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teaching:topics:calculus:notation [2024/05/10 10:44] simon [Leibniz] |
teaching:topics:calculus:notation [2024/05/10 10:51] simon [Leibniz] |
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\[\frac{\rm d}{{\rm d}t}(4t^2+3)\] | \[\frac{\rm d}{{\rm d}t}(4t^2+3)\] | ||
is between the values of the expression and the variable. | is between the values of the expression and the variable. | ||
- | \begin{align*}%% | + | \[\text{repeated:}\quad\frac{{\rm d}^2}{{\rm d}x^2}(3x^3+5)\qquad\text{means:}\quad\frac{\rm d}{{\rm d}x}\left(\frac{{\rm d}}{{\rm d}x}(3x^3+5)\right).\] |
- | \text{repeated:}\quad\frac{{\rm d}^2}{{\rm d}x^2}(3x^3+5) &&& \text{means:}\quad\frac{\rm d}{{\rm d}x}\left(\frac{{\rm d}}{{\rm d}x}(3x^3+5)\right) \\ \\ | + | \[\int_a^b\!x^2 + 4x\,{\rm d}x\] |
- | \int_a^b\!x^2 + 4x\,{\rm d}x &&& \text{is the definite integral of the expression taken from }a\text{ to }b | + | is the definite integral of the expression taken from `a` to `b`. The sum of the infinitessimal slices under that curve. |
- | %%\end{align*} | + | |
The kind of calculus that we deal with at school treats each of these as a single symbol, but it does | The kind of calculus that we deal with at school treats each of these as a single symbol, but it does | ||
make sense to break them down into their parts, considering say \(\ {\rm d}x\ \) on its own. Doing so | make sense to break them down into their parts, considering say \(\ {\rm d}x\ \) on its own. Doing so |