{"id":470,"date":"2021-02-04T15:29:25","date_gmt":"2021-02-04T15:29:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=470"},"modified":"2021-03-31T17:17:43","modified_gmt":"2021-03-31T17:17:43","slug":"problem-set-the-chain-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-the-chain-rule\/","title":{"raw":"Problem Set: The Chain Rule","rendered":"Problem Set: The Chain Rule"},"content":{"raw":"

For the following exercises (1-6), given [latex]y=f(u)[\/latex] and [latex]u=g(x)[\/latex], find [latex]\\frac{dy}{dx}[\/latex] by using Leibniz\u2019s notation for the chain rule: [latex]\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}[\/latex].<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]y=3u-6, \\,\\,\\, u=2x^2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex]y=6u^3, \\,\\,\\, u=7x-4[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736659466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736659466\"]\r\n

[latex]\\frac{dy}{dx} = 18u^2 \\cdot 7=18(7x-4)^2 \\cdot 7[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]y= \\sin u, \\,\\,\\, u=5x-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>[latex]y= \\cos u, \\,\\,\\, u=\\dfrac{\u2212x}{8}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739281153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739281153\"]\r\n

[latex]\\frac{dy}{dx} = \u2212\\sin u \\cdot \\frac{-1}{8}=\u2212\\sin (\\frac{\u2212x}{8}) \\cdot \\frac{-1}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]y= \\tan u, \\,\\,\\, u=9x+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]y=\\sqrt{4u+3}, \\,\\,\\, u=x^2-6x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739262442\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739262442\"]\r\n

[latex]\\frac{dy}{dx} = \\frac{8x-24}{2\\sqrt{4u+3}}=\\frac{4x-12}{\\sqrt{4x^2-24x+3}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For each of the following exercises (7-14),<\/p>\r\n\r\n

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  1. decompose each function in the form [latex]y=f(u)[\/latex] and [latex]u=g(x)[\/latex], and<\/li>\r\n \t
  2. find [latex]\\frac{dy}{dx}[\/latex] as a function of [latex]x[\/latex].<\/li>\r\n<\/ol>\r\n
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    7.\u00a0<\/strong>[latex]y=(3x-2)^6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    8.\u00a0<\/strong>[latex]y=(3x^2+1)^3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736602866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736602866\"]\r\n

    a. [latex]f(u) = u^3, \\, u=3x^2+1[\/latex];\r\nb. [latex]\\frac{dy}{dx} = 18x(3x^2+1)^2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    9.\u00a0<\/strong>[latex]y= \\sin^5 (x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    10.\u00a0<\/strong>[latex]y=\\left(\\dfrac{x}{7}+\\dfrac{7}{x}\\right)^7[\/latex]<\/p>\r\n\r\n

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    a. [latex]f(u)=u^7, \\, u=\\frac{x}{7}+\\frac{7}{x}[\/latex];\r\nb. [latex]\\frac{dy}{dx} = 7(\\frac{x}{7}+\\frac{7}{x})^6 \\cdot (\\frac{1}{7}-\\frac{7}{x^2})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    11.\u00a0<\/strong>[latex]y= \\tan ( \\sec x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    12.\u00a0<\/strong>[latex]y= \\csc (\\pi x+1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739251149\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739251149\"]\r\n

    a. [latex]f(u)= \\csc u, \\, u=\\pi x+1[\/latex];\r\nb. [latex]\\frac{dy}{dx} = \u2212\\pi \\csc (\\pi x+1) \\cdot \\cot (\\pi x+1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    13.\u00a0<\/strong>[latex]y= \\cot^2 x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    14.\u00a0<\/strong>[latex]y=-6 \\sin^{-3} x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739280445\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739280445\"]\r\n

    a. [latex]f(u)=-6u^{-3}, \\, u= \\sin x[\/latex];\r\nb. [latex]\\frac{dy}{dx} = 18 \\sin^{-4} x \\cdot \\cos x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (15-24), find [latex]\\frac{dy}{dx}[\/latex] for each function.<\/p>\r\n\r\n

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    15.\u00a0<\/strong>[latex]y=(3x^2+3x-1)^4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    16.\u00a0<\/strong>[latex]y=(5-2x)^{-2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736653286\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736653286\"]\r\n

    [latex]\\frac{dy}{dx} = \\frac{4}{(5-2x)^3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    17.\u00a0<\/strong>[latex]y= \\cos^3 (\\pi x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    18.\u00a0<\/strong>[latex]y=(2x^3-x^2+6x+1)^3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739195192\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739195192\"]\r\n

    [latex]\\frac{dy}{dx} = 6(2x^3-x^2+6x+1)^2(3x^2-x+3)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    19.\u00a0<\/strong>[latex]y=\\dfrac{1}{\\sin^2 (x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    20.\u00a0<\/strong>[latex]y=(\\tan x+ \\sin x)^{-3}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738989496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738989496\"]\r\n

    [latex]\\frac{dy}{dx} = -3(\\tan x+ \\sin x)^{-4} \\cdot (\\sec^2 x+ \\cos x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    21.\u00a0<\/strong>[latex]y=x^2 \\cos^4 x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    22.\u00a0<\/strong>[latex]y= \\sin (\\cos 7x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738989676\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738989676\"]\r\n

    [latex]\\frac{dy}{dx} = -7 \\cos (\\cos 7x) \\cdot \\sin 7x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    23.\u00a0<\/strong>[latex]y=\\sqrt{6+ \\sec \\pi x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    24.\u00a0<\/strong>[latex]y= \\cot^3 (4x+1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736585152\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736585152\"]\r\n

    [latex]\\frac{dy}{dx} = -12 \\cot^2 (4x+1) \\cdot \\csc^2 (4x+1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    25.\u00a0<\/strong>Let [latex]y=(f(x))^3[\/latex] and suppose that [latex]f^{\\prime}(1)=4[\/latex] and [latex]\\frac{dy}{dx}=10[\/latex] for [latex]x=1[\/latex]. Find [latex]f(1)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    26.\u00a0<\/strong>Let [latex]y=(f(x)+5x^2)^4[\/latex] and suppose that [latex]f(-1)=-4[\/latex] and [latex]\\frac{dy}{dx}=3[\/latex] when [latex]x=-1[\/latex]. Find [latex]f^{\\prime}(-1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739374485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739374485\"]\r\n

    [latex]10\\frac{3}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    27.\u00a0<\/strong>Let [latex]y=(f(u)+3x)^2[\/latex] and [latex]u=x^3-2x[\/latex]. If [latex]f(4)=6[\/latex] and [latex]\\frac{dy}{dx}=18[\/latex] when [latex]x=2[\/latex], find [latex]f^{\\prime}(4)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    28. [T]<\/strong> Find the equation of the tangent line to [latex]y=\u2212\\sin \\left(\\dfrac{x}{2}\\right)[\/latex] at the origin. Use a calculator to graph the function and the tangent line together.<\/p>\r\n[reveal-answer q=\"fs-id1169739374682\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739374682\"]\r\n

    [latex]y=-\\frac{1}{2}x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    29. [T]<\/strong> Find the equation of the tangent line to [latex]y=\\left(3x+\\dfrac{1}{x}\\right)^2[\/latex] at the point [latex](1,16)[\/latex]. Use a calculator to graph the function and the tangent line together.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    30.\u00a0<\/strong>Find the [latex]x[\/latex]-coordinates at which the tangent line to [latex]y=\\left(x-\\dfrac{6}{x}\\right)^8[\/latex] is horizontal.<\/p>\r\n[reveal-answer q=\"fs-id1169738991998\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738991998\"]\r\n

    [latex]x= \\pm \\sqrt{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    31. [T]<\/strong> Find an equation of the line that is normal to [latex]g(\\theta)= \\sin^2 (\\pi \\theta)[\/latex] at the point [latex]\\left(\\frac{1}{4},\\frac{1}{2}\\right)[\/latex]. Use a calculator to graph the function and the normal line together.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (32-39), use the information in the following table to find [latex]h^{\\prime}(a)[\/latex] at the given value for [latex]a[\/latex].<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    [latex]x[\/latex]<\/th>\r\n[latex]f(x)[\/latex]<\/th>\r\n[latex]f^{\\prime}(x)[\/latex]<\/th>\r\n[latex]g(x)[\/latex]<\/th>\r\n[latex]g^{\\prime}(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    0<\/td>\r\n2<\/td>\r\n5<\/td>\r\n0<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
    1<\/td>\r\n1<\/td>\r\n\u22122<\/td>\r\n3<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
    2<\/td>\r\n4<\/td>\r\n4<\/td>\r\n1<\/td>\r\n\u22121<\/td>\r\n<\/tr>\r\n
    3<\/td>\r\n3<\/td>\r\n\u22123<\/td>\r\n2<\/td>\r\n3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
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    32.\u00a0<\/strong>[latex]h(x)=f(g(x)); \\,\\,\\, a=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739243384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739243384\"]\r\n

    10<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    33.\u00a0<\/strong>[latex]h(x)=g(f(x)); \\,\\,\\, a=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    34.\u00a0<\/strong>[latex]h(x)=(x^4+g(x))^{-2}; \\,\\,\\, a=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736662118\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662118\"]\r\n

    [latex]-\\frac{1}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    35.\u00a0<\/strong>[latex]h(x)=\\left(\\dfrac{f(x)}{g(x)}\\right)^2; \\,\\,\\, a=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    36.\u00a0<\/strong>[latex]h(x)=f(x+f(x)); \\,\\,\\, a=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736608228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736608228\"]\r\n

    -4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    37.\u00a0<\/strong>[latex]h(x)=(1+g(x))^3; \\, a=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    38.\u00a0<\/strong>[latex]h(x)=g(2+f(x^2)); \\, a=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736608364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736608364\"]\r\n

    -12<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    39.\u00a0<\/strong>[latex]h(x)=f(g(\\sin x)); \\, a=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    40. [T]<\/strong> The position function of a freight train is given by [latex]s(t)=100(t+1)^{-2}[\/latex], with [latex]s[\/latex] in meters and [latex]t[\/latex] in seconds. At time [latex]t=6[\/latex] s, find the train\u2019s<\/p>\r\n\r\n

      \r\n \t
    1. velocity and<\/li>\r\n \t
    2. acceleration.<\/li>\r\n \t
    3. Using a. and b. is the train speeding up or slowing down?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169736592256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736592256\"]\r\n

      a. [latex]-\\frac{200}{343}[\/latex] m\/s;\r\nb. [latex]\\frac{600}{2401} \\, \\text{m\/s}^2[\/latex];\r\nc. The train is slowing down since velocity and acceleration have opposite signs.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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      41. [T]<\/strong> A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where [latex]t[\/latex] is measured in seconds and [latex]s[\/latex] is in inches:<\/p>\r\n

      [latex]s(t)=-3 \\cos \\left(\\pi t+\\dfrac{\\pi}{4}\\right)[\/latex].<\/p>\r\n\r\n

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      1. Determine the position of the spring at [latex]t=1.5[\/latex] s.<\/li>\r\n \t
      2. Find the velocity of the spring at [latex]t=1.5[\/latex] s.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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        42. [T]<\/strong> The total cost to produce [latex]x[\/latex] boxes of Thin Mint Girl Scout cookies is [latex]C[\/latex] dollars, where [latex]C=0.0001x^3-0.02x^2+3x+300[\/latex]. In [latex]t[\/latex] weeks production is estimated to be [latex]x=1600+100t[\/latex] boxes.<\/p>\r\n\r\n

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        1. Find the marginal cost [latex]C^{\\prime}(x)[\/latex].<\/li>\r\n \t
        2. Use Leibniz\u2019s notation for the chain rule, [latex]\\frac{dC}{dt}=\\frac{dC}{dx} \\cdot \\frac{dx}{dt}[\/latex], to find the rate with respect to time [latex]t[\/latex] that the cost is changing.<\/li>\r\n \t
        3. Use b. to determine how fast costs are increasing when [latex]t=2[\/latex] weeks. Include units with the answer.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169736616383\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736616383\"]\r\n

          a. [latex]C^{\\prime}(x)=0.0003x^2-0.04x+3[\/latex]\r\nb. [latex]\\frac{dC}{dt}=100 \\cdot (0.0003x^2-0.04x+3)[\/latex]\r\nc. Approximately $90,300 per week<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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          43. [T]<\/strong> The formula for the area of a circle is [latex]A=\\pi r^2[\/latex], where [latex]r[\/latex] is the radius of the circle. Suppose a circle is expanding, meaning that both the area [latex]A[\/latex] and the radius [latex]r[\/latex] (in inches) are expanding.<\/p>\r\n\r\n

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          1. Suppose [latex]r=2-\\dfrac{100}{(t+7)^2}[\/latex] where [latex]t[\/latex] is time in seconds. Use the chain rule [latex]\\frac{dA}{dt}=\\frac{dA}{dr} \\cdot \\frac{dr}{dt}[\/latex] to find the rate at which the area is expanding.<\/li>\r\n \t
          2. Use a. to find the rate at which the area is expanding at [latex]t=4[\/latex] s.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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            44. [T]<\/strong> The formula for the volume of a sphere is [latex]S=\\frac{4}{3}\\pi r^3[\/latex], where [latex]r[\/latex] (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.<\/p>\r\n\r\n

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            1. Suppose [latex]r=\\dfrac{1}{(t+1)^2}-\\dfrac{1}{12}[\/latex] where [latex]t[\/latex] is time in minutes. Use the chain rule [latex]\\frac{dS}{dt}=\\frac{dS}{dr} \\cdot \\frac{dr}{dt}[\/latex] to find the rate at which the snowball is melting.<\/li>\r\n \t
            2. Use a. to find the rate at which the volume is changing at [latex]t=1[\/latex] min.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169739066789\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739066789\"]\r\n

              a. [latex]\\frac{dS}{dt}=-\\frac{8\\pi r^2}{(t+1)^3}[\/latex]\r\nb. The volume is decreasing at a rate of [latex]-\\frac{\\pi}{36} \\, \\text{ft}^3\/\\text{min}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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              45. [T]<\/strong> The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function [latex]T(x)=94-10 \\cos \\left[\\dfrac{\\pi}{12}(x-2)\\right][\/latex], where [latex]x[\/latex] is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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              46. [T]<\/strong> The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function [latex]D(t)=5 \\sin \\left(\\dfrac{\\pi}{6} t-\\dfrac{7\\pi}{6}\\right)+8[\/latex], where [latex]t[\/latex] is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.<\/p>\r\n[reveal-answer q=\"fs-id1169739296729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739296729\"]\r\n

              [latex] \\approx 2.3[\/latex] ft\/hr<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

              For the following exercises (1-6), given [latex]y=f(u)[\/latex] and [latex]u=g(x)[\/latex], find [latex]\\frac{dy}{dx}[\/latex] by using Leibniz\u2019s notation for the chain rule: [latex]\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}[\/latex].<\/p>\n

              \n
              \n

              1.\u00a0<\/strong>[latex]y=3u-6, \\,\\,\\, u=2x^2[\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              2.\u00a0<\/strong>[latex]y=6u^3, \\,\\,\\, u=7x-4[\/latex]<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              [latex]\\frac{dy}{dx} = 18u^2 \\cdot 7=18(7x-4)^2 \\cdot 7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              3.\u00a0<\/strong>[latex]y= \\sin u, \\,\\,\\, u=5x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              4.\u00a0<\/strong>[latex]y= \\cos u, \\,\\,\\, u=\\dfrac{\u2212x}{8}[\/latex]<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              [latex]\\frac{dy}{dx} = \u2212\\sin u \\cdot \\frac{-1}{8}=\u2212\\sin (\\frac{\u2212x}{8}) \\cdot \\frac{-1}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              \n
              \n

              5.\u00a0<\/strong>[latex]y= \\tan u, \\,\\,\\, u=9x+2[\/latex]<\/p>\n<\/div>\n<\/div>\n

              \n
              \n

              6.\u00a0<\/strong>[latex]y=\\sqrt{4u+3}, \\,\\,\\, u=x^2-6x[\/latex]<\/p>\n

              Show Solution<\/span><\/p>\n
              \n

              [latex]\\frac{dy}{dx} = \\frac{8x-24}{2\\sqrt{4u+3}}=\\frac{4x-12}{\\sqrt{4x^2-24x+3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              For each of the following exercises (7-14),<\/p>\n

                \n
              1. decompose each function in the form [latex]y=f(u)[\/latex] and [latex]u=g(x)[\/latex], and<\/li>\n
              2. find [latex]\\frac{dy}{dx}[\/latex] as a function of [latex]x[\/latex].<\/li>\n<\/ol>\n
                \n
                \n

                7.\u00a0<\/strong>[latex]y=(3x-2)^6[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                8.\u00a0<\/strong>[latex]y=(3x^2+1)^3[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                a. [latex]f(u) = u^3, \\, u=3x^2+1[\/latex];
                \nb. [latex]\\frac{dy}{dx} = 18x(3x^2+1)^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                9.\u00a0<\/strong>[latex]y= \\sin^5 (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                10.\u00a0<\/strong>[latex]y=\\left(\\dfrac{x}{7}+\\dfrac{7}{x}\\right)^7[\/latex]<\/p>\n

                \n
                Show Solution<\/span><\/p>\n
                \n

                a. [latex]f(u)=u^7, \\, u=\\frac{x}{7}+\\frac{7}{x}[\/latex];
                \nb. [latex]\\frac{dy}{dx} = 7(\\frac{x}{7}+\\frac{7}{x})^6 \\cdot (\\frac{1}{7}-\\frac{7}{x^2})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                11.\u00a0<\/strong>[latex]y= \\tan ( \\sec x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                12.\u00a0<\/strong>[latex]y= \\csc (\\pi x+1)[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                a. [latex]f(u)= \\csc u, \\, u=\\pi x+1[\/latex];
                \nb. [latex]\\frac{dy}{dx} = \u2212\\pi \\csc (\\pi x+1) \\cdot \\cot (\\pi x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                13.\u00a0<\/strong>[latex]y= \\cot^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                14.\u00a0<\/strong>[latex]y=-6 \\sin^{-3} x[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                a. [latex]f(u)=-6u^{-3}, \\, u= \\sin x[\/latex];
                \nb. [latex]\\frac{dy}{dx} = 18 \\sin^{-4} x \\cdot \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                For the following exercises (15-24), find [latex]\\frac{dy}{dx}[\/latex] for each function.<\/p>\n

                \n
                \n

                15.\u00a0<\/strong>[latex]y=(3x^2+3x-1)^4[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                16.\u00a0<\/strong>[latex]y=(5-2x)^{-2}[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]\\frac{dy}{dx} = \\frac{4}{(5-2x)^3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                17.\u00a0<\/strong>[latex]y= \\cos^3 (\\pi x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                18.\u00a0<\/strong>[latex]y=(2x^3-x^2+6x+1)^3[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]\\frac{dy}{dx} = 6(2x^3-x^2+6x+1)^2(3x^2-x+3)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                19.\u00a0<\/strong>[latex]y=\\dfrac{1}{\\sin^2 (x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                20.\u00a0<\/strong>[latex]y=(\\tan x+ \\sin x)^{-3}[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]\\frac{dy}{dx} = -3(\\tan x+ \\sin x)^{-4} \\cdot (\\sec^2 x+ \\cos x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                21.\u00a0<\/strong>[latex]y=x^2 \\cos^4 x[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                22.\u00a0<\/strong>[latex]y= \\sin (\\cos 7x)[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]\\frac{dy}{dx} = -7 \\cos (\\cos 7x) \\cdot \\sin 7x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                23.\u00a0<\/strong>[latex]y=\\sqrt{6+ \\sec \\pi x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                24.\u00a0<\/strong>[latex]y= \\cot^3 (4x+1)[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]\\frac{dy}{dx} = -12 \\cot^2 (4x+1) \\cdot \\csc^2 (4x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                25.\u00a0<\/strong>Let [latex]y=(f(x))^3[\/latex] and suppose that [latex]f^{\\prime}(1)=4[\/latex] and [latex]\\frac{dy}{dx}=10[\/latex] for [latex]x=1[\/latex]. Find [latex]f(1)[\/latex].<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                26.\u00a0<\/strong>Let [latex]y=(f(x)+5x^2)^4[\/latex] and suppose that [latex]f(-1)=-4[\/latex] and [latex]\\frac{dy}{dx}=3[\/latex] when [latex]x=-1[\/latex]. Find [latex]f^{\\prime}(-1)[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]10\\frac{3}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                27.\u00a0<\/strong>Let [latex]y=(f(u)+3x)^2[\/latex] and [latex]u=x^3-2x[\/latex]. If [latex]f(4)=6[\/latex] and [latex]\\frac{dy}{dx}=18[\/latex] when [latex]x=2[\/latex], find [latex]f^{\\prime}(4)[\/latex].<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                28. [T]<\/strong> Find the equation of the tangent line to [latex]y=\u2212\\sin \\left(\\dfrac{x}{2}\\right)[\/latex] at the origin. Use a calculator to graph the function and the tangent line together.<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]y=-\\frac{1}{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                29. [T]<\/strong> Find the equation of the tangent line to [latex]y=\\left(3x+\\dfrac{1}{x}\\right)^2[\/latex] at the point [latex](1,16)[\/latex]. Use a calculator to graph the function and the tangent line together.<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                30.\u00a0<\/strong>Find the [latex]x[\/latex]-coordinates at which the tangent line to [latex]y=\\left(x-\\dfrac{6}{x}\\right)^8[\/latex] is horizontal.<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]x= \\pm \\sqrt{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                31. [T]<\/strong> Find an equation of the line that is normal to [latex]g(\\theta)= \\sin^2 (\\pi \\theta)[\/latex] at the point [latex]\\left(\\frac{1}{4},\\frac{1}{2}\\right)[\/latex]. Use a calculator to graph the function and the normal line together.<\/p>\n<\/div>\n<\/div>\n

                For the following exercises (32-39), use the information in the following table to find [latex]h^{\\prime}(a)[\/latex] at the given value for [latex]a[\/latex].<\/p>\n\n\n\n\n\n\n\n\n
                [latex]x[\/latex]<\/th>\n[latex]f(x)[\/latex]<\/th>\n[latex]f^{\\prime}(x)[\/latex]<\/th>\n[latex]g(x)[\/latex]<\/th>\n[latex]g^{\\prime}(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
                0<\/td>\n2<\/td>\n5<\/td>\n0<\/td>\n2<\/td>\n<\/tr>\n
                1<\/td>\n1<\/td>\n\u22122<\/td>\n3<\/td>\n0<\/td>\n<\/tr>\n
                2<\/td>\n4<\/td>\n4<\/td>\n1<\/td>\n\u22121<\/td>\n<\/tr>\n
                3<\/td>\n3<\/td>\n\u22123<\/td>\n2<\/td>\n3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                \n
                \n

                32.\u00a0<\/strong>[latex]h(x)=f(g(x)); \\,\\,\\, a=0[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                10<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                33.\u00a0<\/strong>[latex]h(x)=g(f(x)); \\,\\,\\, a=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                34.\u00a0<\/strong>[latex]h(x)=(x^4+g(x))^{-2}; \\,\\,\\, a=1[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                [latex]-\\frac{1}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                35.\u00a0<\/strong>[latex]h(x)=\\left(\\dfrac{f(x)}{g(x)}\\right)^2; \\,\\,\\, a=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                36.\u00a0<\/strong>[latex]h(x)=f(x+f(x)); \\,\\,\\, a=1[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                -4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                37.\u00a0<\/strong>[latex]h(x)=(1+g(x))^3; \\, a=2[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                38.\u00a0<\/strong>[latex]h(x)=g(2+f(x^2)); \\, a=1[\/latex]<\/p>\n

                Show Solution<\/span><\/p>\n
                \n

                -12<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                \n
                \n

                39.\u00a0<\/strong>[latex]h(x)=f(g(\\sin x)); \\, a=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                40. [T]<\/strong> The position function of a freight train is given by [latex]s(t)=100(t+1)^{-2}[\/latex], with [latex]s[\/latex] in meters and [latex]t[\/latex] in seconds. At time [latex]t=6[\/latex] s, find the train\u2019s<\/p>\n

                  \n
                1. velocity and<\/li>\n
                2. acceleration.<\/li>\n
                3. Using a. and b. is the train speeding up or slowing down?<\/li>\n<\/ol>\n
                  Show Solution<\/span><\/p>\n
                  \n

                  a. [latex]-\\frac{200}{343}[\/latex] m\/s;
                  \nb. [latex]\\frac{600}{2401} \\, \\text{m\/s}^2[\/latex];
                  \nc. The train is slowing down since velocity and acceleration have opposite signs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  41. [T]<\/strong> A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where [latex]t[\/latex] is measured in seconds and [latex]s[\/latex] is in inches:<\/p>\n

                  [latex]s(t)=-3 \\cos \\left(\\pi t+\\dfrac{\\pi}{4}\\right)[\/latex].<\/p>\n

                    \n
                  1. Determine the position of the spring at [latex]t=1.5[\/latex] s.<\/li>\n
                  2. Find the velocity of the spring at [latex]t=1.5[\/latex] s.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                    \n
                    \n

                    42. [T]<\/strong> The total cost to produce [latex]x[\/latex] boxes of Thin Mint Girl Scout cookies is [latex]C[\/latex] dollars, where [latex]C=0.0001x^3-0.02x^2+3x+300[\/latex]. In [latex]t[\/latex] weeks production is estimated to be [latex]x=1600+100t[\/latex] boxes.<\/p>\n

                      \n
                    1. Find the marginal cost [latex]C^{\\prime}(x)[\/latex].<\/li>\n
                    2. Use Leibniz\u2019s notation for the chain rule, [latex]\\frac{dC}{dt}=\\frac{dC}{dx} \\cdot \\frac{dx}{dt}[\/latex], to find the rate with respect to time [latex]t[\/latex] that the cost is changing.<\/li>\n
                    3. Use b. to determine how fast costs are increasing when [latex]t=2[\/latex] weeks. Include units with the answer.<\/li>\n<\/ol>\n
                      Show Solution<\/span><\/p>\n
                      \n

                      a. [latex]C^{\\prime}(x)=0.0003x^2-0.04x+3[\/latex]
                      \nb. [latex]\\frac{dC}{dt}=100 \\cdot (0.0003x^2-0.04x+3)[\/latex]
                      \nc. Approximately $90,300 per week<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                      \n
                      \n

                      43. [T]<\/strong> The formula for the area of a circle is [latex]A=\\pi r^2[\/latex], where [latex]r[\/latex] is the radius of the circle. Suppose a circle is expanding, meaning that both the area [latex]A[\/latex] and the radius [latex]r[\/latex] (in inches) are expanding.<\/p>\n

                        \n
                      1. Suppose [latex]r=2-\\dfrac{100}{(t+7)^2}[\/latex] where [latex]t[\/latex] is time in seconds. Use the chain rule [latex]\\frac{dA}{dt}=\\frac{dA}{dr} \\cdot \\frac{dr}{dt}[\/latex] to find the rate at which the area is expanding.<\/li>\n
                      2. Use a. to find the rate at which the area is expanding at [latex]t=4[\/latex] s.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                        \n
                        \n

                        44. [T]<\/strong> The formula for the volume of a sphere is [latex]S=\\frac{4}{3}\\pi r^3[\/latex], where [latex]r[\/latex] (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.<\/p>\n

                          \n
                        1. Suppose [latex]r=\\dfrac{1}{(t+1)^2}-\\dfrac{1}{12}[\/latex] where [latex]t[\/latex] is time in minutes. Use the chain rule [latex]\\frac{dS}{dt}=\\frac{dS}{dr} \\cdot \\frac{dr}{dt}[\/latex] to find the rate at which the snowball is melting.<\/li>\n
                        2. Use a. to find the rate at which the volume is changing at [latex]t=1[\/latex] min.<\/li>\n<\/ol>\n
                          Show Solution<\/span><\/p>\n
                          \n

                          a. [latex]\\frac{dS}{dt}=-\\frac{8\\pi r^2}{(t+1)^3}[\/latex]
                          \nb. The volume is decreasing at a rate of [latex]-\\frac{\\pi}{36} \\, \\text{ft}^3\/\\text{min}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                          \n
                          \n

                          45. [T]<\/strong> The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function [latex]T(x)=94-10 \\cos \\left[\\dfrac{\\pi}{12}(x-2)\\right][\/latex], where [latex]x[\/latex] is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.<\/p>\n<\/div>\n<\/div>\n

                          \n
                          \n

                          46. [T]<\/strong> The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function [latex]D(t)=5 \\sin \\left(\\dfrac{\\pi}{6} t-\\dfrac{7\\pi}{6}\\right)+8[\/latex], where [latex]t[\/latex] is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.<\/p>\n

                          Show Solution<\/span><\/p>\n
                          \n

                          [latex]\\approx 2.3[\/latex] ft\/hr<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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                          Candela Citations<\/h3>\n\t\t\t\t\t
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