Agreed. I learned calculus at a small liberal art college. First, we learned about functions, then limits, then derivatives, then integrals, then the fundamental theorem. Derivatives were explained as the limit of the slope of a function at a single point. Integrals were the limit of the area under a function. It seemed very intuitive to me.
I never took a trigonometry class, but didn’t really need it for calculus, except for trying to figure out what the various trig functions actually were. I think many students would benefit from learning calculus before trig.
I’ve recently read articles about a study showing that women are significantly more likely than men to be discouraged from pursuing STEM majors after attempting Calculus I and feeling they don’t understand it—an issue of low confidence.
Think about building a shape with big blocks, then building it with smaller and smaller blocks. Calculus is just a tool for thinking about that process with infinitely small blocks.
Never mind. If you had set some target gap between your understanding and theirs, let’s call it epsilon, then no matter how narrow that target was I’m sure there would have come a time when the gap was smaller than it.
;) I see what you did there. In all reality, the way calculus is taught in university is similar to the way trig, algebra, etc. are taught in high school. All are relatively intuitive and can be mastered through simply practicing.
After all, pretty much every college student in the world takes some level of calculus, so it can’t be too crazy.
But to do that, he’d have to have had some way of calculating the rate of change the rest of the class’ learning was experiencing, and . . . well . . .
Have any of you used calculus outside of school? I havent. Even working as a wireless engineer, I just had formulas to use instead of doing actual calculus.
Maybe not 50% of students but if you download the degrees awarded more than 50% of graduates should require calculus, obviously I don’t know the degree requirements.
That is a bold statement and one not yet proven. I will agree that Leibniz published before Newton, but that’s not quite the same thing as inventing it.
When I took calculus, the teacher read us the following poem on the first day of the class. In a fairly simple image, it encapsulates the concept of infinitesimals, fundamental to calculus. I’ve never forgotten it: “I know that little-bits are small. But a lot of little-bits is ALL.”
*“Will you have some pie?”
Said Jane. Said I,
“Well just a little. Just a bit.”
But I found when I had eaten it
That just one little-bit wouldn’t do.
So I told Jane to make it two.
Then was I happy with what I got?
Well, little-bits can’t make a lot.
For little-bits are small, you see.
So I told Jane to make it three.
Three little-bits are not much more
Than two. So I said, "Make it four.
And I ate them up. Then asked for five.
Then six. Till Jane said “Sakes alive,
Here are TWO MORE and that makes .
If you don’t STOP you’ll eat the plate!”
“Eight little-bits,” I said are fine.
But would you care to make it nine?"
Said Jane, "I’d make it forty-four.
But sad to say, there are no more.
By little-bit and little-bit
You’ve eaten all there was of it.
I know that little-bits are small.
But a lot of little-bits is ALL.
And little by little by little, you see,
Gets down to NONE at all for me!
That’s why I hope that when I bake
Another pie, you will just take
One great big fat thick lot of it,
And let me have a little-bit!"*
I resort to the Laplace transforms (s-domain analysis) often in circuit design but I don’t call it solving a differential equation. I just call it cheating. Last time I solved a differential equation was in Math 315 back at the university.
I see what you did there. In all reality, the way calculus is taught in university is similar to the way trig, algebra, etc. are taught in high school. All are relatively intuitive and can be mastered through simply practicing.