The Definite Integral as a Limit(1966)
\n", "\n", "Please watch the video above. Can you find a better alternative -- poor video is getting old...
\n", "''')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "1. Use our formulas from class to evaluate the following sums:\n", "\n", " 1. $$\\sum_{i = 1}^{20} 100(i^2 - 5i + 1)$$\n", " \n", " 2. $$\\sum_{i = 1}^{50} (i^2 - 10i)$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "2. Use Riemann sums to approximate the area under the curve for the given function on given interval:\n", "\n", " 1. $n = 4$ for $f(x) = \\frac{1}{x-1}$ on $[2, 4]$\n", " \n", " 2. $n = 4$ for $f(x) = \\cos(\\pi x)$ on $[0, 2\\pi]$.\n", " \n", " 3. $n = 8$ for $f(x) = x^2 - 2x + 1$ on $[0, 2]$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "3. Let $r_j$ denote the total rainfall in New York City on the $j$th day of the year in 2020. Interpret \n", "\n", "$$\\sum_{j = 1}^{31} r_j$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "4. Below is the data from our Central Park weather station. We have isolated the Precipiation column, and plotted both it and the total precipitation for the month of January in the year 2020. How could we make the second plot using the first? (**HINT**: Look up what the `.cumsum()` method does in Pandas, or just think about what we're doing in class!)\n", "\n", "| DATE | PRCP | |
|---|---|---|
| 0 | 1870-01-01 | 0.08 |
| 1 | 1870-01-02 | 1.18 |
| 2 | 1870-01-03 | 0.00 |
| 3 | 1870-01-04 | 0.00 |
| 4 | 1870-01-05 | 0.00 |
\n",
" 