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_sources/differential-equations/slope-fields.ipynb

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_sources/getting-started/functions.ipynb

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_sources/getting-started/mathematics.ipynb

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"dy = 1 + 2*x\n",
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"print(dy)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "84566773-1eb4-4bc2-87a4-dbc88ea70d76",
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"metadata": {},
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"source": [
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"## Mathematical Functions\n",
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"\n",
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"All the standard mathematical functions are available in NumPy:\n",
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"\n",
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"| Function | NumPy Syntax |\n",
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"| :---: | :---: |\n",
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"| $\\sin(x)$ | `np.sin(x)` |\n",
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"| $\\cos(x)$ | `np.cos(x)` |\n",
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"| $\\tan(x)$ | `np.tan(x)` |\n",
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"| $\\tan^{-1}(x)$ | `np.arctan(x)` |\n",
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"| $e^x$ | `np.exp(x)` |\n",
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"| $\\ln(x)$ | `np.log(x)` |\n",
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"| $\\log_{10}(x)$ | `np.log10(x)` |\n",
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"| $\\sqrt{x}$ | `np.sqrt(x)` |\n",
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"\n",
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"Let's compute some examples:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 22,
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"id": "f2878313-b169-43f9-9c5f-bb59a66711a5",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.7071067811865476"
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]
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},
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"execution_count": 22,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"np.cos(np.pi/4)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 23,
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"id": "8a283317-4c79-48ad-98d4-aa937348bf45",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.7071067811865475"
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]
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},
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"execution_count": 23,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"1/np.sqrt(2)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 24,
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"id": "edd24472-840f-4e87-9a16-361dd7069e2d",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.6931471805599453"
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]
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},
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"execution_count": 24,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"np.log(2)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 26,
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"id": "a7360663-65c5-4377-b2f2-4d41813baa35",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.7853981633974483"
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]
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},
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"execution_count": 26,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"np.arctan(1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 27,
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"id": "8f2fcd44-5770-4ce2-ad0a-ae86bc742c99",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.3010299956639812"
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]
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},
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"execution_count": 27,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"np.log10(2)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 28,
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"id": "80592a00-61ad-4473-a2e0-cefb431ecbcd",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"2.0"
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]
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},
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"execution_count": 28,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"10**0.3010299956639812"
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]
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},
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{
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"cell_type": "markdown",
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"id": "90b77e2c-8c79-4d07-a9c7-790990db361b",
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"metadata": {},
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"source": [
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"## Custom Functions\n",
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"\n",
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"Create custom functions using the `lambda` keyword. For example, let's create the function:\n",
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"\n",
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"$$\n",
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"f(x) = \\frac{1}{1 + x^2}\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 29,
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"id": "5033800c-5425-48fc-a9fe-520f4744e578",
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"f = lambda x: 1/(1 + x**2)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "cb6f8769-4731-44fa-a444-83f12fbf6316",
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"metadata": {},
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"source": [
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"Compute some values of $f(x)$:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 30,
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"id": "4ac74bd9-130c-4d93-b35d-44e7b391376c",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1.0"
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]
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},
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"execution_count": 30,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"f(0)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 31,
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"id": "57c4bd22-3064-4546-a400-dcbdab424d89",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.5"
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]
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},
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"execution_count": 31,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"f(1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 32,
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"id": "8cabfb40-2d2c-46ea-8154-5e57445eb0c8",
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.2"
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]
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},
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"execution_count": 32,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"f(2)"
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]
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}
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],
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"metadata": {

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