{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "# Function Analysis Part 1 Assignment" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "## Analyze the following functions using the steps from class.\n", "\n", "## Some steps have been done for you - make sure you run the function definitions again." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "## Question 1\n", "\n", "[5 points] $\\quad\\displaystyle f(x)=\\frac{9x-1}{x^2+110}$" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\frac{9 \\, x - 1}{x^{2} + 110}\\)" ], "text/latex": [ "$\\displaystyle \\frac{9 \\, x - 1}{x^{2} + 110}$" ], "text/plain": [ "(9*x - 1)/(x^2 + 110)" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "f(x)=(9*x-1)/(x^2+110)\n", "show(f(x))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 1: Find the domain of $f$.\n", "\n", "The domain is $\\R$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 2: Find the derivative $f'$." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle -\\frac{2 \\, {\\left(9 \\, x - 1\\right)} x}{{\\left(x^{2} + 110\\right)}^{2}} + \\frac{9}{x^{2} + 110}\\)" ], "text/latex": [ "$\\displaystyle -\\frac{2 \\, {\\left(9 \\, x - 1\\right)} x}{{\\left(x^{2} + 110\\right)}^{2}} + \\frac{9}{x^{2} + 110}$" ], "text/plain": [ "-2*(9*x - 1)*x/(x^2 + 110)^2 + 9/(x^2 + 110)" ] }, "execution_count": 12, "metadata": { }, "output_type": "execute_result" } ], "source": [ "df(x)=derivative(f(x),x)\n", "show(df(x))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 3: Find the critical points of $f$ (where $f'$ is $0$ or undefined).\n", "\n" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[x == -1/9*sqrt(8911) + 1/9, x == 1/9*sqrt(8911) + 1/9]" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solve(df(x)==0,x)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "You should convert the solutions above to decimal.\n", "\n" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-10.3775659112312" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(-1/9*sqrt(8911)+1/9)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "10.5997881334535" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(1/9*sqrt(8911)+1/9)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 4: See if the sign of $f'$ actually changes at the critical points of $f$, and determine whether $f$ has a local maximum or local minimum at these points.\n", "\n" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.000479282004635637" ] }, "execution_count": 16, "metadata": { }, "output_type": "execute_result" } ], "source": [ "df(-10.5)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.000312453902207771" ] }, "execution_count": 17, "metadata": { }, "output_type": "execute_result" } ], "source": [ "df(-10.3)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.000584342440431880" ] }, "execution_count": 18, "metadata": { }, "output_type": "execute_result" } ], "source": [ "df(10.45)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.000377214272833236" ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "df(10.7)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 5: Find the second derivative $f''$." ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\frac{8 \\, {\\left(9 \\, x - 1\\right)} x^{2}}{{\\left(x^{2} + 110\\right)}^{3}} - \\frac{2 \\, {\\left(9 \\, x - 1\\right)}}{{\\left(x^{2} + 110\\right)}^{2}} - \\frac{36 \\, x}{{\\left(x^{2} + 110\\right)}^{2}}\\)" ], "text/latex": [ "$\\displaystyle \\frac{8 \\, {\\left(9 \\, x - 1\\right)} x^{2}}{{\\left(x^{2} + 110\\right)}^{3}} - \\frac{2 \\, {\\left(9 \\, x - 1\\right)}}{{\\left(x^{2} + 110\\right)}^{2}} - \\frac{36 \\, x}{{\\left(x^{2} + 110\\right)}^{2}}$" ], "text/plain": [ "8*(9*x - 1)*x^2/(x^2 + 110)^3 - 2*(9*x - 1)/(x^2 + 110)^2 - 36*x/(x^2 + 110)^2" ] }, "execution_count": 20, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2f(x)=derivative(f(x),x,2)\n", "show(d2f(x))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 6: Find the critical points of $f'$ (where $f''$ is $0$ or undefined)." ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[x == -1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(I*sqrt(3) + 1) - 8911/162*(-I*sqrt(3) + 1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3) + 1/9, x == -1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(-I*sqrt(3) + 1) - 8911/162*(I*sqrt(3) + 1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3) + 1/9, x == (8911/81*I*sqrt(110) + 8911/729)^(1/3) + 8911/81/(8911/81*I*sqrt(110) + 8911/729)^(1/3) + 1/9]" ] }, "execution_count": 21, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solve(d2f(x)==0,x)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "You should convert the solutions above to decimal." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.0370358054710737 + 5.32907051820075e-15*I" ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(-1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(I*sqrt(3) +1)-8911/162*(-I*sqrt(3) +1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3)+ 1/9)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.0370358054710737 + 5.32907051820075e-15*I" ] }, "execution_count": 22, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(-1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(I*sqrt(3) +1)-8911/162*(-I*sqrt(3) +1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3)+ 1/9)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.0370358054710737 + 5.32907051820075e-15*I" ] }, "execution_count": 23, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(-1/2*(8911/81*I*sqrt(110) + 8911/729)^(1/3)*(I*sqrt(3) +1)-8911/162*(-I*sqrt(3) +1)/(8911/81*I*sqrt(110) + 8911/729)^(1/3)+ 1/9)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.04083299973307" ] }, "execution_count": 7, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(1/6*sqrt(39)+1)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.0408329997330663" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(-1/6*sqrt(39)+1)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 7: See if the sign of $f''$ actually changes at the critical points of $f'$, and determine whether $f$ has an inflection point at these points." ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.000165289256198347" ] }, "execution_count": 25, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2f(0)\n", "N(1/6050)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.000280946661976095" ] }, "execution_count": 26, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2f(0.1)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.0000253027535196296" ] }, "execution_count": 27, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2f(-18.2)" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.000134953803168578" ] }, "execution_count": 28, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2f(-17.2)" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.0000301219469454848" ] }, "execution_count": 29, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2f(18.1)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.0000113426926653792" ] }, "execution_count": 30, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2f(18.4)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "#inflection points at x=-18.02, 0.037, 18.31" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 8: Find the $x$- and $y$-intercepts." ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[x == (1/9)]" ] }, "execution_count": 31, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solve(f(x)==0,x)" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-1/110" ] }, "execution_count": 32, "metadata": { }, "output_type": "execute_result" } ], "source": [ "f(0)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 9: Determine the end behavior." ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 33, "metadata": { }, "output_type": "execute_result" } ], "source": [ "limit(f(x),x=infinity)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 34, "metadata": { }, "output_type": "execute_result" } ], "source": [ "limit(f(x),x=-infinity)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 10: Make an informed graph. Mark any $x$- and $y$-intercepts, relative maxima and minima, and inflection points." ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.433627684569615" ] }, "execution_count": 35, "metadata": { }, "output_type": "execute_result" } ], "source": [ "f(-10.38)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "f(10.6)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "f(0.037)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "f(-18.02)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "f(18.31)" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "execution_count": 36, "metadata": { }, "output_type": "execute_result" } ], "source": [ "plot (f(x), xmin=-100, xmax=100, ymin=-0.5, ymax=0.5)+point([(-10.38,-0.43), (10.60,0.42), (0.037, -0.006), (-18.02, -0.375), (18.31, 0.368), (1/9,0),(0, -1/110)], color='black', size=20)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 11: Discuss absolute max/min, increasing/decreasing, concave up/down." ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "#Absolute Max=0.42, at x=10.60\n", "#Absolute Min=-0.43, at x=10.38\n", "#inc (-10.38, 10.60)\n", "#dec (-infinity, -10.38) (10.60, infinity)\n", "#cu (-18.02, 0.037), (18.31, infinity)\n", "#cd (-infinity, -18.02), (0.037, 18.31)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "## Question 2\n", "\n", "[5 points] $\\quad g(x)=2x^4-8x^3-x^2+30x$" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle 2 \\, x^{4} - 8 \\, x^{3} - x^{2} + 30 \\, x\\)" ], "text/latex": [ "$\\displaystyle 2 \\, x^{4} - 8 \\, x^{3} - x^{2} + 30 \\, x$" ], "text/plain": [ "2*x^4 - 8*x^3 - x^2 + 30*x" ] }, "execution_count": 38, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(x)=2*x^4-8*x^3-x^2+30*x\n", "show(g(x))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 1: Find the domain of $g$.\n", "\n", "The domain is $\\R$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 2: Find the derivative $g'$." ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle 8 \\, x^{3} - 24 \\, x^{2} - 2 \\, x + 30\\)" ], "text/latex": [ "$\\displaystyle 8 \\, x^{3} - 24 \\, x^{2} - 2 \\, x + 30$" ], "text/plain": [ "8*x^3 - 24*x^2 - 2*x + 30" ] }, "execution_count": 39, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dg(x)=derivative(g(x),x)\n", "show(dg(x))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 3: Find the critical points of $g$ (where $g'$ is $0$ or undefined)." ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[x == (5/2), x == -1, x == (3/2)]" ] }, "execution_count": 40, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solve(dg(x)==0,x)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 4: See if the sign of $g'$ actually changes at the critical points of $g$, and determine whether $g$ has a local maximum or local minimum at these points." ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-6" ] }, "execution_count": 41, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dg(2)" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "24" ] }, "execution_count": 42, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dg(3)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "dg(1)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-6" ] }, "execution_count": 43, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dg(2)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "dg(-1.5)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "24.0000000000000" ] }, "execution_count": 44, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dg(-0.5)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 5: Find the second derivative $g''$." ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle 24 \\, x^{2} - 48 \\, x - 2\\)" ], "text/latex": [ "$\\displaystyle 24 \\, x^{2} - 48 \\, x - 2$" ], "text/plain": [ "24*x^2 - 48*x - 2" ] }, "execution_count": 45, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2g(x)=derivative(g(x),x,2)\n", "show(d2g(x))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 6: Find the critical points of $g'$ (where $g''$ is $0$ or undefined)." ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[x == -1/6*sqrt(39) + 1, x == 1/6*sqrt(39) + 1]" ] }, "execution_count": 46, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solve(d2g(x)==0,x)" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.0408329997330663" ] }, "execution_count": 47, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(-1/6*sqrt(39)+1)" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.04083299973307" ] }, "execution_count": 48, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(1/6 *sqrt(39)+1)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 7: See if the sign of $g''$ actually changes at the critical points of $g'$, and determine whether $g$ has an inflection point at these points." ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.460000000000000" ] }, "execution_count": 49, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2g(-0.05)" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-0.538400000000000" ] }, "execution_count": 51, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2g(-0.03)" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-2" ] }, "execution_count": 52, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2g(2)" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "70" ] }, "execution_count": 53, "metadata": { }, "output_type": "execute_result" } ], "source": [ "d2g(3)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "#inflection point at x=2.041" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 8: Find the $x$- and $y$-intercepts.\n", "\n", "[Caution: $g$ has two x-intercepts. When you solve $g(x)=0$, CoCalc will give you four answers, but only two are real. Convert to decimal, and watch out for scientific notation.]" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[x == -1/6*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3)*(I*sqrt(3) + 1) - 5/3*(7/4)^(2/3)*(-I*sqrt(3) + 1)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3, x == -1/6*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3)*(-I*sqrt(3) + 1) - 5/3*(7/4)^(2/3)*(I*sqrt(3) + 1)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3, x == 1/3*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 10/3*(7/4)^(2/3)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3, x == 0]" ] }, "execution_count": 56, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solve(g(x)==0,x)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(0)" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.83551650478519 + 0.967641208530388*I" ] }, "execution_count": 58, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(-1/6*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3)*(I*sqrt(3) + 1) - 5/3*(7/4)^(2/3)*(-I*sqrt(3) + 1)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3)" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-1.67103300957038 - 2.74364199474365e-17*I" ] }, "execution_count": 59, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N( -1/6*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3)*(-I*sqrt(3) + 1) - 5/3*(7/4)^(2/3)*(I*sqrt(3) + 1)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3)" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.83551650478519 - 0.967641208530389*I" ] }, "execution_count": 61, "metadata": { }, "output_type": "execute_result" } ], "source": [ "N(1/3*(7/4)^(1/3)*(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 10/3*(7/4)^(2/3)/(9*sqrt(23)*sqrt(2) - 74)^(1/3) + 4/3)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 9: Determine the end behavior." ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "+Infinity" ] }, "execution_count": 63, "metadata": { }, "output_type": "execute_result" } ], "source": [ "limit(g(x), x=infinity)" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "+Infinity" ] }, "execution_count": 64, "metadata": { }, "output_type": "execute_result" } ], "source": [ "limit(g(x), x=-infinity)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 10: Make an informed graph. Mark any $x$- and $y$-intercepts, relative maxima and minima, and inflection points." ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-21" ] }, "execution_count": 65, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(-1)" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "21.8750000000000" ] }, "execution_count": 66, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(2.5)" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "25.8750000000000" ] }, "execution_count": 67, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(1.5)" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-1.22511575945134" ] }, "execution_count": 68, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(-0.0408)" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "23.7540842405487" ] }, "execution_count": 69, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(2.0408)" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "execution_count": 70, "metadata": { }, "output_type": "execute_result" } ], "source": [ "plot(g(x),xmin=-2,xmax=4,ymin=-30,ymax=30)+point([(-1,-21),(2.5,21.875),(0,0),(-1.67,0),(1.5,25.875),(-0.0408,-1.225),(2.0408,23.7541)],color='black',size=50)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "Step 11: Discuss absolute max/min, increasing/decreasing, concave up/down." ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "#Aaron: Here are the correct answers:\n", "#no absolute max\n", "#absolute min = -21 at x = -1\n", "#increasing: (-1,1.5), (2.5,oo)\n", "#decreasing: (-oo,-1), (1.5,2.5)\n", "#concave up: (-oo,-0.04), (2.04,oo)\n", "#concave down: (-0.04,2.04)\n" ] } ], "metadata": { "kernelspec": { "argv": [ "sage-10.0", "--python", "-m", "sage.repl.ipython_kernel", "--matplotlib=inline", "-f", 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