New answers tagged calculus
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Differentiation and integration considered by all scientists throughout the ages as one of the best sciences that guided the mind of man over all times
The fields of the use of calculus are very wide. It enters into many fields and are not limited to specific people or to those who use it only. But to almost all human beings. Here are some examples of its ...
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I think that it makes sense to introduce continuity in the same lesson that you introduce limits.
Here is a sketch of a lesson plan:
Give them this link
https://www.desmos.com/calculator/rlu2zgcjyf
Group work:
Is $f(2)$ defined or undefined?
As $t$ approaches $2$, what are the values of $f(t)$ approaching?
Make a table of values for $t = 1.9, 1.99, 1.999$ ...
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Ibrahim:
I would not think of it as either/or. Really, you should be able to understand and use the concepts in various ways. Algebraic symbol manipulation, "word problems" (to include not just physics, but chem, econ, business, bio, etc.), as well as graphical views. Perhaps even other conceptions (e.g. 500+ years ago, algebra was less ...
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To provide a different angle from the great replies already offered, I think other factors include:
the technology required for a computational approach has only (relatively) recently become readily accessible to a wide audience (& still isn't in many places);
depending on the approach, the use of software can hijack the course, making it more about ...
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You have asked two very different questions. I'll leave the differential equations for someone else. There is one particular application of integration which is my favorite last problem to do in Calc I. (We got behind this semester, and I was very sad not to have time for this. It feels like a perfect grand finale to me.)
You probably learned the formula for ...
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Many traditional calculus classes completely omit all mention of numerical calculation and approximation (in any concrete sense). It is certainly viable to cut Newton's method and it might make sense if students can't properly manipulate logarithms and trigonometric functions.
On the other hand, numerical calculation and approximation introduce lots of ...
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Below is an image that shows the net change in $F$ over each interval as a (finite) "differential," which is the same in this case as a term in a Riemann sum for the integral. The net change is given by the change in $y$ for the black, dashed secant line and is equal to the change in $y$ along the purple, solid tangent line obtained via the Mean ...
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Answer:
I feel like a lot of these suggestions are quite complicated. Not complicated for us, but complicated for new-to-the-topic students. If you must (but see below) go with a life example, I think doing something like Zeno's paradox of motion might be helpful. first half of the trip, half of the second half of the trip, etc. (1/2^x)
Too-long-for-...
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As an amateur photographer, my first thought was the lens position for focus on an object. It should be easy to intuitively explain that the closer the object the further the image because the lens only bends the rays so much. As the object gets very far away, the location of the image moves toward a limit, which is the focal length of the lens. Like the ...
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A nice easy limit that can be seen in everyday life.
Suppose your student looks along a perfectly level road, with perfect visibility. They look down at an angle that allows them to see the road 10m ahead of them. Then 20m ahead. Then 30m ahead. And so on, for any amount of distance ahead.
You can easily show, draw, find an equation, even tabulate, and ...
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I have actually just finished teaching a first semester of calculus, and when I started it I was concerned about examples to start with. And as far as I can see, first two semesters of differential and integral calculus are about approximations: finite difference is an approximation of derivative (and vice versa!), Riemann sum is an approximation of Riemann ...
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Maybe these are too easy but...
If you drop a bouncy ball, eventually you will stop hearing it bounce!
Well, we can model the ball by accounting for the predominant mode of energy loss via collision with the floor but ignoring air resistance and assuming constant gravitational acceleration. Then on each bounce with impact speed $v$, its rebound speed is $v·r$...
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I recently encountered -
$3^{x-1}+2^{x}=5$
I spent too much time trying to use logs to isolate X and got nowhere. I then rearranged to look at it as
$y=3^{x-1}+2^{x}-5$
and that's when I realized that this was a classic case of the need for Newton's Method. I'd say that I'd strongly advise against underestimating the value of this technique.
(FWIW - After ...
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The stability of the solar system is a classic real-life problem about time going to $+\infty$. It’s not an easy problem in which to understand the mathematics, not even close to the introductory level of $1/x$. Its pedagogical value lies in the problem statement being easy to understand and to relate to, and it is a problem to which people have devoted a ...
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Deeba and Rushkady Go to Town: A Fanciful Real-Life Story
A festival was just starting in town, and Deeba and Rushkady walked toward the square headed to the Infinite Pancake Eating Contest. The contest would end only when all participants agreed it was over. Deeba said, “Rushkady, since you always eat pancakes twice as a fast as me, no matter how fast or ...
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If you are willing to take some time to explain the model and do some simulation, I really like to show students a logistic growth model (in discrete time). The basic setup is something like the following:
Assume that there is a population of rabbits living in some area. We are going to make a few assumptions about this population:
the rabbits breed once ...
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