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"Everyone knows that nearly no engineers use calculus in their day job. Therefore, the purpose and measure of success isn't whether an engineer uses calculus in

"Everyone knows that nearly no engineers use calculus in their day job. Therefore, the purpose and measure of success isn't whether an engineer uses calculus in their day job. It is a great signal of technical problem solving ability, it is really powerful having a large pool of people who are trained in working with rates, areas and margins." I'm not saying that calculus is useless in an engineer's education - far from it. I'm arguing that knowing the algebraic tricks required to take the limit of (x^2)sin(2x)/sqrt(x) as x->0 is useless. It's just a display of the mastery of very intricate algebra, exactly the same as all the other displays of mastery of intricate algebra she has had to master and will continue to be required to master. What she actually needs is a solid understanding of the physical meaning of derivatives and integrals, how we arrive at them via numerical approximations, the fact that the velocity of a rigid body can be described as the integral over the forces that body is experiencing, etc. That connection is literally never made in calculus. The connection is of course made in the parallel calculus-based physics class she is taking. Which has its own problems, but is far superior as an introduction to calculus than the actual calculus class is.

RhysU 4y ago
> The connection is of course made in the parallel calculus-based physics class she is taking... The physicists as always (a) are pragmatically out-front but (b) don't worry about whether their tools are internally sound so long as they make useful predictions. Sounds about right. It is important for the aforementioned daughter to know this cultural difference between physicists and mathematicians if she is to be a good engineer. I believe this can only be learned firsthand if it is to be appreciated.
ivansavz 4y ago
It sounds like your daughter might enjoy reading my MATH+PHYS book. It covers mechanics and calculus in an integrated manner. I don't do all the proofs, so might not be a good fit for a proofs-based, course, but lots of illustrations and intuition. Preview here: https://minireference.com/static/excerpts/noBSmathphys_v5_preview.pdf https://minireference.com/static/excerpts/noBSmathphys_v5_pr... Concept map here: https://minireference.com/static/conceptmaps/math_and_physics_concepts.pdf https://minireference.com/static/conceptmaps/math_and_physic... Also be sure to check out the SymPy, in case you don't know already: https://minireference.com/static/tutorials/sympy_tutorial.pdf https://minireference.com/static/tutorials/sympy_tutorial.pd... very good for checking answers...
- whitepirate20 4y ago
  Looks great. I added it to my cart for a future purchase and recommended the linear algebra one to my friend who just started a new degree.
Chinjut 4y ago
I wouldn't describe it as a useless trick to know that sin(2x) ≈ 2x for small x. I'd describe this as a basic part of understanding what it means to say sin'(0) = 1.
- sokoloff 4y ago
  > sin'(0) = 1. = 0, not 1, unless I'm misunderstanding your notation (or blowing trig).
  - MikeDelta 4y ago
    The ' stands for (first-order) derivative. The second would be '' and so on. sin(0)=0, sin'(0)=1
    - sokoloff 4y ago
      Ah, thanks! I was thinking it was inverse (or typo); didn’t get to d sin(x)/x.
one1one1 4y ago
You don't need tricks to solve the limit of (x^2)sin(2x)/sqrt(x) as x->0, only to know equivalent infinitesimals, so that sin(2x) can be replaced by 2x in this limit and 2x^3/sqrt(x) can be simplified to 2x^(3-1/2) so the limit is zero. Moreover, you should know that x^2 and sin(2x) are really smaller compared to sqrt(x) so that the limit is zero. Knowing how to compare two infinitesimals can help you to replace a difficult process for an easier one, just like in differential equations sin(x) is replaced by x for small vibrations. That kind of knowledge allows you obtain an approximate solution in many real processes. So I don't consider that knowledge a bag of tricks. Furthermore, replacing sin(2x) for 2x is just the process of taking the linear approximation (the tangent line near x=0 for y=sin(2x)), so this is just an example of the power of calculus to replace a difficult question with an easier one. Also you learn to explore the world near a point (like a miscroscope) by the process of taking the linear approximation or taylor expansion of a function near a point, then you iterate this procedure to nearby points just like what happens when you solve a differential equation by Euler method. So I should say that there is a framework to solve problems via approximate solutions and computing limits is just a good first step to enter the realm of Calculus.
- one1one1 4y ago
  If your daughter is not going to learn that 1) x^2 is much smaller than sqrt(x) for x small. Or to know how 2) how to computer the tangent line of y=sin(2x), then I think she will not be able to appreciate how to approximate the solution for a problem and bound the error of that solution, and that are very useful skills. If the learns those two facts I think she will be able to solve that limit and appreciate how rules compose to apply your knowledge more broadly.

RhysU 4y ago

> The connection is of course made in the parallel calculus-based physics class she is taking... The physicists as always (a) are pragmatically out-front but (b) don't worry about whether their tools are internally sound so long as they make useful predictions. Sounds about right. It is important for the aforementioned daughter to know this cultural difference between physicists and mathematicians if she is to be a good engineer. I believe this can only be learned firsthand if it is to be appreciated.

ivansavz 4y ago

It sounds like your daughter might enjoy reading my MATH+PHYS book. It covers mechanics and calculus in an integrated manner. I don't do all the proofs, so might not be a good fit for a proofs-based, course, but lots of illustrations and intuition. Preview here: https://minireference.com/static/excerpts/noBSmathphys_v5_preview.pdf https://minireference.com/static/excerpts/noBSmathphys_v5_pr... Concept map here: https://minireference.com/static/conceptmaps/math_and_physics_concepts.pdf https://minireference.com/static/conceptmaps/math_and_physic... Also be sure to check out the SymPy, in case you don't know already: https://minireference.com/static/tutorials/sympy_tutorial.pdf https://minireference.com/static/tutorials/sympy_tutorial.pd... very good for checking answers...

whitepirate20 4y ago
Looks great. I added it to my cart for a future purchase and recommended the linear algebra one to my friend who just started a new degree.

Chinjut 4y ago

I wouldn't describe it as a useless trick to know that sin(2x) ≈ 2x for small x. I'd describe this as a basic part of understanding what it means to say sin'(0) = 1.

sokoloff 4y ago
> sin'(0) = 1. = 0, not 1, unless I'm misunderstanding your notation (or blowing trig).
- MikeDelta 4y ago
  The ' stands for (first-order) derivative. The second would be '' and so on. sin(0)=0, sin'(0)=1
  - sokoloff 4y ago
    Ah, thanks! I was thinking it was inverse (or typo); didn’t get to d sin(x)/x.

one1one1 4y ago

You don't need tricks to solve the limit of (x^2)sin(2x)/sqrt(x) as x->0, only to know equivalent infinitesimals, so that sin(2x) can be replaced by 2x in this limit and 2x^3/sqrt(x) can be simplified to 2x^(3-1/2) so the limit is zero. Moreover, you should know that x^2 and sin(2x) are really smaller compared to sqrt(x) so that the limit is zero. Knowing how to compare two infinitesimals can help you to replace a difficult process for an easier one, just like in differential equations sin(x) is replaced by x for small vibrations. That kind of knowledge allows you obtain an approximate solution in many real processes. So I don't consider that knowledge a bag of tricks. Furthermore, replacing sin(2x) for 2x is just the process of taking the linear approximation (the tangent line near x=0 for y=sin(2x)), so this is just an example of the power of calculus to replace a difficult question with an easier one. Also you learn to explore the world near a point (like a miscroscope) by the process of taking the linear approximation or taylor expansion of a function near a point, then you iterate this procedure to nearby points just like what happens when you solve a differential equation by Euler method. So I should say that there is a framework to solve problems via approximate solutions and computing limits is just a good first step to enter the realm of Calculus.

one1one1 4y ago
If your daughter is not going to learn that 1) x^2 is much smaller than sqrt(x) for x small. Or to know how 2) how to computer the tangent line of y=sin(2x), then I think she will not be able to appreciate how to approximate the solution for a problem and bound the error of that solution, and that are very useful skills. If the learns those two facts I think she will be able to solve that limit and appreciate how rules compose to apply your knowledge more broadly.