{"id":469,"date":"2021-02-04T15:29:18","date_gmt":"2021-02-04T15:29:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=469"},"modified":"2021-04-08T19:23:09","modified_gmt":"2021-04-08T19:23:09","slug":"problem-set-derivatives-of-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-derivatives-of-trigonometric-functions\/","title":{"raw":"Problem Set: Derivatives of Trigonometric Functions","rendered":"Problem Set: Derivatives of Trigonometric Functions"},"content":{"raw":"

For the following exercises (1-10), find [latex]\\frac{dy}{dx}[\/latex] for the given functions.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]y=x^2- \\sec x+1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736597682\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736597682\"][latex]\\frac{dy}{dx}=2x- \\sec x \\tan x[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex]y=3 \\csc x+\\dfrac{5}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]y=x^2 \\cot x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736658563\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658563\"]\r\n

[latex]\\frac{dy}{dx}=2x \\cot x-x^2 \\csc^2 x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>[latex]y=x-x^3 \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]y=\\dfrac{\\sec x}{x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739275191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739275191\"]\r\n

[latex]\\frac{dy}{dx}=\\frac{x \\sec x \\tan x- \\sec x}{x^2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]y= \\sin x \\tan x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>[latex]y=(x+ \\cos x)(1- \\sin x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739303655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739303655\"]\r\n

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

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8.\u00a0<\/strong>[latex]y=\\dfrac{\\tan x}{1- \\sec x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]y=\\dfrac{1- \\cot x}{1+ \\cot x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739350855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739350855\"]\r\n

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

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10.\u00a0<\/strong>[latex]y= \\cos x(1+ \\csc x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (11-16), find the equation of the tangent line to each of the given functions at the indicated values of [latex]x[\/latex]. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.<\/p>\r\n\r\n

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11. [T]\u00a0<\/strong>[latex]f(x)=\u2212\\sin x, \\,\\,\\, x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739266640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739266640\"]\r\n

[latex]y=\u2212x[\/latex]<\/p>\r\n\"The<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. [T]\u00a0<\/strong>[latex]f(x)= \\csc x, \\,\\,\\, x=\\frac{\\pi}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [T]\u00a0<\/strong>[latex]f(x)=1+ \\cos x, \\,\\,\\, x=\\frac{3\\pi}{2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736655158\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736655158\"]\r\n

[latex]y=x+\\frac{2-3\\pi}{2}[\/latex]<\/p>\r\n\"The<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [T]\u00a0<\/strong>[latex]f(x)= \\sec x, \\,\\,\\, x=\\frac{\\pi}{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [T]\u00a0<\/strong>[latex]f(x)=x^2- \\tan x, \\,\\,\\, x=0[\/latex]<\/p>\r\n[reveal-answer q=\"780193\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"780193\"]\r\n\r\n[latex]y=\u2212x[\/latex]\r\n\r\n\"The<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [T]\u00a0<\/strong>[latex]f(x)=5 \\cot x, \\,\\,\\, x=\\frac{\\pi}{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (17-22), find [latex]\\frac{d^2 y}{dx^2}[\/latex] for the given functions.<\/p>\r\n\r\n

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17.\u00a0<\/strong>[latex]y=x \\sin x- \\cos x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736662274\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662274\"]\r\n

[latex]\\frac{d^2 y}{dx^2} = 3 \\cos x-x \\sin x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n18.\u00a0<\/strong>[latex]y= \\sin x \\cos x[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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19.\u00a0<\/strong>[latex]y=x-\\frac{1}{2} \\sin x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736662404\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662404\"]\r\n

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

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20.\u00a0<\/strong>[latex]y=\\frac{1}{x}+ \\tan x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>[latex]y=2 \\csc x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739303458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739303458\"]\r\n

[latex]\\frac{d^2 y}{dx^2} = 2\\csc x( \\csc^2 x + \\cot^2 x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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23.\u00a0<\/strong>Find all [latex]x[\/latex] values on the graph of [latex]f(x)=-3 \\sin x \\cos x[\/latex] where the tangent line is horizontal.<\/p>\r\n[reveal-answer q=\"fs-id1169739335862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739335862\"]\r\n

[latex]x = \\frac{(2n+1)\\pi}{4}[\/latex], where [latex]n[\/latex] is an integer<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>Find all [latex]x[\/latex] values on the graph of [latex]f(x)=x-2 \\cos x[\/latex] for [latex]0<x<2\\pi[\/latex] where the tangent line has a slope of 2.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>Let [latex]f(x)= \\cot x[\/latex]. Determine the point(s) on the graph of [latex]f[\/latex] for [latex]0<x<2\\pi[\/latex] where the tangent line is parallel to the line [latex]y=-2x[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736613904\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736613904\"]\r\n

[latex](\\frac{\\pi}{4},1), \\, (\\frac{3\\pi}{4},-1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n26. [T]<\/strong> A mass on a spring bounces up and down in simple harmonic motion, modeled by the function [latex]s(t)=-6 \\cos t[\/latex] where [latex]s[\/latex] is measured in inches and [latex]t[\/latex] is measured in seconds. Find the rate at which the spring is oscillating at [latex]t=5[\/latex] s.\r\n\r\n<\/div>\r\n<\/div>\r\n
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\r\n\r\n27.<\/strong> Let the position of a swinging pendulum in simple harmonic motion be given by\u00a0[latex]s(t)=a \\cos t+b \\sin t[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are constants, [latex]t[\/latex] measures time in seconds, and [latex]s[\/latex] measures position in centimeters, If the position is [latex]0[\/latex]cm and the velocity is [latex]3[\/latex]cm\/s when [latex]t=0[\/latex], find the values of [latex]a[\/latex] and [latex]b[\/latex].\r\n\r\n[reveal-answer q=\"fs-id1169739341388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739341388\"]\r\n

[latex]a=0, \\, b=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>After a diver jumps off a diving board, the edge of the board oscillates with position given by [latex]s(t)=-5 \\cos t[\/latex] cm at [latex]t[\/latex] seconds after the jump.<\/p>\r\n\r\n

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  1. Sketch one period of the position function for [latex]t\\ge 0[\/latex].<\/li>\r\n \t
  2. Find the velocity function.<\/li>\r\n \t
  3. Sketch one period of the velocity function for [latex]t\\ge 0[\/latex].<\/li>\r\n \t
  4. Determine the times when the velocity is 0 over one period.<\/li>\r\n \t
  5. Find the acceleration function.<\/li>\r\n \t
  6. Sketch one period of the acceleration function for [latex]t\\ge 0[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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    29.\u00a0<\/strong>The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by [latex]y=10+5 \\sin x[\/latex] where [latex]y[\/latex] is the number of hamburgers sold and [latex]x[\/latex] represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find [latex]y^{\\prime}[\/latex] and determine the intervals where the number of burgers being sold is increasing.<\/p>\r\n[reveal-answer q=\"fs-id1169739353429\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739353429\"]\r\n

    [latex]y^{\\prime}=5 \\cos (x)[\/latex], increasing on [latex](0,\\frac{\\pi}{2}), \\, (\\frac{3\\pi}{2},\\frac{5\\pi}{2})[\/latex], and [latex](\\frac{7\\pi}{2},12)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    30. [T]<\/strong> The amount of rainfall per month in Phoenix, Arizona, can be approximated by [latex]y(t)=0.5+0.3 \\cos t[\/latex], where [latex]t[\/latex] is the number of months since January. Find [latex]y^{\\prime}[\/latex] and use a calculator to determine the intervals where the amount of rain falling is decreasing.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (31-33), use the quotient rule to derive the given equations.<\/p>\r\n\r\n

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    31.\u00a0<\/strong>[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    32.\u00a0<\/strong>[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    33.\u00a0<\/strong>[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    34.\u00a0<\/strong>Use the definition of derivative and the identity<\/p>\r\n

    [latex]\\cos (x+h)= \\cos x \\cos h- \\sin x \\sin h[\/latex]\u00a0 to prove that\u00a0 [latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (35-39), find the requested higher-order derivative for the given functions.<\/p>\r\n\r\n

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    35.\u00a0<\/strong>[latex]\\frac{d^3 y}{dx^3}[\/latex] of [latex]y=3 \\cos x[\/latex]<\/p>\r\n\r\n

    [reveal-answer q=\"501872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"501872\"][latex]\\frac{d^3 y}{dx^3} = 3 \\sin x[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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    36.\u00a0<\/strong>[latex]\\frac{d^2 y}{dx^2}[\/latex] of [latex]y=3 \\sin x+x^2 \\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    37.\u00a0<\/strong>[latex]\\frac{d^4 y}{dx^4}[\/latex] of [latex]y=5 \\cos x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736592517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736592517\"]\r\n

    [latex]\\frac{d^4 y}{dx^4} = 5 \\cos x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    38.\u00a0<\/strong>[latex]\\frac{d^2 y}{dx^2}[\/latex] of [latex]y= \\sec x+ \\cot x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    39.\u00a0<\/strong>[latex]\\frac{d^3 y}{dx^3}[\/latex] of [latex]y=x^{10}- \\sec x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739301990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739301990\"]\r\n

    [latex]\\frac{d^3 y}{dx^3} = 720x^7-5 \\tan (x) \\sec^3 (x)- \\tan^3 (x) \\sec (x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

    For the following exercises (1-10), find [latex]\\frac{dy}{dx}[\/latex] for the given functions.<\/p>\n

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    1.\u00a0<\/strong>[latex]y=x^2- \\sec x+1[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
    [latex]\\frac{dy}{dx}=2x- \\sec x \\tan x[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    2.\u00a0<\/strong>[latex]y=3 \\csc x+\\dfrac{5}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

    Show Solution<\/span><\/p>\n
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    [latex]\\frac{dy}{dx}=2x \\cot x-x^2 \\csc^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    4.\u00a0<\/strong>[latex]y=x-x^3 \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    5.\u00a0<\/strong>[latex]y=\\dfrac{\\sec x}{x}[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]\\frac{dy}{dx}=\\frac{x \\sec x \\tan x- \\sec x}{x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    6.\u00a0<\/strong>[latex]y= \\sin x \\tan x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    7.\u00a0<\/strong>[latex]y=(x+ \\cos x)(1- \\sin x)[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]\\frac{dy}{dx}=(1- \\sin x)(1- \\sin x)- \\cos x(x+ \\cos x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    8.\u00a0<\/strong>[latex]y=\\dfrac{\\tan x}{1- \\sec x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    9.\u00a0<\/strong>[latex]y=\\dfrac{1- \\cot x}{1+ \\cot x}[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]\\frac{dy}{dx}=\\frac{2 \\csc^2 x}{(1+ \\cot x)^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

    For the following exercises (11-16), find the equation of the tangent line to each of the given functions at the indicated values of [latex]x[\/latex]. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.<\/p>\n

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    11. [T]\u00a0<\/strong>[latex]f(x)=\u2212\\sin x, \\,\\,\\, x=0[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]y=\u2212x[\/latex]<\/p>\n

    \"The<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    12. [T]\u00a0<\/strong>[latex]f(x)= \\csc x, \\,\\,\\, x=\\frac{\\pi}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    13. [T]\u00a0<\/strong>[latex]f(x)=1+ \\cos x, \\,\\,\\, x=\\frac{3\\pi}{2}[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]y=x+\\frac{2-3\\pi}{2}[\/latex]<\/p>\n

    \"The<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    14. [T]\u00a0<\/strong>[latex]f(x)= \\sec x, \\,\\,\\, x=\\frac{\\pi}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    15. [T]\u00a0<\/strong>[latex]f(x)=x^2- \\tan x, \\,\\,\\, x=0[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]y=\u2212x[\/latex]<\/p>\n

    \"The<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    16. [T]\u00a0<\/strong>[latex]f(x)=5 \\cot x, \\,\\,\\, x=\\frac{\\pi}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

    For the following exercises (17-22), find [latex]\\frac{d^2 y}{dx^2}[\/latex] for the given functions.<\/p>\n

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    17.\u00a0<\/strong>[latex]y=x \\sin x- \\cos x[\/latex]<\/p>\n

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    [latex]\\frac{d^2 y}{dx^2} = 3 \\cos x-x \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    18.\u00a0<\/strong>[latex]y= \\sin x \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    19.\u00a0<\/strong>[latex]y=x-\\frac{1}{2} \\sin x[\/latex]<\/p>\n

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    [latex]\\frac{d^2 y}{dx^2} = \\frac{1}{2} \\sin x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    20.\u00a0<\/strong>[latex]y=\\frac{1}{x}+ \\tan x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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    21.\u00a0<\/strong>[latex]y=2 \\csc x[\/latex]<\/p>\n

    Show Solution<\/span><\/p>\n
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    [latex]\\frac{d^2 y}{dx^2} = 2\\csc x( \\csc^2 x + \\cot^2 x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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    23.\u00a0<\/strong>Find all [latex]x[\/latex] values on the graph of [latex]f(x)=-3 \\sin x \\cos x[\/latex] where the tangent line is horizontal.<\/p>\n

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    [latex]x = \\frac{(2n+1)\\pi}{4}[\/latex], where [latex]n[\/latex] is an integer<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    24.\u00a0<\/strong>Find all [latex]x[\/latex] values on the graph of [latex]f(x)=x-2 \\cos x[\/latex] for [latex]0\n<\/div>\n<\/div>\n

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    25.\u00a0<\/strong>Let [latex]f(x)= \\cot x[\/latex]. Determine the point(s) on the graph of [latex]f[\/latex] for [latex]0\n

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    [latex](\\frac{\\pi}{4},1), \\, (\\frac{3\\pi}{4},-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    26. [T]<\/strong> A mass on a spring bounces up and down in simple harmonic motion, modeled by the function [latex]s(t)=-6 \\cos t[\/latex] where [latex]s[\/latex] is measured in inches and [latex]t[\/latex] is measured in seconds. Find the rate at which the spring is oscillating at [latex]t=5[\/latex] s.<\/p>\n<\/div>\n<\/div>\n

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    27.<\/strong> Let the position of a swinging pendulum in simple harmonic motion be given by\u00a0[latex]s(t)=a \\cos t+b \\sin t[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are constants, [latex]t[\/latex] measures time in seconds, and [latex]s[\/latex] measures position in centimeters, If the position is [latex]0[\/latex]cm and the velocity is [latex]3[\/latex]cm\/s when [latex]t=0[\/latex], find the values of [latex]a[\/latex] and [latex]b[\/latex].<\/p>\n

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    [latex]a=0, \\, b=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    28.\u00a0<\/strong>After a diver jumps off a diving board, the edge of the board oscillates with position given by [latex]s(t)=-5 \\cos t[\/latex] cm at [latex]t[\/latex] seconds after the jump.<\/p>\n

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    1. Sketch one period of the position function for [latex]t\\ge 0[\/latex].<\/li>\n
    2. Find the velocity function.<\/li>\n
    3. Sketch one period of the velocity function for [latex]t\\ge 0[\/latex].<\/li>\n
    4. Determine the times when the velocity is 0 over one period.<\/li>\n
    5. Find the acceleration function.<\/li>\n
    6. Sketch one period of the acceleration function for [latex]t\\ge 0[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
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      29.\u00a0<\/strong>The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by [latex]y=10+5 \\sin x[\/latex] where [latex]y[\/latex] is the number of hamburgers sold and [latex]x[\/latex] represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find [latex]y^{\\prime}[\/latex] and determine the intervals where the number of burgers being sold is increasing.<\/p>\n

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      [latex]y^{\\prime}=5 \\cos (x)[\/latex], increasing on [latex](0,\\frac{\\pi}{2}), \\, (\\frac{3\\pi}{2},\\frac{5\\pi}{2})[\/latex], and [latex](\\frac{7\\pi}{2},12)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      30. [T]<\/strong> The amount of rainfall per month in Phoenix, Arizona, can be approximated by [latex]y(t)=0.5+0.3 \\cos t[\/latex], where [latex]t[\/latex] is the number of months since January. Find [latex]y^{\\prime}[\/latex] and use a calculator to determine the intervals where the amount of rain falling is decreasing.<\/p>\n<\/div>\n<\/div>\n

      For the following exercises (31-33), use the quotient rule to derive the given equations.<\/p>\n

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      31.\u00a0<\/strong>[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      32.\u00a0<\/strong>[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      33.\u00a0<\/strong>[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      34.\u00a0<\/strong>Use the definition of derivative and the identity<\/p>\n

      [latex]\\cos (x+h)= \\cos x \\cos h- \\sin x \\sin h[\/latex]\u00a0 to prove that\u00a0 [latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex].<\/p>\n<\/div>\n<\/div>\n

      For the following exercises (35-39), find the requested higher-order derivative for the given functions.<\/p>\n

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      35.\u00a0<\/strong>[latex]\\frac{d^3 y}{dx^3}[\/latex] of [latex]y=3 \\cos x[\/latex]<\/p>\n

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      [latex]\\frac{d^3 y}{dx^3} = 3 \\sin x[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      36.\u00a0<\/strong>[latex]\\frac{d^2 y}{dx^2}[\/latex] of [latex]y=3 \\sin x+x^2 \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      37.\u00a0<\/strong>[latex]\\frac{d^4 y}{dx^4}[\/latex] of [latex]y=5 \\cos x[\/latex]<\/p>\n

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      [latex]\\frac{d^4 y}{dx^4} = 5 \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      38.\u00a0<\/strong>[latex]\\frac{d^2 y}{dx^2}[\/latex] of [latex]y= \\sec x+ \\cot x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      39.\u00a0<\/strong>[latex]\\frac{d^3 y}{dx^3}[\/latex] of [latex]y=x^{10}- \\sec x[\/latex]<\/p>\n

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      [latex]\\frac{d^3 y}{dx^3} = 720x^7-5 \\tan (x) \\sec^3 (x)- \\tan^3 (x) \\sec (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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