teaching:topics:calculus:limits
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teaching:topics:calculus:limits [2020/09/13 18:53] simon |
teaching:topics:calculus:limits [2022/07/26 13:19] simon |
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| \(\sum\limits_{k=1}^{5}\) called "sum" which means adding together each of the values of the sequence made by setting (in this example) `k` to each integer from 1 to 5 in the expression (or function) after it ... | \(\sum\limits_{k=1}^{5}\) called "sum" which means adding together each of the values of the sequence made by setting (in this example) `k` to each integer from 1 to 5 in the expression (or function) after it ... | ||
| \[W_5 = \sum_{k=1}^{5}\frac{200}{2^k}\] | \[W_5 = \sum_{k=1}^{5}\frac{200}{2^k}\] | ||
| - | We are using this new variable, here `k`, like we use the index or counter in a computer program. Now we can talk about distance travelled after counting off `n` intervals in the wolf's pursuit ... | + | We are using this new variable, here `k`, like we use the index or counter in procedural computer programs. Now we can talk about distance travelled after counting off `n` intervals in the wolf's pursuit ... |
| \[W_n = \sum_{k=1}^n\frac{200}{2^k}\] | \[W_n = \sum_{k=1}^n\frac{200}{2^k}\] | ||
| Lastly we want to be able to write the sum if we kept adding up and somehow (?!?) included **all** of the infinite number of smaller and smaller distances: | Lastly we want to be able to write the sum if we kept adding up and somehow (?!?) included **all** of the infinite number of smaller and smaller distances: | ||
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| What do we mean when we write \(\quad\lim\limits_{x\to n}{\rm f}(x)\) ? | What do we mean when we write \(\quad\lim\limits_{x\to n}{\rm f}(x)\) ? | ||
| - | Then ... what does it mean to write \(\quad\lim\limits_{x\to\infty}{\rm f}(x)\) ? | + | Then ... what does it mean to write \(\quad\lim\limits_{x\to\infty}\,{\rm f}(x)\) ? |
| Now ... try to graph the function: \(\displaystyle\quad {\rm f}(x) = \frac{x^2-a^2}{x-a},\quad\) say when `\ a = 2` | Now ... try to graph the function: \(\displaystyle\quad {\rm f}(x) = \frac{x^2-a^2}{x-a},\quad\) say when `\ a = 2` | ||
teaching/topics/calculus/limits.txt ยท Last modified: 2022/07/26 13:20 by simon
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