Added the notes for the session 5

session04/4. Recap exercises.ipynb

@@ -1 +1 @@
-{"cells":[{"cell_type":"markdown","metadata":{"executionInfo":{"elapsed":1297,"status":"ok","timestamp":1676635374594,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"UCsbknx_PJBW","outputId":"60405cf4-8ad8-4b65-ccc9-f2941e9eef2b"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"KaV3ZOBkYEXs","outputId":"c4273427-fbf4-4f4e-f63e-dfbac077c187"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{c \\left(x^{5} + 1\\right)}{5}$"],"text/plain":["c*(x**5 + 1)/5"]},"execution_count":18,"metadata":{},"output_type":"execute_result"}],"source":["import sympy as sp\n","\n","x, c, u = sp.symbols('x c u')\n","\n","sp.simplify(sp.integrate(c*u**4, (u, -1, x)))"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"d2nsEQXFXrls"},"outputs":[],"source":["def f(u):\n","    return c*u**4"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"23GYLoXAXrlt","outputId":"015ac4ef-64c0-401d-9a8b-38ce373ffdf3"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{c \\left(x^{5} + 1\\right)}{5}$"],"text/plain":["c*(x**5 + 1)/5"]},"execution_count":20,"metadata":{},"output_type":"execute_result"}],"source":["sp.simplify(sp.integrate(f(u), (u, -1, x)))"]},{"cell_type":"markdown","metadata":{"id":"mJdI3ZtAXrlt"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"executionInfo":{"elapsed":411,"status":"ok","timestamp":1676635469952,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"yB-AWtjoQa4A","outputId":"2735032d-f670-4ca4-abcf-42167142bb95"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{5}{2}$"],"text/plain":["5/2"]},"execution_count":23,"metadata":{},"output_type":"execute_result"}],"source":["sp.solve(sp.integrate(c*x**4, (x, -1, 1)) -1, c)[0]"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"T6t0kAsfXrlu","outputId":"9cb48181-3a8e-46c6-8045-674d9a107e11"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{5}{2}$"],"text/plain":["5/2"]},"execution_count":22,"metadata":{},"output_type":"execute_result"}],"source":["sp.solve(sp.integrate(f(u), (u, -1, 1)) -1, c)[0]"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"XbUS5tSfRw_C"},"outputs":[],"source":["from sympy import Rational as R\n","def f(x):\n","    return R(5,2) * x**4"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"7lCOtWEpUG-e"},"outputs":[],"source":["def F(x):\n","    return R(1,2) *(x**5 + 1)"]},{"cell_type":"markdown","metadata":{"id":"MgAvo6TWXrlv"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"6aFfufYJXrlw","outputId":"627f25ad-6d81-4ea2-c595-7033aad8ee3d"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{1}{32}$"],"text/plain":["1/32"]},"execution_count":31,"metadata":{},"output_type":"execute_result"}],"source":["F(R(1,2)) - F(R(-1,2))"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"aE0-0Rm6Xrlw","outputId":"e87d515a-a30e-4cae-a1d2-5b3a95016f75"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{1}{32}$"],"text/plain":["1/32"]},"execution_count":32,"metadata":{},"output_type":"execute_result"}],"source":["sp.integrate(f(x), (x, R(-1,2), R(1,2)))"]},{"cell_type":"markdown","metadata":{"id":"0KRF2d0gXrlw"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"1ESma27RXrlx","outputId":"909b8e63-e6a1-45ee-ad12-8ace2b800d39"},"outputs":[{"data":{"text/latex":["$\\displaystyle 0$"],"text/plain":["0"]},"execution_count":34,"metadata":{},"output_type":"execute_result"}],"source":["EX = sp.integrate(x*f(x), (x, -1, 1))\n","EX"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"oALp_12jXrlx","outputId":"b79b6402-7ca8-4c86-b827-9a69a87f6fc7"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{5}{7}$"],"text/plain":["5/7"]},"execution_count":35,"metadata":{},"output_type":"execute_result"}],"source":["sp.integrate(x**2 * f(x), (x, -1, 1)) - EX**2"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"vLJ5BAlCK8GH"},"outputs":[],"source":["def F(x):\n","  return R(1,2)*((x**5)+1)"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":51},"executionInfo":{"elapsed":463,"status":"ok","timestamp":1677506667902,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"jXrP9eIULKqF","outputId":"de1bd990-7c2d-4a7d-afff-3782611ebbbc"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{1}{32}$"],"text/plain":["1/32"]},"execution_count":17,"metadata":{},"output_type":"execute_result"}],"source":["F(R(1,2)) - F(R(-1, 2))"]},{"cell_type":"markdown","metadata":{"id":"kSV2LHcRXrly"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":56},"executionInfo":{"elapsed":3,"status":"ok","timestamp":1677507737319,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"OB7sSXOUMMY7","outputId":"28bdeae2-58c4-43a3-b574-7e9874406149"},"outputs":[{"data":{"text/latex":["$\\displaystyle 4$"],"text/plain":["4"]},"execution_count":41,"metadata":{},"output_type":"execute_result"}],"source":["# a)\n","from sympy import *\n","x = symbols('x')\n","integrate(x*R(1,4)*x*exp(-x/2), (x, 0, oo))"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"jSBGUxpcYF3H","outputId":"daddbdb0-2238-4c0a-8121-e0bfad0f9de4"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\infty$"],"text/plain":["oo"]},"execution_count":42,"metadata":{},"output_type":"execute_result"}],"source":["# b)\n","integrate(x*5*x**-2, (x, 5, oo))"]},{"cell_type":"markdown","metadata":{"id":"y8FPClS8Xrl0"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"markdown","metadata":{"id":"Ubs6T6mwXrl0"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"markdown","metadata":{"id":"0RzkugNaXrl0"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"executionInfo":{"elapsed":243,"status":"ok","timestamp":1677511949281,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"b1BnKgh2e-Sq","outputId":"6fbeeca0-cb1e-4eda-efa9-6e308f7ff1f8"},"outputs":[{"data":{"text/plain":["{a: 0.600000000000000, b: 1.20000000000000}"]},"execution_count":43,"metadata":{},"output_type":"execute_result"}],"source":["a, b = symbols('a b')\n","eq1 = Eq(0.5*a +0.25*b, 3/5)\n","eq2 = Eq(a+1/3 * b, 1)\n","\n","solve((eq1, eq2), (a, b))"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"SagHwTZlXrl1","outputId":"b04d7ff7-2fc7-4342-bc79-b0775c8dcf3c"},"outputs":[{"data":{"text/plain":["{a: 0.600000000000000, b: 1.20000000000000}"]},"execution_count":44,"metadata":{},"output_type":"execute_result"}],"source":["solve((0.5*a +0.25*b - 3/5, a+1/3 * b - 1), (a, b))"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"f0Jo9PaJXrl1","outputId":"2c2b0c4c-6b6e-4420-e79f-dba8d4a3c0c4"},"outputs":[{"data":{"text/plain":["{a: 0.600000000000000, b: 1.20000000000000}"]},"execution_count":45,"metadata":{},"output_type":"execute_result"}],"source":["solve((integrate((a+b*x**2), (x, 0, 1)) - 1, integrate(x*(a+b*x**2), (x, 0, 1)) - 3/5), (a, b))"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":51},"executionInfo":{"elapsed":238,"status":"ok","timestamp":1677658993965,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"pHvgw1zxPgaV","outputId":"7865426e-ab64-42fe-90a9-aa29eb2ef54b"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{17}{72}$"],"text/plain":["17/72"]},"execution_count":5,"metadata":{},"output_type":"execute_result"}],"source":["import sympy as sp\n","x, y = sp.symbols('x y')\n","\n","fxy = x + y\n","\n","sp.integrate((x*y**2) * fxy, (x, 0, 1), (y, 0, 1))"]},{"cell_type":"code","execution_count":null,"metadata":{},"outputs":[],"source":[]}],"metadata":{"colab":{"provenance":[]},"kernelspec":{"display_name":"Python 3 (ipykernel)","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.11.1"}},"nbformat":4,"nbformat_minor":0}
+{"cells":[{"attachments":{},"cell_type":"markdown","metadata":{"executionInfo":{"elapsed":1297,"status":"ok","timestamp":1676635374594,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"UCsbknx_PJBW","outputId":"60405cf4-8ad8-4b65-ccc9-f2941e9eef2b"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":1,"metadata":{"id":"KaV3ZOBkYEXs","outputId":"c4273427-fbf4-4f4e-f63e-dfbac077c187"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{c \\left(x^{5} + 1\\right)}{5}$"],"text/plain":["c*(x**5 + 1)/5"]},"execution_count":1,"metadata":{},"output_type":"execute_result"}],"source":["import sympy as sp\n","\n","x, c, u = sp.symbols('x c u')\n","\n","sp.simplify(sp.integrate(c*u**4, (u, -1, x)))"]},{"cell_type":"code","execution_count":2,"metadata":{"id":"d2nsEQXFXrls"},"outputs":[],"source":["def f(u):\n","    return c*u**4"]},{"cell_type":"code","execution_count":3,"metadata":{"id":"23GYLoXAXrlt","outputId":"015ac4ef-64c0-401d-9a8b-38ce373ffdf3"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{c \\left(x^{5} + 1\\right)}{5}$"],"text/plain":["c*(x**5 + 1)/5"]},"execution_count":3,"metadata":{},"output_type":"execute_result"}],"source":["sp.simplify(sp.integrate(f(u), (u, -1, x)))"]},{"attachments":{},"cell_type":"markdown","metadata":{"id":"mJdI3ZtAXrlt"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":4,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"executionInfo":{"elapsed":411,"status":"ok","timestamp":1676635469952,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"yB-AWtjoQa4A","outputId":"2735032d-f670-4ca4-abcf-42167142bb95"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{5}{2}$"],"text/plain":["5/2"]},"execution_count":4,"metadata":{},"output_type":"execute_result"}],"source":["sp.solve(sp.integrate(c*x**4, (x, -1, 1)) -1, c)[0]"]},{"cell_type":"code","execution_count":5,"metadata":{"id":"T6t0kAsfXrlu","outputId":"9cb48181-3a8e-46c6-8045-674d9a107e11"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{5}{2}$"],"text/plain":["5/2"]},"execution_count":5,"metadata":{},"output_type":"execute_result"}],"source":["sp.solve(sp.integrate(f(u), (u, -1, 1)) -1, c)[0]"]},{"cell_type":"code","execution_count":6,"metadata":{"id":"XbUS5tSfRw_C"},"outputs":[],"source":["from sympy import Rational as R\n","def f(x):\n","    return R(5,2) * x**4"]},{"cell_type":"code","execution_count":7,"metadata":{"id":"7lCOtWEpUG-e"},"outputs":[],"source":["def F(x):\n","    return R(1,2) *(x**5 + 1)"]},{"attachments":{},"cell_type":"markdown","metadata":{"id":"MgAvo6TWXrlv"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":8,"metadata":{"id":"6aFfufYJXrlw","outputId":"627f25ad-6d81-4ea2-c595-7033aad8ee3d"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{1}{32}$"],"text/plain":["1/32"]},"execution_count":8,"metadata":{},"output_type":"execute_result"}],"source":["F(R(1,2)) - F(R(-1,2))"]},{"cell_type":"code","execution_count":9,"metadata":{"id":"aE0-0Rm6Xrlw","outputId":"e87d515a-a30e-4cae-a1d2-5b3a95016f75"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{1}{32}$"],"text/plain":["1/32"]},"execution_count":9,"metadata":{},"output_type":"execute_result"}],"source":["sp.integrate(f(x), (x, R(-1,2), R(1,2)))"]},{"attachments":{},"cell_type":"markdown","metadata":{"id":"0KRF2d0gXrlw"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":10,"metadata":{"id":"1ESma27RXrlx","outputId":"909b8e63-e6a1-45ee-ad12-8ace2b800d39"},"outputs":[{"data":{"text/latex":["$\\displaystyle 0$"],"text/plain":["0"]},"execution_count":10,"metadata":{},"output_type":"execute_result"}],"source":["EX = sp.integrate(x*f(x), (x, -1, 1))\n","EX"]},{"cell_type":"code","execution_count":11,"metadata":{"id":"oALp_12jXrlx","outputId":"b79b6402-7ca8-4c86-b827-9a69a87f6fc7"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{5}{7}$"],"text/plain":["5/7"]},"execution_count":11,"metadata":{},"output_type":"execute_result"}],"source":["sp.integrate(x**2 * f(x), (x, -1, 1)) - EX**2"]},{"cell_type":"code","execution_count":12,"metadata":{"id":"vLJ5BAlCK8GH"},"outputs":[],"source":["def F(x):\n","  return R(1,2)*((x**5)+1)"]},{"cell_type":"code","execution_count":13,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":51},"executionInfo":{"elapsed":463,"status":"ok","timestamp":1677506667902,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"jXrP9eIULKqF","outputId":"de1bd990-7c2d-4a7d-afff-3782611ebbbc"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{1}{32}$"],"text/plain":["1/32"]},"execution_count":13,"metadata":{},"output_type":"execute_result"}],"source":["F(R(1,2)) - F(R(-1, 2))"]},{"attachments":{},"cell_type":"markdown","metadata":{"id":"kSV2LHcRXrly"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":14,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":56},"executionInfo":{"elapsed":3,"status":"ok","timestamp":1677507737319,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"OB7sSXOUMMY7","outputId":"28bdeae2-58c4-43a3-b574-7e9874406149"},"outputs":[{"data":{"text/latex":["$\\displaystyle 4$"],"text/plain":["4"]},"execution_count":14,"metadata":{},"output_type":"execute_result"}],"source":["# a)\n","from sympy import *\n","x = symbols('x')\n","integrate(x*R(1,4)*x*exp(-x/2), (x, 0, oo))"]},{"cell_type":"code","execution_count":15,"metadata":{"id":"jSBGUxpcYF3H","outputId":"daddbdb0-2238-4c0a-8121-e0bfad0f9de4"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\infty$"],"text/plain":["oo"]},"execution_count":15,"metadata":{},"output_type":"execute_result"}],"source":["# b)\n","integrate(x*5*x**-2, (x, 5, oo))"]},{"attachments":{},"cell_type":"markdown","metadata":{"id":"y8FPClS8Xrl0"},"source":["![image.png](attachment:image.png)"]},{"attachments":{},"cell_type":"markdown","metadata":{"id":"Ubs6T6mwXrl0"},"source":["![image.png](attachment:image.png)"]},{"attachments":{},"cell_type":"markdown","metadata":{"id":"0RzkugNaXrl0"},"source":["![image.png](attachment:image.png)"]},{"cell_type":"code","execution_count":16,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"executionInfo":{"elapsed":243,"status":"ok","timestamp":1677511949281,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"b1BnKgh2e-Sq","outputId":"6fbeeca0-cb1e-4eda-efa9-6e308f7ff1f8"},"outputs":[{"data":{"text/plain":["{a: 0.600000000000000, b: 1.20000000000000}"]},"execution_count":16,"metadata":{},"output_type":"execute_result"}],"source":["a, b = symbols('a b')\n","eq1 = Eq(0.5*a +0.25*b, 3/5)\n","eq2 = Eq(a+1/3 * b, 1)\n","\n","solve((eq1, eq2), (a, b))"]},{"cell_type":"code","execution_count":17,"metadata":{"id":"SagHwTZlXrl1","outputId":"b04d7ff7-2fc7-4342-bc79-b0775c8dcf3c"},"outputs":[{"data":{"text/plain":["{a: 0.600000000000000, b: 1.20000000000000}"]},"execution_count":17,"metadata":{},"output_type":"execute_result"}],"source":["solve((0.5*a +0.25*b - 3/5, a+1/3 * b - 1), (a, b))"]},{"cell_type":"code","execution_count":18,"metadata":{"id":"f0Jo9PaJXrl1","outputId":"2c2b0c4c-6b6e-4420-e79f-dba8d4a3c0c4"},"outputs":[{"data":{"text/plain":["{a: 0.600000000000000, b: 1.20000000000000}"]},"execution_count":18,"metadata":{},"output_type":"execute_result"}],"source":["solve((integrate((a+b*x**2), (x, 0, 1)) - 1, integrate(x*(a+b*x**2), (x, 0, 1)) - 3/5), (a, b))"]},{"cell_type":"code","execution_count":19,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":51},"executionInfo":{"elapsed":238,"status":"ok","timestamp":1677658993965,"user":{"displayName":"Richard Brooks","userId":"04496048306261208300"},"user_tz":-60},"id":"pHvgw1zxPgaV","outputId":"7865426e-ab64-42fe-90a9-aa29eb2ef54b"},"outputs":[{"data":{"text/latex":["$\\displaystyle \\frac{17}{72}$"],"text/plain":["17/72"]},"execution_count":19,"metadata":{},"output_type":"execute_result"}],"source":["import sympy as sp\n","x, y = sp.symbols('x y')\n","\n","fxy = x + y\n","\n","sp.integrate((x*y**2) * fxy, (x, 0, 1), (y, 0, 1))"]},{"cell_type":"code","execution_count":null,"metadata":{},"outputs":[],"source":[]}],"metadata":{"colab":{"provenance":[]},"kernelspec":{"display_name":"Python 3 (ipykernel)","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.10.9"}},"nbformat":4,"nbformat_minor":0}