions of the form F(x) + c , where c is any real number, contains all (and
only those) functions whose derivative is f(x).
8. For physics we use differentiation for velociy, acceleration, etc. But what
other numerical applications use differentiation - like statistics for example?
Why do we sometimes go up to 'n' number of derivatives for n>1? What is a
reason to find the nth derivative of a function?
We shall soon see (Look at section 8.5 for a preview.) how integration
plays the role of defining/determining probabilities and mean values. (And,
we wouldn’t know how to evaluate many integrals without the Fundamental
Theorem of Calculus, which links integration to differentiation.) We’ll also
see how the nth derivatives are used to write an infinite series
representation for a function. (Look at section 11.10 for a preview.)
9. Why is the chain rule necessary when differentiating a compound
function? What role does it play in the resulting derivative?
A flippant answer, though true, is that if you don’t use the chain rule you
aren’t likely to find the derivative of a composite function. For example,
suppose the concentration of some drug in the blood is given by
C(t)= Ae−3t , where A is some constant, C(t) might be in parts per
million (ppm), and t is in hours. If you claim that C'(t)= Ae−3t , ignoring
the chain rule, then you won’t be finding out how fast the concentration is
changing in ppm per hour. To get that rate by using the chain rule to get
C'(t)= −3Ae−3t ppm per hour. It is easier to verify that you definitely
don’t get the right answer if you ignore the chain rule with a simpler
example, say f (x) = (3x + 2)2 . Ignoring the chain rule would give you
2(3x + 2) for a derivative, but if you expand the function first to get
f (x) = (3x + 2)2 = 9x 2 + 12x + 4 , then it is clear that the derivative should be
18x + 12 . If you go to the next question, note that the incorrect claim
C'(t)= Ae−3t is not the derivative of C(t) with respect to t, but it is the
derivative of C(t) with respect to u=-3t.
10. When we take the derivative with respect to something other than x,
say
d
1
[ln(3x + 1)] =
for example, how does this affect the
d(3x + 1)
3x + 1
interpretation of the graph of the derivative?
If you graph the derivative as a function of the variable with respect to
which you differentiated, with that variable (3x+1 in your example), then
you can interpret the graph in the same way as usual. Taking your
example, graph the function 1/u against u: The fact that this derivative is
positive and decreasing for u positive means that ln u is an increasing
function, increasing at a decreasing rate. To bring x into the picture
requires knowing how u is changing with respect to changes in x; i.e.,
bringing in the “other link” in the chain of the chain rule.
11. How can you tell if the integral of a given function exists?
A superb question! It is well beyond the scope of the course to answer, but
along the lines of questions that mathematicians asked and then generated
much mathematics to answer.
You can interpret the question in different ways. If you equate ‘integral’ with
‘antiderivative,’ then a question might be whether there is an antiderivative
that is some combination of algebraic and transcendental functions. Just
formulating the question is hard! If you interpret the question as, “How can
you tell if the definite integral exists?” there are simple sufficient conditions,
like, “The definite integral of f exists if f is continuous over the interval over
which the integral is taken.” But necessary conditions? Way harder!
12. So the idea of replacing a complicated integral by a simpler integral is
to change the original variable x to a new variable u that is a function of x.
Can an integral be evaluated if it has other variables in addition to a
variable x?
For functions of two variables, there are double integrals! See Calculus III!
13. So I know that loga(x)= ln(x)/ln(a) but can I use the quotient rule from
there to determine the derivative?
Yes! Or, more simply, use the rule for constant multiples (in this case, the
constant multiple is 1/lna) of a function.
14. Because you are reducing the powers in a derivative, aren't you finding
something different? For example. if you have a distance formula to find the
position of something, wont taking the derivative cause you to find
something related to the position instead of the position? so i guess my
question would be: when you take a derivative of a function, how does it
affect what you were solving for originally?
Yes! When you take the derivative of a function you get something
different, not just an alternate representation of the function. You get
something that gives you information about the function, point-by-point how
fast the function is changing and in what direction.
15. We've seen some equations that are impossible to integrate
(like ecos(2x) ) and I'm sure there are some equations for which you can't
take the derivative, though no examples are coming to mind at the
moment. Is there some generalizable rule in common to know when
equations either cannot be derived or cannot be integrated, or do you just
have to examine the problem?
You can put together any kind of function involving algebraic operations
(plus compositions of functions) and the elementary transcendental
functions and the “rules of differentiation” are sufficient to determine the
derivative of that function. However, you can put together a pretty simple
function, like your example, for which there are no rules for antidifferentiating, nor even any rules to tell you when such a thing is
impossible.
16. For the examples we have done in class, we have taken the integral of
a function, then to check have taken the derivative. Are there any
situations in which you would be able to go one way (take either the
integral or the derivative) and not be able to check the answer by doing the
other?
Given my answer to the previous question, you can see how the two
processes are vastly different, yet intimately intertwined. The fact that there
are rules for differentiation makes it possible, practical, and critical to check
the result of anti-differentiating by taking the derivative of that result. The
fact that there are not rules for anti-differentiation makes it neither possible
nor practical to check the result of differentiating by anti-differentiating.
17. The way that I see it, integrals and derivatives are like opposite
operations of each other. However, if that is the case, then why don't the
same rules for differentiating apply to integrating. Like if you have an
integrals that is a fraction, why wouldn't the "opposite quotient rule" work?
Yes, integrals and derivatives are opposite operations of each other – or, I
would say, inverse operations. That being the case, I would not expect the
rules of differentiating to apply to integrating. Here’s an analogy: Squaring
and taking the square root are inverse operations – and you don’t have
anything like similar rules for performing each of the two operations.
Climbing a mountain and descending a mountain (or a tree, for that matter)
often present different challenges. And did you ever watch a cat descend
from a tree?
18. Is it possible to find the integral of 1/sinx ?
Yes, the trick is to think of 1/sinx as cscx, and then multiply by 1 – in the
form of (cscx-cotx)/(cscx-cotx). You get a fraction whose numerator is the
derivative of the denominator, so that you can integrate the result by
making a direct substitution.
19. Derivatives are a rate of change. Are integrals of a rate of change also,
just moving in the opposite direction?
No; you’re right that derivatives are a rate of change – more precisely, the
limit of a quotient of differences (See page 158.) Definite integrals,
indicating their nature of being something of an inverse to derivatives, are
limits (of a complex sort) of a sum of products. (See page 380.)
20. is it possible to tell which integration technique will be effective just by
nature of the equation?
After lots of practice and reflection on the results of practice, you can start
making good guesses about which integration technique is likely to work.
21. I'm a little confused about direct substitution. How do we know which
part of the expression should be substituted by "u" and once we have
named "u" and "du," how do we know where to plug it back into the
equation?
An answer to your first question is in the answer to the previous question:
There aren’t any rules! It’s a matter of practice, reflection, and persistence.
As for your second question, here are some examples to illustrate how
easy it is sometimes and, on the other hand, …:
cos( x )
1
dx , substitute u = x . Then du =
dx . So,
x
2 x
1
dx by 2du to get
you can replace cos( x ) by cos(u) and replace
x
cos( x )
∫ x dx = ∫ 2 cos(u)du , which is easy to integrate directly.
a. For the integral
∫
b. For the integral
∫
above du =
1
2 x
x cos( x )dx , again substitute u = x . As in part a
dx , But there is no square root in the denominator of
the integrand, so what to do? Just stick it in there where it’s needed –
and too balance things, stick it in the numerator as well: We get
∫
x cos( x )dx =
∫
x cos( x )
dx = ∫ 2u 2 cos(u)du , which, though it isn’t easy, is
x
do-able using integration by parts – twice!
One essential in this direct substitution is that there must be a “du” in the
new integral – if not, you don’t really have an integral.
As I noted in class, there is some mystery to this process, which can be
justified in a rigorous way, but not without an enormous amount of work.
So, when dx and du appear in the integral, we treat them as meaning,
:integrate with respect to (dx or du), but then we treat them outside the
integral as differentials and almost just like algebraic symbols. (However,
the d and the x in dx never get separated – likewise the d and the u in du.)
22. guess I am having trouble with the traveling between the two lands of x
and u. How is it that we can automatically plug x back into the solution of
the solved integral after using substitution?
The metaphorical view: In a sense, when we travel to u-land, we’re really in
the same place but viewing the land through funny glasses that make us
see things in terms of u instead of x. We can solve the problem because
this change in perspective allows us to see more clearly. Once this new
perspective has enabled us to solve the problem, we take the glasses off to
see the answer in its original perspective.
We can make the substitution process precise and justify it rigorously, but,
for now, it makes sense to apply the process and then verify that the
solution is valid by differentiating the answer.
SEQUENCES – Domain
1. Why, when the definition of sequence “… a function whose domain is the
set of positive integers,“ is talking about functions, does the domain have to
be a set of positive integers?
Well, it doesn’t quite have to be a domain of positive integers, but if you
want a sequence to be a list of numbers that has a beginning, to be able
to talk about the first term in the sequence, the second term, the third
term, and for every given term that there be an immediate next term – if
you want sequences to have all of those properties – then the positive
integers are a natural choice.
2. I understand why the domain of a sequence must be positive, but why
can't it include fractions, why must it consist only of integers?How about
including all the negative integers – or just include 0? And what about
irrational numbers … or imaginary numbers?
If you look at the answer to question 1, you’ll see that fractions, positive
rational numbers, with their usual ordering fail to have all the required
properties (the items in bold font). Bonus Q: Which property (or properties)
above does the set of positive rational numbers, with its usual ordering, <,
fail to have? Double Bonus Q: Find a way to order the set of positive
rational numbers so that all the above properties are satisfied.
Likewise, using all integers won’t work if you want all the “bold” properties
listed in #1 to hold. Just including 0 along with the strictly positive integers
works, though – or you could use -17, -16, -15, -14, … , -2, -1, 0, 1, 2, 3, …
As for irrational numbers and imaginary numbers, an argument you
construct to show that the rational numbers won’t work should show that
neither irrational nor imaginary numbers work either.
See below for more questions.
3. "If a sequence is supposed to be a list of numbers written in definite
order and can also be defined as a function whose domain is the set of
positive integers, what is a list of positive integers which is written in no
definitive order. i.e., numbers appearing to be written in no special order
but which are still positive integers."
A list written in no particular order, the order not mattering, is a set, rather
than a sequence.
"Can it still be considered a sequence just because the domain is a set of
positive numbers? "
Once you have no definite order, there is no sequence. The reason that
any function whose domain is the set of positive integers is a sequence is
that the range of that function "inherits" the order of the positive integers;
i.e., f(n) becomes an .
SEQUENCES – Going to Infinity
4. How is it that the statement, “ lim an = ∞ ,” says that the nth term an is
n→∞
getting larger? Why does the statement have an equals sign in it?
If I expand on the statement, adding words to explain some of the
compact notation, I could write that what lim an = ∞ , means is:
n→∞
As n gets larger and larger [ n → ∞ ], the term [ an ] of the sequence
gets larger and larger, without bound [ = ∞ ].
The symbol = is, indeed, strange in the sense that infinity is not a real
number; the expression, " = ∞ " symbolizes the idea, “larger and larger,
without bound.”
5. Parts (b) and (c) of problem 1 on page 710 show the limit as n
approaches infinity of an as either approaching a certain number or
continuing to get larger. Will it ever be the case that a sequence has no
limit as n approaches infinity?
Well, strictly speaking, if
lim a
n→∞ n
= ∞ , we say that the sequence has no
limit. There are possibilities other than a numerical limit or infinite limit.
For example, a sequence that alternates between two numbers will also
have neither a numerical nor infinite limit.
6. Can both of the following statements be true:
lim a
n→∞ n
=∞ ?
lim a
n→∞ n
= 8 and
No; not if each instance of the symbol an refers to the same sequence. Our
textbook refers to two different sequences when making the above two
statements.
7. If a sequence is going to infinity (an indeterminate value) how is that we
can quantify it? How can I get some intuition about this idea?
First of all, when a sequence goes to infinity, that is not called
indeterminate. What "going to infinity" means is that no matter how large a
number B you are given, you can go far enough out in the sequence
(quantitatively, there is a number N), so that all the values in the sequence
beyond N (whenever n > N) are larger than the original number ( an > B)
you were given. See the textbook for two briefer statements that describe a
sequence "going to infinity."
Here’s another approach, using a specific sequence: To get intuition about
this formal definition, think about some sequence that approaches infinity,
say an = (1.01)n . For the expression, "for every positive number M," think
really big number like 1,000,000,000 = (10)^9, one billion in American
English. Intuitively, the question is: Will the values in the sequence ever get
larger than this number M=109 ? More formally, "Can we find a number N,
so that all the values in the sequence 'beyond' N are larger than this
number M?" Even more formally, does there exist a number N so that
whenever n > N, this forces an > M. In the case of the example sequence,
and the example number M =109 , can we assure that a = (1.01)n > 109
n
= M for all n beyond some number N. We can answer this question in the
affirmative by solving the inequality for n. To do so, use properties of < and
of logarithms.
Note that "indeterminate" refers to things like a sequence of ratios where
both the numerator and denominator approach infinity. Such a sequence
can go to infinity, go to zero, or go to any number in between, thus has
indeterminate form. An example is the sequence whose nth term is
8n/(n+5). This sequence "goes to 8;" i.e., it has the limit 8 as n approaches
infinity.
SEQUENCES –Limits Graphically
8. When representing the equation in (b) as a graph, what does the limit
look like visually?
Below is just one possibility. It shows the first 15 points on the graph of an
increasing sequence. It’s meant to “suggest” that the sequence is
approaching the number 8. What that would mean is that no matter how
small a horizontal band you took about the y=8 line, you could find a
Number N so that for any integer n > N the points on the graph (n, an )
would lie within that horizontal band.
10
8
5
-10
-5
5
-5
-10
See below for more questions.
10
15
SEQUENCES – When Do They Have a Limit?
9. Do all infinite sequences have a limit, or are there only certain
sequences that have limits? If so, how does one identify which sequences
have limits?
Some sequences have a limit, while others do not, and it is our job to
determine which ones have a limit, which ones do not have a limit, and, in
the case of the ones that do have a limit, to determine what that limit is. To
do such a job is the subject of section 1 of chapter 11 - the theorems of
11.1 tell us how, if it is possible, to determine whether a limit exists and
what that limit is if it does exist. For example, today in class I referred to the
theorem that says, "If |r|<1, then r n approaches 0 as n approaches infinity."
This theorem tells us about limits for infinitely many cases (all the values of
r) of sequences and enables us to tell what certain infinite series converge
to.
SEQUENCES – Why Study Them?
10. Why do we study sequences and what is a real-life application of
sequences?
Putting together a few things I have mentioned in class: Limits are at the
heart of calculus and what distinguishes it from algebra; derivatives,
integrals, and infinite series all involve limits of one kind or another. The
idea of limit of a sequence to my mind has an appealing and intriguing
definition, one that you can build on in future mathematics courses.
Limit of a sequence is a building block for infinite series, which is a building
block for representing functions as power series, Fourier series, and
wavelets, the last of these now being the standard for JPEG.
In Calculus I you may have studied Newton’s Method for approximating a
solution to an equation. That method produces a sequence that
approaches a solution to an equation.
Another real-life application of sequences lies in models for the long-term
predator-prey relationship, with the values in the sequence being ordered
pairs consisting of the number of predators and the number of prey at
points in time. (This is a bit of a generalization of the sequences of
numbers that we are studying.)
SEQUENCES – How Does a Sequence Approach Its Limit?
11. In problem 1b on p. 710 the answer suggests that as n gets larger and
larger, the term an will get very close to 8. But how can just knowing the
limit of this sequence tell me how this sequence approaches its limit?
In answer to your question, knowing the limit of a sequence doesn't actually
tell you anything about "how" the sequence approaches its limit. As you
suggest, "how" is important, particularly how fast the sequence approaches
its limit, which is important when the sequence is used to approximate
something and you need to know how far "out" in the sequence to go if you
want to be within, say, 0.001 of the number you are approximating. The
subject of "how fast" is studied in real analysis, Math 301, here at MHC.
THE “WRITE A QUESTION” HOMEWORK
12. I understand what you asked me to write a question about, so I thought
I would instead ask a different, but related question. Do you have a
problem with that?
Yes, I do have a problem with that. Here's why: I'm try to get everyone in
Calculus II – 02 to sort of "push the envelope" of their understanding
through question-asking. The idea is to test the limits (no pun intended) of
your understanding, not to stay at the initial understanding that allows you
limited flexibility in using a concept. I can understand that you may think
this a waste of time, but my experience is that the process is worthwhile,
and in most cases, I think that anyone could ask a question of genuine
interest to her with some thinking.