I've liked the previous posts from betterexplained, but this one seems to be extremely unintuitive to me. I was teaching this recently, and the simpler idea that e is the function which is its own derivative is a much simpler way of expressing a meaningful truth about it, and following it through, leads you to exactly the same limit as a formal definition. It also tails very neatly into why exponentials are used in physics for decay curves. That said, obvious to me and one student is not necessarily obvious to everyone! Anyone else have an opinion on this?
Your explanation appears more elegant, but it isn't simpler. I as an algebra student can grasp nearly all of the lesson from betterexplained. I haven't yet learned derivatives, so your explanation doesn't much sense. I very much appreciate the lower knowledge level explanation, as all my teacher would say when we covered natural log was that e was a magic number used in growth equations.
Well, for me this post is the only one that made sence and thanks to it I now have a clear understanding what e is. "e is the function which is its own derivative" is way less intuitive.
Perhaps this would be more intuitive if shown graphically. Something like; draw a curve where the gradient is the value. As the value goes up, so does the gradient, so does the value and so on. What you have drawn, e^x is the function where the value is the gradient - exponential growth.
Thanks for this - this is exactly how you would say it to someone who doesn't know what a derivative is. Apologies for assuming anyone interested in this would know calculus - poor form indeed! Explaining in terms of derivatives definitely isn't simple if you don't know what a derivative is.
Thanks for the replies. For those who like the betterexplained explanation:
1) Did you understand the bit with the limits, or did you find the thing made enough sense without that? I would have thought that would be harder than introductory calculus, but if not, I may start using it.
2) Do you think it made it easier to get to the point parallel described? I ask because regardless of other interpretations, my students need to understand what e means in calculus.
1) I'm probably missing some truths by not knowing how limits work, but his use of limits didn't impede my understanding of the larger point. He quickly moves each time from an equation with limits to a more basic algebra expression.
2) I understand what parallel is saying after looking up "gradient"..
Comments
I've liked the previous posts from betterexplained, but this one seems to be extremely unintuitive to me. I was teaching this recently, and the simpler idea that e is the function which is its own derivative is a much simpler way of expressing a meaningful truth about it, and following it through, leads you to exactly the same limit as a formal definition. It also tails very neatly into why exponentials are used in physics for decay curves. That said, obvious to me and one student is not necessarily obvious to everyone! Anyone else have an opinion on this?
Your explanation appears more elegant, but it isn't simpler. I as an algebra student can grasp nearly all of the lesson from betterexplained. I haven't yet learned derivatives, so your explanation doesn't much sense. I very much appreciate the lower knowledge level explanation, as all my teacher would say when we covered natural log was that e was a magic number used in growth equations.
Well, for me this post is the only one that made sence and thanks to it I now have a clear understanding what e is. "e is the function which is its own derivative" is way less intuitive.
Perhaps this would be more intuitive if shown graphically. Something like; draw a curve where the gradient is the value. As the value goes up, so does the gradient, so does the value and so on. What you have drawn, e^x is the function where the value is the gradient - exponential growth.
Thanks for this - this is exactly how you would say it to someone who doesn't know what a derivative is. Apologies for assuming anyone interested in this would know calculus - poor form indeed! Explaining in terms of derivatives definitely isn't simple if you don't know what a derivative is.
That sounds pretty damn intuitive. Just imagining the drawing you're describing has given me some insight.
Thanks for the replies. For those who like the betterexplained explanation:
1) Did you understand the bit with the limits, or did you find the thing made enough sense without that? I would have thought that would be harder than introductory calculus, but if not, I may start using it.
2) Do you think it made it easier to get to the point parallel described? I ask because regardless of other interpretations, my students need to understand what e means in calculus.
Thanks again!
1) I'm probably missing some truths by not knowing how limits work, but his use of limits didn't impede my understanding of the larger point. He quickly moves each time from an equation with limits to a more basic algebra expression.
2) I understand what parallel is saying after looking up "gradient"..