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(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 1 2 1 3 1 4 ∑n x n = x+ 2 x + x + x + ... , you can see that it doesn’t actually depend on 3 2 n =1 any variable n. It is only if you consider the function that tells you the nth 1 term of the series, f (x, n) = x n that you might consider differentiating or n integrating with respect to either ‘x’ or ‘n’. 2. I have a question regarding question number seven in the review sheet ln x you gave us today. Using the fact provided ( Loga x = ), I found that ln a d 1 [Loga x] = by using the quotient rule (or the rule for the derivative of a dx x ln a constant multiple of a function). Assuming I am correct in my derivative, 1 then I would get: ∫ x ln a dx =Log x + c a 1 However, I would have no idea how to antidifferentiate without simply x ln a d 1 knowing that [Loga x] = . Is this some rule of derivatives that I don't dx x ln a 1 know? Or is there an easier way to antidifferentiate ? x ln a 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 1 1 1 ln x differentiation, you could get the result ∫ 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 ? d n Yet (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 ∫ ysin(π y)dy . Now 1 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 ) = xsin(π x) . dx 1 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 ∫ f (x)dx , why must x be greater than or equal to the a 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 Michaelis- Menten 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, d 1 say [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 anti- differentiating, 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 a. For the integral x ∫ dx , substitute u = x . Then du = 2 x dx . So, 1 you can replace cos( x ) by cos(u) and replace dx by 2du to get x cos( x ) ∫ x dx = ∫ 2 cos(u)du , which is easy to integrate directly. b. For the integral ∫ x cos( x )dx , again substitute u = x . As in part a 1 above du = dx , But there is no square root in the denominator of 2 x 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 ) ∫ x cos( x )dx = ∫ x dx = ∫ 2u 2 cos(u)du , which, though it isn’t easy, is 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 = 8 and lim a n→∞ n =∞ ? 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 10 15 -5 -10 See below for more questions. 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.