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Here's another exercise (resp. exam question) that tests understanding: given a sketch of a curve in a graph, roughly sketch the derivative (or integral). The n

Here's another exercise (resp. exam question) that tests understanding: given a sketch of a curve in a graph, roughly sketch the derivative (or integral). The number of otherwise good students who go "but I can't do the derivative without the formula?" suggests we need more questions like this.

xnorswap 4y ago
Also a good question to test intuition is being asked to sketch e^sin(x) and sin(e^x).
- bheadmaster 4y ago
  Intuition tells me e^sin(x) would look similar to an ordinary sinusiod, except its range would be between e^-1 and e, and its shape would not be smooth as a sinusiod. I have no idea what the shape would look like, and I'm a visual learner when it comes to mathematics. I think most of these questions are not measuring intuition per se, but rather has the tested person previously seen such functions plotted on a graph. Either that, or my mathematical intuition has got rusty from years of code monkeying.
  - soVeryTired 4y ago
    It wouldn’t look too dissimilar to a standard sinusoid, just shifted upwards a bit. In the range [-1, 1], e^x is reasonably similar to 1 + x.
  - madcaptenor 4y ago
    You're basically right. Here's the graph: https://www.desmos.com/calculator/lf3xerswdy https://www.desmos.com/calculator/lf3xerswdy
    - bheadmaster 4y ago
      Ah, now it makes sense. The slope of the sine is "squashed" down by the x->e^x mapping. Perhaps (regularly) seeing functions plotted on graphs is a necessary precondition to maintain intuition :)
- jhart99 4y ago
  Seems a little much for a regular calculus class. Wouldn't that fit better in a complex analysis class?
  - jetunsaure 4y ago
    No, this would be a good exercise for first-year calculus. It's a very reasonable question, nothing out of the ordinary.
    - KMag 4y ago
      In fact, I was taught the graphical intuition via a few days of these sorts of exercises, before being introduced to the formulas. It worked really well, at least for 9th grade students in an acceleration program at the University of Minnesota (UMTYMP).
- anthk 4y ago
  TCL, package require math, a few lines of puts writting to a file and then gnuplot that.
  - yberreby 4y ago
    How does that test intuition?
  - EForEndeavour 4y ago
    That's one of many ways to check one's answer. The whole point is to be able to intuit a rough sketch of what the graph should be, not to use software tools to build a pixel-perfect render of the plot.
- h4ch1 4y ago
  I really enjoy https://graphtoy.com/ https://graphtoy.com/ by the great Inigo Quilez to quickly verify my mental models for mathematical equations
  - Timon3 4y ago
    Thank you for sharing that! I usually go with Desmos, but for a quick plot this seems even nicer, especially given the time coordinate!
bluepod4 4y ago
I recall a similar exercise on my AP Calc exam from years ago. Instead of sketching the derivative based on the graph of a function, we had to sketch the function based on a table of data which described the function as well as its first and second derivatives in terms of value, existence, and sign at various points and intervals.
amelius 4y ago
The problem is that most teachers cannot come up with questions that go outside the small number of cases which the student trains on. I mean, coming up with fundamentally new questions is very hard work once the low hanging fruit is gone.
- red_admiral 4y ago
  That or the point of the class is to prepare for a test set by someone other than the teacher (such as in UK A-levels), where you know the questions will mostly be of the form "derivative of 3x^2-5x+1" or similar.
- AlecSchueler 4y ago
  Sounds like something ChatGPT could do quite quickly though.
  - petschge 4y ago
    Quickly maybe. But to do it well you need a deep understanding of the material which is something that large language models lack.
    - whoami_-h 4y ago
      [dead]
pavlov 4y ago
I learned basics of calculus as a teenager making animations with a pirated copy of Adobe After Effects. It’s a motion graphics package where you can animate any property of an element over time using both absolute values and velocity curves (i.e. the derivative). It shows both curves next to each other, so you can tweak either and see how the other changes. Seeing graphics animate according to a derivative that you just plotted yourself is really useful to develop practical intuition about what it means. After Effects is too expensive and complex for high schools, but maybe some kind of modern Logo-style environment that combines coding and animation could be useful for calculus beginners. (And linear algebra too — another field where the basics have a direct intuitive application in computer graphics.)
- irchans 4y ago
  This is a good idea. If I ever teach basic calc again, I will use it. Thanks! (The last time I taught math was vector calc 4 years ago. The last time I taught the first semester intro to calc was 30 years ago OMG-LOL.)
- mrguyorama 4y ago
  Geogebra and friends are more powerful and open-ended, education focused systems. I played around with it a bit in freshman year Geometry.
angry_moose 4y ago
I remember there being a distinct lack of "closing the loop" on concepts in university math. Day 1 of the class: The derivative calculates the slope of a function Day 2: The integral calculates the area under the curve of the function Days 3-89: Rote exercises deriving and integrating increasingly obscure functions Day 90: Final Exam Spending a few days at the end re-exploring the "big picture day-1" to tie together all of the various strands of knowledge you accumulate over the semester would have made all of it so much more effective.
- nerdponx 4y ago
  This, bigtime. I loved the 3blue1brown series an example of how to do this better.
- deanCommie 4y ago
  It's interesting because I was going to argue with you until I applied the same idea to a subject I couldn't care less about if I tried in school, but now think is deeply fascinating: History. See, I loved math. So all through Calculus I could easily remember the big picture. Every time I practiced an Integral I imagined curves and calculating the areas under them and the visual problem and the relevance of what the curve represented in real life and the massive amount of applications it could be used for in the real world lit up my neurons like fireworks in the sky. (Then I went into computer science, and wound up never needing calclus again, based on the type of work I happen to be doing, but alas) But where I really would have wished a constant reinforcement of the "big picture" is History. Cuz it always seemed so pointless and useless. Why are we studying these old dead people, and everything they did. Who cares? They're old, and they're dead, and nothing they did matters to us anymore. Until you grow up, and go from your 20s to your 40s, and suddenly realize oh shit we're living THROUGH history. We're creating history NOW. We're making choices, and we're making mistakes just like those old dead people in history. Old dead people that weren't really any less developed or evolved primates than us. Just equally victims of their circumstance like us, and also agents of change like us. Suddenly history seems much more significant.
  - postsantum 4y ago
ThrowawayTestr 4y ago
Sketching equations and their derivatives was how we started pre-calc in high school
andruby 4y ago
My school teacher asked us this exact exercise several times. He always made sure to link abstract concepts to real applications, as well as showing us how the mathematical concepts were discovered. (note: this was not in the US, but in the early 2000's in a small European country)
deleted 4y ago
[deleted]
Paul-Craft 4y ago
Don't people do this routinely anymore? I remember routinely doing "sketch the derivative" type exercises in calc class pretty frequently.
Spivak 4y ago
This style question is actually on the AP calc exams. When I came across them I thought they were cute.

xnorswap 4y ago

Also a good question to test intuition is being asked to sketch e^sin(x) and sin(e^x).

bheadmaster 4y ago
Intuition tells me e^sin(x) would look similar to an ordinary sinusiod, except its range would be between e^-1 and e, and its shape would not be smooth as a sinusiod. I have no idea what the shape would look like, and I'm a visual learner when it comes to mathematics. I think most of these questions are not measuring intuition per se, but rather has the tested person previously seen such functions plotted on a graph. Either that, or my mathematical intuition has got rusty from years of code monkeying.
- soVeryTired 4y ago
  It wouldn’t look too dissimilar to a standard sinusoid, just shifted upwards a bit. In the range [-1, 1], e^x is reasonably similar to 1 + x.
- madcaptenor 4y ago
  You're basically right. Here's the graph: https://www.desmos.com/calculator/lf3xerswdy https://www.desmos.com/calculator/lf3xerswdy
  - bheadmaster 4y ago
    Ah, now it makes sense. The slope of the sine is "squashed" down by the x->e^x mapping. Perhaps (regularly) seeing functions plotted on graphs is a necessary precondition to maintain intuition :)
jhart99 4y ago
Seems a little much for a regular calculus class. Wouldn't that fit better in a complex analysis class?
- jetunsaure 4y ago
  No, this would be a good exercise for first-year calculus. It's a very reasonable question, nothing out of the ordinary.
  - KMag 4y ago
    In fact, I was taught the graphical intuition via a few days of these sorts of exercises, before being introduced to the formulas. It worked really well, at least for 9th grade students in an acceleration program at the University of Minnesota (UMTYMP).
anthk 4y ago
TCL, package require math, a few lines of puts writting to a file and then gnuplot that.
- yberreby 4y ago
  How does that test intuition?
- EForEndeavour 4y ago
  That's one of many ways to check one's answer. The whole point is to be able to intuit a rough sketch of what the graph should be, not to use software tools to build a pixel-perfect render of the plot.
h4ch1 4y ago
I really enjoy https://graphtoy.com/ https://graphtoy.com/ by the great Inigo Quilez to quickly verify my mental models for mathematical equations
- Timon3 4y ago
  Thank you for sharing that! I usually go with Desmos, but for a quick plot this seems even nicer, especially given the time coordinate!

bluepod4 4y ago

I recall a similar exercise on my AP Calc exam from years ago. Instead of sketching the derivative based on the graph of a function, we had to sketch the function based on a table of data which described the function as well as its first and second derivatives in terms of value, existence, and sign at various points and intervals.

amelius 4y ago

The problem is that most teachers cannot come up with questions that go outside the small number of cases which the student trains on. I mean, coming up with fundamentally new questions is very hard work once the low hanging fruit is gone.

red_admiral 4y ago
That or the point of the class is to prepare for a test set by someone other than the teacher (such as in UK A-levels), where you know the questions will mostly be of the form "derivative of 3x^2-5x+1" or similar.
AlecSchueler 4y ago
Sounds like something ChatGPT could do quite quickly though.
- petschge 4y ago
  Quickly maybe. But to do it well you need a deep understanding of the material which is something that large language models lack.
  - whoami_-h 4y ago
    [dead]

pavlov 4y ago

I learned basics of calculus as a teenager making animations with a pirated copy of Adobe After Effects. It’s a motion graphics package where you can animate any property of an element over time using both absolute values and velocity curves (i.e. the derivative). It shows both curves next to each other, so you can tweak either and see how the other changes. Seeing graphics animate according to a derivative that you just plotted yourself is really useful to develop practical intuition about what it means. After Effects is too expensive and complex for high schools, but maybe some kind of modern Logo-style environment that combines coding and animation could be useful for calculus beginners. (And linear algebra too — another field where the basics have a direct intuitive application in computer graphics.)

irchans 4y ago
This is a good idea. If I ever teach basic calc again, I will use it. Thanks! (The last time I taught math was vector calc 4 years ago. The last time I taught the first semester intro to calc was 30 years ago OMG-LOL.)
mrguyorama 4y ago
Geogebra and friends are more powerful and open-ended, education focused systems. I played around with it a bit in freshman year Geometry.

angry_moose 4y ago

I remember there being a distinct lack of "closing the loop" on concepts in university math. Day 1 of the class: The derivative calculates the slope of a function Day 2: The integral calculates the area under the curve of the function Days 3-89: Rote exercises deriving and integrating increasingly obscure functions Day 90: Final Exam Spending a few days at the end re-exploring the "big picture day-1" to tie together all of the various strands of knowledge you accumulate over the semester would have made all of it so much more effective.

nerdponx 4y ago
This, bigtime. I loved the 3blue1brown series an example of how to do this better.
deanCommie 4y ago
It's interesting because I was going to argue with you until I applied the same idea to a subject I couldn't care less about if I tried in school, but now think is deeply fascinating: History. See, I loved math. So all through Calculus I could easily remember the big picture. Every time I practiced an Integral I imagined curves and calculating the areas under them and the visual problem and the relevance of what the curve represented in real life and the massive amount of applications it could be used for in the real world lit up my neurons like fireworks in the sky. (Then I went into computer science, and wound up never needing calclus again, based on the type of work I happen to be doing, but alas) But where I really would have wished a constant reinforcement of the "big picture" is History. Cuz it always seemed so pointless and useless. Why are we studying these old dead people, and everything they did. Who cares? They're old, and they're dead, and nothing they did matters to us anymore. Until you grow up, and go from your 20s to your 40s, and suddenly realize oh shit we're living THROUGH history. We're creating history NOW. We're making choices, and we're making mistakes just like those old dead people in history. Old dead people that weren't really any less developed or evolved primates than us. Just equally victims of their circumstance like us, and also agents of change like us. Suddenly history seems much more significant.
- postsantum 4y ago

ThrowawayTestr 4y ago

Sketching equations and their derivatives was how we started pre-calc in high school

andruby 4y ago

My school teacher asked us this exact exercise several times. He always made sure to link abstract concepts to real applications, as well as showing us how the mathematical concepts were discovered. (note: this was not in the US, but in the early 2000's in a small European country)

deleted 4y ago

[deleted]

Paul-Craft 4y ago

Don't people do this routinely anymore? I remember routinely doing "sketch the derivative" type exercises in calc class pretty frequently.

Spivak 4y ago

This style question is actually on the AP calc exams. When I came across them I thought they were cute.