1.\u00a0<\/strong>[latex]x^2-y^2=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 2.\u00a0<\/strong>[latex]6x^2+3y^2=12[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738106228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738106228\"]\r\n [latex]\\frac{dy}{dx}=\\frac{-2x}{y}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 3.\u00a0<\/strong>[latex]x^2 y=y-7[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 4.\u00a0<\/strong>[latex]3x^3+9xy^2=5x^3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738223311\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738223311\"]\r\n [latex]\\frac{dy}{dx}=\\frac{x}{3y}-\\frac{y}{2x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 5.\u00a0<\/strong>[latex]xy- \\cos (xy)=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 6.\u00a0<\/strong>[latex]y\\sqrt{x+4}=xy+8[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738237518\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738237518\"]\r\n [latex]\\frac{dy}{dx}=\\large \\frac{y-\\frac{y}{2\\sqrt{x+4}}}{\\sqrt{x+4}-x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 7.\u00a0<\/strong>[latex]\u2212xy-2=\\dfrac{x}{7}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 8.\u00a0<\/strong>[latex]y \\sin(xy)=y^2+2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169737911399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737911399\"]\r\n [latex]\\frac{dy}{dx}=\\large \\frac{y^2 \\cos(xy)}{2y- \\sin(xy)-xy \\cos xy}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 9.\u00a0<\/strong>[latex](xy)^2+3x=y^2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 10.\u00a0<\/strong>[latex]x^3 y+xy^3=-8[\/latex]<\/p>\r\n[reveal-answer q=\"608460\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"608460\"]\r\n[latex]\\frac{dy}{dx}=\\large \\frac{-3x^2 y-y^3}{x^3+3xy^2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n For the following exercises (11-16), find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.<\/p>\r\n\r\n 11. [T]\u00a0\u00a0<\/strong>[latex]x^4 y-xy^3=-2, \\,\\,\\, (-1,-1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 12. [T]\u00a0\u00a0<\/strong>[latex]x^2 y^2+5xy=14, \\,\\,\\, (2,1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738226137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738226137\"] 13. [T]\u00a0\u00a0<\/strong>[latex]\\tan (xy)=y, \\,\\,\\, (\\frac{\\pi}{4},1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 14. [T]\u00a0\u00a0<\/b>[latex]xy^2 + \\sin(\\pi y)-2x^2=10, \\,\\,\\, (2,-3)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738234517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738234517\"] 15. [T]\u00a0\u00a0<\/strong>[latex]\\frac{x}{y}+5x-7=-\\frac{3}{4}y, \\,\\,\\, (1,2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 16. [T]\u00a0\u00a0<\/strong>[latex]xy+ \\sin (x)=1, \\,\\,\\, (\\frac{\\pi}{2},0)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738221212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738221212\"] 18.\u00a0<\/strong>For the equation [latex]x^2+2xy-3y^2=0[\/latex],<\/p>\r\n\r\n a. [latex]y=\u2212x+2[\/latex]\r\nb. [latex](3,-1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 19.\u00a0<\/strong>Find all points on the graph of [latex]y^3-27y=x^2-90[\/latex] at which the tangent line is vertical.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 20.\u00a0<\/strong>For the equation [latex]x^2+xy+y^2=7[\/latex],<\/p>\r\n\r\n a. [latex](\\pm \\sqrt{7},0)[\/latex]\r\nb. Slope is -2 at both intercepts\r\nc. They are parallel since the slope is the same at both intercepts.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 21.\u00a0<\/strong>Find the equation of the tangent line to the graph of the equation [latex]\\sin^{-1} x+\\sin^{-1} y=\\frac{\\pi}{6}[\/latex] at the point [latex](0,\\frac{1}{2})[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 22.\u00a0<\/strong>Find the equation of the tangent line to the graph of the equation [latex]\\tan^{-1}(x+y)=x^2+\\frac{\\pi}{4}[\/latex] at the point [latex](0,1)[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738201952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738201952\"]\r\n [latex]y=\u2212x+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 23.\u00a0<\/strong>Find [latex]y^{\\prime}[\/latex] and [latex]y''[\/latex] for [latex]x^2+6xy-2y^2=3[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 24. [T]<\/strong> The number of cell phones produced when [latex]x[\/latex] dollars is spent on labor and [latex]y[\/latex] dollars is spent on capital invested by a manufacturer can be modeled by the equation [latex]60x^{\\frac{3}{4}}y^{\\frac{1}{4}}=3240[\/latex].<\/p>\r\n\r\n a. -0.5926\r\nb. When $81 is spent on labor and $16 is spent on capital, the amount spent on capital is decreasing by $0.5926 per $1 spent on labor.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 25. [T]<\/strong> The number of cars produced when [latex]x[\/latex] dollars is spent on labor and [latex]y[\/latex] dollars is spent on capital invested by a manufacturer can be modeled by the equation [latex]30x^{\\frac{1}{3}}y^{\\frac{2}{3}}=360[\/latex].<\/p>\r\n (Both [latex]x[\/latex] and [latex]y[\/latex] are measured in thousands of dollars.)<\/p>\r\n\r\n 26.\u00a0<\/strong>The volume of a right circular cone of radius [latex]x[\/latex] and height [latex]y[\/latex] is given by [latex]V=\\frac{1}{3}\\pi x^2 y[\/latex]. Suppose that the volume of the cone is [latex]85\\pi \\, \\text{cm}^3[\/latex]. Find [latex]\\frac{dy}{dx}[\/latex] when [latex]x=4[\/latex] and [latex]y=16[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737954152\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737954152\"]\r\n [latex]\\frac{dy}{dx}=-8[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n For the following exercises (27-28), consider a closed rectangular box with a square base with side [latex]x[\/latex] and height [latex]y[\/latex].<\/p>\r\n\r\n 28.\u00a0<\/strong>If the surface area of the rectangular box is 78 square feet, find [latex]\\frac{dy}{dx}[\/latex] when [latex]x=3[\/latex] feet and [latex]y=5[\/latex] feet.<\/p>\r\n[reveal-answer q=\"fs-id1169738071332\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738071332\"]\r\n [latex]\\frac{dy}{dx}=-2.67[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n For the following exercises (29-31), use implicit differentiation to determine [latex]y^{\\prime}[\/latex]. Does the answer agree with the formulas we have previously determined?<\/p>\r\n\r\n 29.\u00a0<\/strong>[latex]x= \\sin y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 30.\u00a0<\/strong>[latex]x= \\cos y[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738071429\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738071429\"]\r\n [latex]y^{\\prime}=-\\frac{1}{\\sqrt{1-x^2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n 31.\u00a0<\/strong>[latex]x= \\tan y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":" For the following exercises (1-10), use implicit differentiation to find [latex]\\frac{dy}{dx}[\/latex].<\/p>\n 1.\u00a0<\/strong>[latex]x^2-y^2=4[\/latex]<\/p>\n<\/div>\n<\/div>\n 2.\u00a0<\/strong>[latex]6x^2+3y^2=12[\/latex]<\/p>\n [latex]\\frac{dy}{dx}=\\frac{-2x}{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n 3.\u00a0<\/strong>[latex]x^2 y=y-7[\/latex]<\/p>\n<\/div>\n<\/div>\n 4.\u00a0<\/strong>[latex]3x^3+9xy^2=5x^3[\/latex]<\/p>\n [latex]\\frac{dy}{dx}=\\frac{x}{3y}-\\frac{y}{2x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n 5.\u00a0<\/strong>[latex]xy- \\cos (xy)=1[\/latex]<\/p>\n<\/div>\n<\/div>\n 6.\u00a0<\/strong>[latex]y\\sqrt{x+4}=xy+8[\/latex]<\/p>\n [latex]\\frac{dy}{dx}=\\large \\frac{y-\\frac{y}{2\\sqrt{x+4}}}{\\sqrt{x+4}-x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n 7.\u00a0<\/strong>[latex]\u2212xy-2=\\dfrac{x}{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n 8.\u00a0<\/strong>[latex]y \\sin(xy)=y^2+2[\/latex]<\/p>\n [latex]\\frac{dy}{dx}=\\large \\frac{y^2 \\cos(xy)}{2y- \\sin(xy)-xy \\cos xy}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n 9.\u00a0<\/strong>[latex](xy)^2+3x=y^2[\/latex]<\/p>\n<\/div>\n<\/div>\n 10.\u00a0<\/strong>[latex]x^3 y+xy^3=-8[\/latex]<\/p>\n For the following exercises (11-16), find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.<\/p>\n 11. [T]\u00a0\u00a0<\/strong>[latex]x^4 y-xy^3=-2, \\,\\,\\, (-1,-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n 12. [T]\u00a0\u00a0<\/strong>[latex]x^2 y^2+5xy=14, \\,\\,\\, (2,1)[\/latex]<\/p>\n 13. [T]\u00a0\u00a0<\/strong>[latex]\\tan (xy)=y, \\,\\,\\, (\\frac{\\pi}{4},1)[\/latex]<\/p>\n<\/div>\n<\/div>\n 14. [T]\u00a0\u00a0<\/b>[latex]xy^2 + \\sin(\\pi y)-2x^2=10, \\,\\,\\, (2,-3)[\/latex]<\/p>\n 15. [T]\u00a0\u00a0<\/strong>[latex]\\frac{x}{y}+5x-7=-\\frac{3}{4}y, \\,\\,\\, (1,2)[\/latex]<\/p>\n<\/div>\n<\/div>\n 16. [T]\u00a0\u00a0<\/strong>[latex]xy+ \\sin (x)=1, \\,\\,\\, (\\frac{\\pi}{2},0)[\/latex]<\/p>\n 17. [T]<\/strong> The graph of a folium of Descartes with equation [latex]2x^3+2y^3-9xy=0[\/latex] is given in the following graph.<\/p>\n 18.\u00a0<\/strong>For the equation [latex]x^2+2xy-3y^2=0[\/latex],<\/p>\n a. [latex]y=\u2212x+2[\/latex] 19.\u00a0<\/strong>Find all points on the graph of [latex]y^3-27y=x^2-90[\/latex] at which the tangent line is vertical.<\/p>\n<\/div>\n<\/div>\n 20.\u00a0<\/strong>For the equation [latex]x^2+xy+y^2=7[\/latex],<\/p>\n![\"The](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205513\/CNX_Calc_Figure_03_08_202.jpg\")
<\/span>\r\n[latex]y=-\\frac{1}{2}x+2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n![\"The](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205517\/CNX_Calc_Figure_03_08_204.jpg\")
<\/span>\r\n[latex]y=\\large \\frac{1}{\\pi +12}x-\\frac{3\\pi +38}{\\pi +12}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n![\"The](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205521\/CNX_Calc_Figure_03_08_206.jpg\")
<\/span>\r\n[latex]y=0[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n![\"A](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205525\/CNX_Calc_Figure_03_08_207.jpg\")
<\/span>\r\n\r\n \t
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\n[latex]y=-\\frac{1}{2}x+2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/span>
\n[latex]y=\\large \\frac{1}{\\pi +12}x-\\frac{3\\pi +38}{\\pi +12}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/span>
\n[latex]y=0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/span><\/p>\n\n
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\nb. [latex](3,-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n