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Litle help understanding the point behind Calculus .

  1. May 8, 2010 #1
    Litle help understanding the point behind Calculus.....

    Hi guys! Thanks for any help you can give. I'm in college, taking HS Algebra, not too much of a problem, and am taking College algebra and trig over the summer. I have decided to study calculus from the textbook my college (uw-madison) uses (Thomas' Calculus. Any reviews?) over the summer. Right now, I have pretty much spent the last four or so hours on the first section, discussing limits, and I cannot seem to understand the point behind it. I understand algebra, geometry, and trig, but I can't get what they are talking about when discussing limits. Calculus just seems extremely awkward, I feel as if I am missing something important the whole time. Can someone help me out with this? I probably havn't explained this very well, but any help would be appreciated...
     
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  3. May 9, 2010 #2
    Re: Litle help understanding the point behind Calculus.....

    You've explained it perfectly. From what I can tell, you've got a standard Calculus textbook. For the purposes of this discussion, "standard" = "USELESS".

    While I understand their approach, it's a little off-putting to see epsilons and deltas and all sorts of nonsense before you even get to the exciting thing about calculus!

    Here's my summary of limits, which should get you through the basics of differentiation and integration:
    1) the limit as x-->5 of x^2 is just 5^2.
    2) If you end up with division by zero when you try to simply substitute, a "trick" is required. The tricks involve canceling factors, using conjugates of rational expressions, and the limit (as x --> 0) of sin(x)/x.
    *The most significant occurrence of this is in the "difference quotient". The limit as h-->0 (or delta x, or c, or whatever) of the "difference quotient" is the definition of the derivative (i.e. "the CALCULUS operation")

    I say "screw limits". If you know enough to draw a curve, label points (x, f(x)) and (x+h, f(x+h)), find the slope of that line, and make h infinitely small... you've got your head on straight.

    Of course, some dissenters that want to indoctrinate you in the lifeless monotony might come and argue with me, but I just want you to enjoy some pretty sweet math.

    (edit...)
    There's an old book: "What is Calculus About" by Sawyer... accessible and helpful.
     
  4. May 9, 2010 #3
    Re: Litle help understanding the point behind Calculus.....

    This is comforting for where I am now, but doesn't give me much hope for the future :bugeye:
    lol, jk

    a couple of things here...Why exactly are we trying to find this thing called a limit? I'm still not clear on a couple of points....What is a limit and why are we trying to find it? Also, I'm a bit confused, I kind of felt that you could not...or at least..should not just substitute a number into the function like that?

    Thanks!
     
  5. May 9, 2010 #4
    Re: Litle help understanding the point behind Calculus.....

    I can vouch for Thomas Calculus. It's not a bad book - if you know what you're doing with it. The first time I looked at the limits section in that book I was so overwhelmed & skipped half of it. It took quite a while studying from other sources to cut through all the bs of that limits chapter.

    What I would suggest is for you to definitely get a book like "Calculus for the Utterly Confused" and maybe "Calculus Made Easy" - This book only if you're good at algebra & understand exponentials & logs (I didn't & hate this book specifically because of this reason lol but now it's a good book whenever I glance in it).

    The other thing I'll recommend for you to do is watch all of the calculus videos at www.khanacademy.org - Seriously, this will help so much.
     
  6. May 9, 2010 #5
    Re: Litle help understanding the point behind Calculus.....

    When I first took calculus, initially the presentation of limits was confusing and to a certain extent very confusing. However, it's important because taking limits will eventually help you understand the derivative (the instantaneous slope of a function). The first portion of the beginning semester of calculus, from my point of view, is the most difficult. Just remember the basics, and the assortment of mathematical tools you have to simplify limit problems. Once you understand what the derivative entails you will be happy to see that calculus provides a much quicker shortcut to the limit problems you’re dealing. It will make sense soon.

    However, I must say the following:

    1.) Calculus is vastly important. Once you understand it you will realize that it's not difficult at all. Like any other math-practicing problems helps immensely understanding the fundamental theory of calculus. I hear so many people make the mistake of saying that it's useless (most of the time I'm sure it's from people who don't understand it themselves). Calculus is the mathematics of change and the universe is not static. Once your able to use calculus and understand calculus based equations, you're ability to understand dynamical systems grows immensly.

    2.) Initially, calculus deals with the tangent of a curve, and the area of the curve. Don't lose sight of this simple picture. When you're introduced to the complex notation that calculus is composed of, it's very easy to get lost and forget what the underlying principle is.

    3.) Don't be scarred of limits. Like any other math learn to master and how to solve limit problems. As I've progressed to higher math courses, limits are continuously re-introduced.

    Good luck.
     
  7. May 9, 2010 #6

    lurflurf

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    Re: Litle help understanding the point behind Calculus.....

    Limits arise where we have a method to solve a problem, but the methods never finishes it just continues forever. The limit is the answer that would be reached at the end, if there were one, which there is not.
    limit as x-->5 of x^2
    is a poor example in my view, it causes more confusion
    what is being asked is if x were very near to 5 would x^2 be near a particular value and what is that value. You have notices that x^2 is a special type of function called continuous so that
    limit x->a f(x)=f(a)
     
  8. May 9, 2010 #7

    Redbelly98

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    Re: Litle help understanding the point behind Calculus.....

    The point of calculus is to calculate the rate of change of a function (this is equivalent to the slope on a graph of the function). It is useful for solving many problems in science and engineering, which will become clear later if you take calculus-based science or engineering courses.

    Calculating a function's rate of change (or slope) involves taking a limit. So calculus courses begin first with teaching the concept of limits. But soon afterwards, shortcuts to calculating rates of change are taught and the topic of limits is rarely brought up again.
     
  9. May 9, 2010 #8
    Re: Litle help understanding the point behind Calculus.....

    Ok, let me try to get this straight...
    The limit is a point. Right? If we know the limit is 10, right? (thats just a random number I picked) You put, say 1, then 2, then 3, and so on into a function like f(x)=2x, you cannot ever put 10 in? Do people agree?

    I guess I'm just really struggling to see what makes this limit idea worthwhile. From what I can tell, you are just plugging in numbers into an equation or function, while specifically leaving one out, for some unknown reason. My brain keeps coming up with "so what?" Why not plug in this off limits number and just cut with beating around the bush?

    As you can tell, I'm really lost............ :(
     
    Last edited: May 9, 2010
  10. May 9, 2010 #9

    Redbelly98

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    Re: Litle help understanding the point behind Calculus.....

    For that example, you can put any number in for x, so it's not the best example for showing how limits can be useful.

    A better example might be the following:

    [tex]lim_{h \rightarrow 0} \ \frac{(2+h)^2 - 4}{h} [/tex]​

    You can't simply substitute h=0 into the expression here, you need to do some algebra to simplify the expression first.
     
  11. May 9, 2010 #10
    Re: Litle help understanding the point behind Calculus.....

    1. Ok, so you cannot find the limit of just any equation?

    So far, the best explanation I've found is this:

    What I get from this is the limit is the slope of a curved line at a certain point. 2. But, why would we ever want to find this?
     
    Last edited: May 9, 2010
  12. May 9, 2010 #11
    Re: Litle help understanding the point behind Calculus.....

    You cannot always use DIRECT SUBSTITUTION to find the limit. In the example given, you need to rewrite/simplify, as Red said.
     
  13. May 9, 2010 #12
    Re: Litle help understanding the point behind Calculus.....

    maybe im getting ahead of myself. I am going to try a chain of logic here, can you tell me if I am going wrong anywhere?

    1. Calculus is the study of change, rates of change. I don't really understand this. Are we studying the speed something is changing?

    2. The rate of change can be figured out algebraically, or using calculus. Algebra can be used when finding the change over a time interval, but for finding how fast something is changing at an instant, you have to use calc?
     
  14. May 9, 2010 #13
    Re: Litle help understanding the point behind Calculus.....

    Good.
    1. YES, that's what speed is! If this topic is going to have any "real-world" application, we must assume that we have functions that model some physical/natural/economic/etc phenomena.
    Our functions can be polynomials, exponential, logs, trig functions, combinations of these and/or other functions... and we get to:
    2. Finding average rate of change is done by using the slope formula/definition of the endpoints of the time interval. But like you said, the instantaneous rate of change requires more advanced techniques.

    Have you seen the definition of the derivative? Do you understand the connection to this and to the slope of a SECANT line?
    It's very intuitive, but the "text-only" approach is lacking.
     
  15. May 9, 2010 #14
    Re: Litle help understanding the point behind Calculus.....

    Ahh, thank you! I feel like I am finally making some progress on this. No, I have not yet seen the definition of a derivative. Just from flipping through the book, I figured that should come after getting a solid idea of what the limit is. Or is the limit better understood after learning the derivative?
     
  16. May 9, 2010 #15
    Re: Litle help understanding the point behind Calculus.....

    yes and no!

    Here's a layman's approach to the formal limit:
    Given a function (say, f(x) = 3 + 1/x), I say that the limit as x approaches infinity IS THREE. That means that I can get as close to THREE as you want.
    So you give me the smallest positive number you can think of, and I can get closer to 3 than that. Let's say you want me to get .0001 away from 3. Ok, just take the function value at 10,001... f(10,001) = 3.00009
    This value is within .0001 of 3 (the "limit" that I started with). We can repeat this all day, but I can PROVE that I can get closer than you want. (proof omitted!)

    Of course, many times you can just plug in values. Like in the example I gave a while back .... f(x) = x^2. I can get this function's value to be as close to "25" as you'd like, since all I have to do is just plug in 5 to get EXACTLY 25 (=5^2).

    Another example that you've seen FAILS when you try to "just plug in" the value. You get 0/0, which is no bueno.

    Most limits that you will work with can be evaluated with:
    1) direct substitution
    2) assuming that 1/x goes to ZERO when |x| gets big. If we had done this, then 3 + 1/x would go to 3 + 0 = 3 = the LIMIT
     
  17. May 9, 2010 #16
    Re: Litle help understanding the point behind Calculus.....

    So, from what I'm getting, you can find the limit of any function, as the limit is just how a function behaves as the x value approaches a certain number. The function can be any function, and you can still find a limit. There are two types of methods used depending on what the function looks like, if it can be defined for all real numbers, or if there are values for which it is undefined. If it is possible for a function to be undefined, like in a square root or (random value)/(zero), you should algebraically simplify the function, find where the function is undefined, and that is the limit.

    Am I doing this right? :biggrin:
     
  18. May 10, 2010 #17

    Redbelly98

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    Re: Litle help understanding the point behind Calculus.....

    I would say no. The derivative is an example of a limit, so can't be really understood without understanding limits first. Unfortunately, this puts students temporarily in the situation of having to learn about limits without knowing why they are important.

    Yes. That sums it up very well.

    Yes, provided certain conditions are met -- the book you are using probably has a list of the conditions necessary for a limit to exist.

    Pretty much, yes. Limits also involve the notion of continuity of a function, which you can read about in your textbook.

    Pretty much. I'm not sure about the square root example, but the divide-by-zero case is the important one for taking derivatives of functions.

    Another point: it is not "(random value)/zero", but "zero/zero", in these types of limits.
     
  19. May 10, 2010 #18

    HallsofIvy

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    Re: Litle help understanding the point behind Calculus.....

    Take a look at the problem Newton was trying to solve when he invented calculus- the orbits of the planets.

    Imagine that you are in a space ship high above the plane of the ecliptic and you take a "snap shot" of the solar system. Theoretically, at least, you could use that snap shot to calculate the distance of each planet from the sun at that instant. If the gravitational force, and so the acceleration of each planet, depends only on the distance from the sun, which was what Newton believed and was trying to show, then you can calculate the acceleration of each planet at that instant.

    But what does that even MEAN? Acceleration is the rate of change of velocity. In order to have a "change" there has to be a lapse of time- we have to measure the velocities at two different times. Actually, it is worse than that because velocity itself is the rate of change of distance- to measure each velocity we would have to have two different times- to measure acceleration should require at least 4 different times!

    The really crucial point about the (differential) Calculus is that it allows us to make precise the notion of "rate of change at a specific instant".
     
  20. May 10, 2010 #19

    HallsofIvy

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    Re: Litle help understanding the point behind Calculus.....

    I once had to teach a "Calculus for Economics and Business Administration Majors" course from a text chosen by the College of Business Administration. It covered limits on one page, listing four properties of limits:
    1) lim f(x)+ g(x)= lim f(x)+ lim g(x)
    2) lim f(x)- g(x)= lim f(x)- lim g(x)
    3) lim f(x)g(x)= (lim f(x))(lim g(x)) and
    4) lim f(x)/g(x)= (lim f(x))/(lim g(x)) as long as [itex]lim g(x)\ne 0[/itex]

    On the next page, they introduced the derivative completely ignored the point that in the definition of the derivative, the denominator always goes to 0 so that the fourth rule does not apply!
     
  21. May 10, 2010 #20
    Re: Litle help understanding the point behind Calculus.....

    The whole point of calculus is to be able to calculate. That you end up studying limits, derivatives, and integration techniques is besides the point. These are necessary tools you need in order to do calculations. Similarly, you need to learn about your car and the traffic rules, if you want to drive a car safely. But that's not the reason why you want to use a car.

    Without calculus techniqus, you are limited to being able to do artithemtic and computations involiving polynomals and rational functions. You cannot efficiently evaluate trigonometric functions, logarithms, exponential functions etc.
     
  22. May 10, 2010 #21
    Re: Litle help understanding the point behind Calculus.....

    Basically, Calculus is a subject that should be interesting to us if for no other reason than because it is a total departure from the maths that come before it. Yes, the same rules apply and the basic gist of computing problems remains, but what's really interesting is how different in theory and approach Calculus is and should be taught. In a lot of ways its the gateway to upper mathematics and should be treated as such.

    A few things that I can say that may help keep things in perspective:

    1. A function (for many a Calc I class is the first time we see the dreaded "function") is best thought of as a rule. What I mean is that it should be thought of as something new, not an equation nor a graph (obviously) but something different entirely. And I say a rule because in many ways a function dictates how a quantity or an expression behaves.

    2. The limit is a concept that must remain in the forefront of your mind when studying anything Calculus related. And it's actually easy to conceive; a limit is just a way of making something get infinitely and arbitrarily close to something else without actually reaching said something else.

    3. Keep in mind the tangent and area problems, remembering that integration is the inverse of differentiation. Theses two problems from which the calculus was birthed illustrate the need for thinking of something as just the sum of an arbitrarily large number of parts.

    I hope my little two cents helps a bit. Good luck in your studies and most importantly have fun.
     
  23. May 11, 2010 #22
    Re: Litle help understanding the point behind Calculus.....

    To put it succintly, the limit of a function is the value to which the function approaches as the the variable approaches some value.

    As an example, let's say we are given the following function:

    [tex]f(x) = \frac{1}{x}[/tex]

    Just looking at this function, we can tell that it is defined for all x [tex]\neq[/tex] 0. That is to say that the function does not exist at x = 0. However, we may find ourselves in some situation where it would be really, really nice if we had an idea of what was going on at x = 0. But, since we can't know, we'll have to settle with knowing where the function is heading as it becomes arbitrarily close to 0. When we say we want to find out where a function (or a sequence, as might be the case) is heading as it becomes arbitrarily close to some value, incalculable or not, we are asking for the limit of that function at that value.


    Now, I chose the function to be 1/x for a reason: because it is really easy to understand graphically.

    This is the graph of f(x) = 1/x:

    [URL]http://library.thinkquest.org/2647/media/odd1ox.gif[/URL]

    For ease of understanding, let's concentrate on Quadrant I, which contains the positive x-axis. In the first quadrant, we are coming up on 0 from the right, or towards the origin from the positive values of x (Approaching a value from different sides is actually a topic you'll cover when you learn about continuity and differentiablity. For now, all you need to understand is that we are heading closer and closer to 0 from the right of the x-axis.). Start at any positive value of x that you like, and trace the curve with your eyes as x gets smaller and smaller. Where is that curve heading? The fact of the matter is that the curve is never going to touch the y-axis. Not ever. That means that the y-axis is an asymptote, the behavior of the function at x = 0 is asymptotic, and the function just gets bigger and bigger, with no end in sight. In this case, we say that the limit as x goes to 0 from the right is infinity. And we write:

    [tex]\lim_{x \rightarrow 0^+}\frac{1}{x} = \inf[/tex]

    And this is true for any function for which f(x) that gets bigger and bigger with no possible end as x increases or decreases: they all go to infinity. (f(x) = x^2 is an example of a function that goes to infinity as x increases because no matter the value of x, there is always going to be some new value of y that is greater than the last one).

    Now, what happens if we head the other direction? Still focusing on the first quadrant, what if we start at 1 and trace the curve as x gets larger and larger? You've probably noticed that we have another asymptote at the x-axis. That is to say, the values of y get smaller and smaller, and the curve is never going to touch the x-axis. When teaching this to young children, the reaction I get most often is "It goes to infinity again!" But does it? The important thing to remember here is that when graphing a function, we are not so interested in the values that the variable takes on as we are in the values the function takes on as the variable changes. With that in mind, the limit of 1/x as x goes to infinity is 0, because as x increases, the function is getting closer and closer to 0.

    [tex]\lim_{x\to\inf}\frac{1}{x} = 0[/tex]

    Hope this helps!
     
    Last edited by a moderator: Apr 25, 2017
  24. May 12, 2010 #23
    Re: Litle help understanding the point behind Calculus.....

    Limits are related to a special property held by the reals called the least upper bound property. One way to think about limits is that they are a way to make certain kinds of infinity finite.

    They allow you to make sense of situations where the following hold:

    1) Something is impossible to do at the point you're interested in.
    2) The same thing is possible to do at points NEAR the point you're interested in.
    3) The closer you get to the point, the more accurate your results will be.

    This is the situation with a derivative:

    1) [tex]\frac{f(x) - f(y)}{x - y}[/tex] is not defined when [tex]y = x[/tex].
    2) However, it is defined for all other values of y.
    3) The closer y is to x, the more accurate your results.

    Limits also come up in many, many other contexts in math and there are several variations. They are used to define integrals. They are used to define series (taylor series, fourier series, etc) which give rise to many familiar functions, such as e^x, sine, cosine, etc.
     
  25. May 12, 2010 #24
    Re: Litle help understanding the point behind Calculus.....

    limits are useful for defining a function as its variable heads toward -infinity or infinity or as the function evaluated at that variable heads toward some abstract value (like infinity or -infinity) or if by evaluating the limit, you do division by zero.

    calculus is useful for understanding the universe (i do not know of any high level physics that is not calculus based)
     
  26. May 13, 2010 #25
    Re: Litle help understanding the point behind Calculus.....

    No, there may not be a limit at all points. A limit only exists if the function converges to a value as you approach the point. For example, the limits of the expressions [tex]\frac 1 x[/tex] and [tex]sin\;ln\;x[/tex] at x=0 does not exist. Furthermore, a function could have distinct left and right limits, such as the expression [tex]\frac{\sqrt{x^4+6a^3+9a^2}}x[/tex], which has a left limit of -3 and a right limit of 3 at x=0. Or a function could have a left limit but no right limit, or vice versa.
     
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