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Q. | The advice this program gives is very vague. Why should I have to do it for you?!
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A. | A fair cop! This program is for learning to use some diagrams, which are meant to help you understand the process of working backwards through the product-rule and the chain-rule. If it helps you in this way as it's meant to, you may not mind being required to make many of the crucial decisions. However, if you do mind, then of course the program ought ideally to be able to make these decisions. And it will acquire this ability as it develops. There is an abundance of excellent software online - for example at Wolfram - that will provide a complete written working of any given calculus problem. When our program is finished, it will have a similar power, added to its own unique advantages! | ||
| Which are? |
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A. | People who prefer these diagrams to a normal written working find that they can plan, revise, navigate and check much more easily - adding up to a greater overall feeling of understanding. | ||
| What's so wrong with the formulas? |
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A. | They're powerful vehicles for driving backwards through the product-rule and chain-rule. But some people feel the route to to be a bit dark and treacherous. Taking a map can help! Also, some people are bothered that the dy/dx notation takes them on short-cuts they don't understand, and consequently find hard to navigate. Having said that... of course the aim of the formulas is broadly the same, and some people may find them more natural - especially if they generally prefer words to pictures. | ||
| Why not just draw the map, then? Why try and be high-tec? |
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A. | We couldn't agree more! The Wizards are purely to get you
started drawing the diagrams. But the free-form drawing panel
is an experiment to see if such a thing has any potential as
an adjunct to written work. See what you think, and let us know.
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Q. | But I'm not going to draw one of these diagrams in an exam,
am I? At least, you'll need to show me how to convert it into a normal working.
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A. | We think the diagram should be fine, actually -
unless a question asks very specifically for you to show your
working in a particular style. (We'd be
very keen to air views on this matter, in the users'
Forum.) Make
clear, of course, that a straight line is to be read as
differentiating or integrating (down or up) with respect to the
main variable, and the dotted line similarly with respect to the
dashed balloon.
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Q. | But how can it possibly benefit
my students to be afraid of using ∫ and differentials like dx and dy?
Obviously you can always avoid them in differentiation and
therefore in anti-differentiation if you want to, but why should you want to? Don't
you see their importance in applying calculus to physical or geometric
problems involving infinitessimals, and don't you see why they
are so loved for negotiating useful substitutions in anti-differentiation?
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A. |
I think I do see why, but I think doing without them in this particular method of anti-differentiation won't make me or anybody any less ready to use them in other, related processes. Try it, and see. Also, consider the positive benefit that we're trying to bring... just an overview of the process, so that new-comers can better understand the strategy behind useful substitutions. It just happens that if you make the overview clear, the differentials are (surprisingly) dispensible. But - your views on this always welcome at the Forum or at tom@ballooncalculus.org. | ||
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