11th grade-all tasks
{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# <div dir=\"RTL\"> החבילה SymPy </div>\n",
"______________\n",
"<div dir=\"RTL\">\n",
" SymPy היא ספרייה למתמטיקה סימבולית המיועדת ל- Python. היא כתובה כולה רק ב-Python, כך שאין צורך בסיפריות נוספות כדי להשתמש בה. בכוונת מפתחי ה- Sympy לפתחה עד כדי מערכת המכילה את כל התכונות והיכולות של תוכנות CAS (Computer Algebra System). המפתחים מנסים לשמור את הקוד פשוט ככל האפשר כדי ליצור חבילה קלה להבנה ושאפשר להרחיבה בקלות. \n",
"באופן בסיסי Sympy היא מחשבון סימבולי, אבל יתרונה בכך שאפשר להשתמש בה בתוכנית מחשב.\n",
"\n",
"</div>\n"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"import sympy as sp\n",
"#from sympy import *\n",
"sp.init_printing()"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"import math"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$1.4142135623730951$$"
]
},
"execution_count": 3,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"math.sqrt(2)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\sqrt{2}$$"
]
},
"execution_count": 4,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"sp.sqrt(2)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$1.0471975511966$$"
]
},
"execution_count": 5,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"sp.acos(0.5)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$1.0471975511966$$"
]
},
"execution_count": 6,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"sp.acos(1/2)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\">תרגיל 1</div>\n",
"<div dir=\"RTL\">חשבו: $$acos(-1)$$ </div>\n",
"$$\\sqrt5$$\n",
"$$\\sqrt9$$\n"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$1.22464679914735 \\cdot 10^{-16}$$"
]
},
"execution_count": 12,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"sp.sin(math.pi)"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> Symbols </div>\n",
"<div dir=\"RTL\"> האוביקט הבסיסי ב- Sympy הוא ה- Symbol אוביקט זה מייצג משתנה מתמטי. יצירת אובייקט זה נעשיית באמצעות הפונקציה Symbol </div>"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\sin^{2}{\\left (\\alpha \\right )} + \\sin^{2}{\\left (\\beta \\right )}$$"
]
},
"execution_count": 13,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"x,y,z=sp.symbols('x y z')\n",
"alpha,beta,gamma=sp.symbols('alpha,beta,gamma')\n",
"sp.sin(alpha)**2+sp.sin(beta)**2"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\"> תרגיל 2 </div>\n",
"<div dir=\"RTL\">בעזרת הפונקציה symbols צרו את המשתנים mu ו- sigma (החליפו את ה- ? בקוד המתאים) </div>"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"ename": "TypeError",
"evalue": "'Symbol' object is not iterable",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mTypeError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m<ipython-input-8-33eeddf750a1>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mmu\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0msigma\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0msp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msymbols\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'?'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 2\u001b[0m \u001b[0mmu\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0msigma\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
"\u001b[1;31mTypeError\u001b[0m: 'Symbol' object is not iterable"
]
}
],
"source": [
"mu,sigma=sp.symbols('?')\n",
"mu,sigma"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\"> תרגיל 3 </div>\n",
"<div dir=\"RTL\">הגדירו את הפונקציה $e^{ {-(x-\\mu)^2}\\over{\\sigma^2}}$ פונקציה זו מכונה פונקציית הפעמון.</div>\n",
"<div dir=\"RTL\"><B> את הפונקציה $e^x$ מגדירים כ- (exp(x </B></div>"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> גזירה של פונקציה</div>"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$3 a x^{2}$$"
]
},
"execution_count": 10,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"a=sp.Symbol('a')\n",
"sp.diff(a*x**3,x)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"ename": "NameError",
"evalue": "name 'bell' is not defined",
"output_type": "error",
"traceback": [
"\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)",
"\u001b[1;32m<ipython-input-11-7a06e4fb7c64>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mbell\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdiff\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[1;31mNameError\u001b[0m: name 'bell' is not defined"
]
}
],
"source": [
"bell.diff(x)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> שרטוט גרף באמצעות sympy.plotting </div>"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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"
},
"execution_count": 12,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"text/plain": [
"<sympy.plotting.plot.Plot at 0xe887d1dac8>"
]
},
"execution_count": 12,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"%matplotlib -- inline\n",
"from sympy.plotting import *\n",
"plot(sp.exp(-(x-3)**2),(x,-1,6))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\"> תרגיל 4 </div>\n",
"<div dir=\"RTL\"> חשבו את הניגזרת השנייה של פונקציית \"הפעמון\"\n",
" ושרטטו גרף שלה (הניחו כי :$\\sigma=3.0$ ו- $\\mu=1.5$ ) <br>\n",
" הערה: חשבו את הנגזרת ולאחר מכן הגדירו את ביטוי הניגזרת מחדש כאשר אתם מציבים ערכים לפרמטרים: $\\mu,\\sigma$. </div>"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\">הפונקציות simplify, subs,expand,evalf </div>"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$a^{3} + a^{2} b - a b^{2} - b^{3}$$"
]
},
"execution_count": 13,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"a,b,c,d=sp.symbols('a b c d')\n",
"expr=(a-b)*(a+b)**2\n",
"sp.expand(expr)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$a^{3} + a^{2} b - a b^{2} - b^{3}$$"
]
},
"execution_count": 14,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"expr.expand()"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left(- \\frac{c d}{2} + 3\\right) \\left(\\frac{c d}{2} + 3\\right)^{2}$$"
]
},
"execution_count": 15,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"new_expr=expr.subs([(a,3),(b,d*c/2)])\n",
"new_expr"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\frac{1}{8} \\left(- c d + 6\\right) \\left(c d + 6\\right)^{2}$$"
]
},
"execution_count": 16,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"sp.simplify(new_expr)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$-26.136$$"
]
},
"execution_count": 17,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"new_expr.subs([(c,2.4),(d,3)]).evalf()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\">תרגיל 5 </div>\n",
"<div dir=\"RTL\"> בעזרת הפונקציה help בדקו מה פעולת הפונקציה simplify.<br>\n",
"\n",
"\n",
"\n",
"מה עושה כל אחת מהפונקציות subs, evalf ו- expand</div>"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": true
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\">תרגיל 6 </div>\n",
"<div dir=\"RTL\"> הפעילו את הפונקציה simplify על הניגזרת השלישית של פונקציית הפעמון </div>"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\"> תרגיל 7 <div/>\n",
"<div dir=\"RTL\">פשטו את הביטוי : $ (x+1)^2 \\over {1-x^2}$ (הפונקציה simplify)"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\">תרגיל 8</div>\n",
"<div dir=\"RTL\"> הפעילו על הביטוי הקודם את הפונקציה expand."
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\"> תרגיל 9 </div>\n",
"<div dir=\"RTL\"> הביטוי שלמטה הוא נוסחת הרון לחישוב שטח משולש שצלעותיו הן a,b ו-s. c הוא חצי היקף המשולש. השתמשו בפונקציה subs וחשבו את שטח המשולש אם נתון ש: a=6, b=7, c=9 </div>\n",
"<div dir=\"RTL\"> קודם הציבו במקום s את מחצית סכום הצלעות ולאחר מכן הציבו את הערכים.</div>"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"a,b,c,s=sp.symbols('a,b,c,s')\n",
"area=sp.sqrt(s*(s-a)*(s-b)*(s-c))\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> בדיקת שוויון </div>"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 20,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"x,y=sp.symbols('x y')\n",
"a=(x-y)**2\n",
"b=x**2-2*x*y+y**2\n",
"a==b"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$0$$"
]
},
"execution_count": 21,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"sp.simplify(a-b)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> יצירת משוואה </div>\n",
"<div dir=\"RTL\"> יוצרים משוואה באמצעות הפונקציה Eq</div>"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$y = 4 x^{2} - x + 3$$"
]
},
"execution_count": 22,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"e=sp.Eq(y,4*x**2-x+3)\n",
"e"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$y$$"
]
},
"execution_count": 23,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"e.lhs"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$4 x^{2} - x + 3$$"
]
},
"execution_count": 24,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"e.rhs"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left [ - \\frac{1}{8} \\sqrt{16 y - 47} + \\frac{1}{8}, \\quad \\frac{1}{8} \\sqrt{16 y - 47} + \\frac{1}{8}\\right ]$$"
]
},
"execution_count": 25,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"sp.solve(e,x)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> תרגיל 10 </div>\n",
"<div dir=\"RTL\">כתבו פונקציה המקבלת שני ביטווים. הפונקציה צריכה tupl המכיל שני ערכים בוליאניים. הראשון מציין האם הביטויים זהים והשני האם הביטויים שווים מתמטית. </div>"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"def equality_exercise(a,b):\n",
" \"\"\"Return a tuple of tow boolean. the first is True if a=b symbolicaly, \n",
" the second is True if a==b mathematically. \n",
" Examples\n",
" ========\n",
" >>> x=symbols('x')\n",
" >>> equality_exercise(x,2)\n",
" (False,False)\n",
" >>> equality_exercise((x+1)**2,x**2+2*x+1)\n",
" (False,True)\n",
" >>> equality_exercise(4*x,4*x)\n",
" (True,True)\n",
" \"\"\"\n",
" return a==b,sp.simplify(a-b)==0\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(False, True)"
]
},
"execution_count": 27,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"expr1=(x-y)**2\n",
"expr2=x**2-2*x*y+y**2\n",
"equality_exercise(expr1,expr2)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> דוגמא </div>\n",
"<div dir=\"RTL\"> קיבול חום סגול של חומר מוגדר ככמות האנרגיה שצריך לספק לכמות חומר שגודלה יחידת מסה אחת כדי לחמם אותה ב- $ 1^0C $. קיבול החום הסגולי של מים הוא\n",
" $ 4.2{ J\\over {gr\\cdot 1^0C}} $ (האות J מציינת את יחידת האנרגיה ג'ול) הספק של גוף חימום שווה לכמות האנרגיה שפולט גוף החימום ביחידת זמן. יחידת ההספק היא ווט (W) והיא שווה לכמות אנרגיה של 1J בשנייה (s). בהמשך דוגמא לחישוב הזמן הדרוש לחימום דוד מים המכיל 150 ליטר מים בטמפרטורה של $ 23^0 C $ לטמפרטורה של $ 60^0C $בעזרת גוף חימום שהספקו 2000W. \n",
" </div>\n",
"<div dir=\"RTL\"><B> הערה: </B> </div>\n",
" <div dir=\"RTL\"> כאשר מצמידים גוף בטמפרטורה גבוהה לגוף בטמפרטורה נמוכה יותר עוברת אנרגיה מהגוף החם לקר עד שהטמפרטורות משתוות. כמות האנרגיה העוברת מהגוף החם לקר מכונה חום. <br></div>\n",
"\n",
"Q - כמות חום\n",
"\n",
"T1- טמפרטורה התחלתית\n",
"\n",
"T2- טמפרטורה סופית\n",
"\n",
"m - מסת המים\n",
"\n",
"c -קיבול חום סגולי של מים\n",
"\n",
"P - הספק גוף החימום\n",
"\n",
"##<div dir=\"RTL\"> חישוב כמות החום הדרושה לחימום המים:</div>\n"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": true
},
"outputs": [
],
"source": [
"c=4.2# J/gr-1C\n",
"m=150*1000 #gr\n",
"T1=23 #C\n",
"T2=60 #C\n",
"P=2000 #w\n",
"Q=c*m*(T2-T1)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=RTL>חישוב הזמן:</div>"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"11655.0 s\n"
]
}
],
"source": [
"t=Q/P\n",
"print( t, \"s\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"<div dir=\"RTL\"> כפי שניתן לראות במקרה זה אין צורך להשתמש ב-Sympy לפתרון התרגיל. </div>"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"\n",
"\n",
"# <div dir=\"RTL\"> השימוש ב- Sympy הוא הדרך הפשוטה לפתרון בעיות מסובכות והדרך המסובכת לפתרון בעיות פשוטות </div>\n",
"\n",
"\n",
"\n",
"## <div dir=\"RTL\"> תרגיל 11 </div>\n",
"\n",
"<div dir=\"RTL\">\tנתון גוף שמסתו m1 בטמפרטורה T1 וקיבול חום C1.<BR>\n",
"מצמידים אותו לגוף שמסתו m2, הטמפרטורה שלו T2 וקיבול החום שלו C2. פתחו בעזרת Sympy<BR> ביטוי לטמפרטורה הסופית של שני הגופים ( הניחו כי כמות החום שפלט האחד שווה לכמות החום שקלט השני).</div>"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> הפונקציה solve </div>\n",
"\n",
"<div dir=\"RTL\"> נתונה המשוואה: $$ x- {1\\over x}+b\\cdot x=7 $$ </br>\n",
"\n",
"\n",
"\n",
"<div dir=\"RTL\">נמצא לה פתרון סימבולי: </div>"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Eq(b*x + x - 1/x, 7)\n"
]
},
{
"data": {
"text/latex": [
"$$\\left [ \\frac{- \\sqrt{4 b + 53} + 7}{2 b + 2}, \\quad \\frac{\\sqrt{4 b + 53} + 7}{2 b + 2}\\right ]$$"
]
},
"execution_count": 33,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"x,b=sp.symbols('x b')\n",
"eq=sp.Eq(x-1/x+b*x,7)\n",
"print(eq)\n",
"ans=sp.solve(eq,x)\n",
"ans"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\"> פיתרון נומרי עבור b=5 </div>"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"-0.129 1.30\n"
]
}
],
"source": [
"print( ans[0].subs(b,5).n(3), ans[1].subs(b,5).n(3))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### <div dir=\"RTL\">יותר מנעלם אחד: </div>"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left [ \\left ( - \\frac{7}{3}, \\quad - \\frac{4}{3}\\right ), \\quad \\left ( - \\frac{5}{3}, \\quad - \\frac{7}{3}\\right ), \\quad \\left ( 0, \\quad 1\\right ), \\quad \\left ( 1, \\quad \\frac{1}{3}\\right )\\right ]$$"
]
},
"execution_count": 35,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"x,y=sp.symbols('x,y')\n",
"sp.solve([sp.Eq(3*(x-y)**2+x-2,y),sp.Eq(y*x+x/3+y,1)],[x,y])"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> תרגיל 12 </div>\n",
"<div dir=\"RTL\">חלקיק נע במהירות קצובה לאורך הקו הישר: $ y=m\\cdot x+n $ . בנקודה (a,b) דיסקה ברדיוס r.</div>\n",
"<div dir=\"RTL\"> פתחו נוסחא באמצעותה ניתן לדעת האם החלקיק מתנגש בדיסקה ואם כן היכן.</div>\n",
"<div dir=\"RTL\"> היכן נקודת ההתנגשות עבור הערכים הבאים: a=2,b=3,m=1,n=0.5, ו- r=5 </div>\n"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# <div dir=\"RTL\"> מציאת נקודות קיצון של פונקציה </div>\n",
"\n",
"<div dir=\"RTL\"> בנקודת קיצון הניגזרת מתאפסת </div>\n",
"\n",
"## $$ f(x)=e^\\frac{-(x^2-ax+b)}{c} $$"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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"
},
"execution_count": 40,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"text/plain": [
"<sympy.plotting.plot.Plot at 0x90935a8198>"
]
},
"execution_count": 40,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"x,a,b,c=sp.symbols('x,a,b,c')\n",
"f=sp.exp(-(x**2-a*x-b)/c)\n",
"plot(f.subs([(a,2),(b,1),(c,3)]),(x,-4,5))"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"sp.solve(f.diff(x),x)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## <div dir=\"RTL\"> תרגיל 12 </div>\n",
"<div dir=\"RTL\"> מזריקים כמות A של תרופה לחולה. הריכוז c של התרופה נימדד ב: $ \\frac{mg}{ml} $.<br> כעבור זמן t מרגע הזרקת התרופה הריכוז ניתן על ידי : $ c(t)=Ate^{-t/3} $. הזמן נימדד בדקות. <br>הריכוז המקסימאלי המותר של התרופה הוא $ 1 \\frac{mg}{ml}$ </div>\n",
"\n",
"<div dir=\"RTL\"> - איזו כמות מקסימאלית A מותר להזריק ? ומתי מתקבל הריכוז המקסימאלי? </div>\n",
"<div dir=\"RTL\"> - שרטטו גרף של הריכוז בדם כתלות בזמן והעריכו באמצעותו מתי מתקבל ריכוז של $ 0.25 \\frac{mg}{ml}$ </div>\n",
"<div dir=\"RTL\"> - מצאו לסעיף הקודם תשובה מדויקת בעזרת הפונקציה nsolv. </div>\n",
"<div dir=\"RTL\"> -כל כמה זמן צריך להזריק לחולה את התרופה כדי שהריכוז לא יהיה מעבר לערך המקסימאלי ולא יפחת מערך מינימאלי של $ 0.25 \\frac{mg}{ml}$"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left [ 3\\right ]$$"
]
},
"execution_count": 21,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"A, t =sp.symbols('A t')\n",
"c_t = A * t * sp.exp(-t/3)\n",
"sp.solve(sp.diff(c_t,t),t)\n"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (Ubuntu Linux)",
"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": "3.5.2"
}
},
"nbformat": 4,
"nbformat_minor": 0
}