MathJax: \sin
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Martin Sleziak
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I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $sin(\alpha)$$\sin(\alpha)$ as the ratio of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $sin(\alpha)$ as the ratio of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $\sin(\alpha)$ as the ratio of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

deleted 1 characters in body
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Alex B.
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I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $sin(\alpha)$ as the rationratio of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $sin(\alpha)$ as the ration of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $sin(\alpha)$ as the ratio of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

added 368 characters in body
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Alex B.
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I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $sin(\alpha)$ as the ration of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

I agree with Mark that your reservation against using "advanced" notions such as limits and derivative seem to be more applicable to high school mathematics, than to the undergraduate syllabus, since limits should arguably be one of the first things maths students learn at university. In particular, it's not clear to me how you would have defined the real numbers without recourse to limits.

But to address the actual question regardless of its motivation, what exactly are the drawbacks in starting with option 1? You can then prove addition laws for sin and cos using one of the various proofs by picture (which I think are actually really pretty). From there, you can derive the derivative of the functions (once you get thus far in your syllabus) using the definition of the derivative and the addition laws. Then, once Taylor series are available to you, you derive the power series and finally the connection to the exponential.

As for the exponential itself, it seems to me that any introduction of the number $e$ that avoids differentiation and integration will be extremely unmotivated and not very illuminating.

Edit: my favourite introduction to the exponential function goes as follows (after having defined the derivative): having differentiated polynomials, one gets the natural urge to differentiate something like $2^x$. If one goes through the definition of the derivative, one gets as an answer some limit (if it exists) times the function itself. Repeating this with $3^x$ gives the same result, but a different limit. A natural question then is: can we choose a base to make this limit 1, so that the derivative of the function gives you the function back? If you go through the algebra, you will arrive at the number $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$.

Edit 2: To address the recent edit of the question: I don't think that using the 1st method is cheating. You can introduce $sin(\alpha)$ as the ration of the side opposite to the angle $\alpha$ to the hypotenuse in a right-angled triangle. They already know from high school that this is well defined and I don't think that that proof uses disguised limits either.

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Alex B.
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Alex B.
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