{"id":154,"date":"2026-01-12T15:55:41","date_gmt":"2026-01-12T15:55:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=154"},"modified":"2026-01-12T15:55:42","modified_gmt":"2026-01-12T15:55:42","slug":"analytical-applications-of-derivatives-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/analytical-applications-of-derivatives-get-stronger-answer-key\/","title":{"raw":"Analytical Applications of Derivatives: Get Stronger Answer Key","rendered":"Analytical Applications of Derivatives: Get Stronger Answer Key"},"content":{"raw":"

Related Rates<\/h2>\r\n
    \r\n \t
  1. [latex]8[\/latex]<\/li>\r\n \t
  2. [latex]\\frac{13}{\\sqrt{10}}[\/latex]<\/li>\r\n \t
  3. [latex]2\\sqrt{3}[\/latex] ft\/sec<\/li>\r\n \t
  4. The distance is decreasing at [latex]390[\/latex] mi\/h.<\/li>\r\n \t
  5. The distance between them shrinks at a rate of [latex]\\frac{1320}{13}\\approx 101.5[\/latex] mph.<\/li>\r\n \t
  6. <\/li>\r\n \t
  7. [latex]\\frac{9}{2}[\/latex] ft\/sec<\/li>\r\n \t
  8. <\/li>\r\n \t
  9. It grows at a rate [latex]\\frac{4}{9}[\/latex] ft\/sec<\/li>\r\n \t
  10. <\/li>\r\n \t
  11. The distance is increasing at [latex]\\frac{135\\sqrt{26}}{26}[\/latex] ft\/sec<\/li>\r\n \t
  12. [latex]-\\frac{5}{6}[\/latex] m\/sec<\/li>\r\n \t
  13. [latex]240\\pi \\, \\text{m}^2[\/latex]\/sec<\/li>\r\n \t
  14. [latex]\\frac{1}{2\\sqrt{\\pi}}[\/latex] cm<\/li>\r\n \t
  15. The area is increasing at a rate [latex]\\frac{(3\\sqrt{3})}{8} \\, \\text{ft}^{2} \/ \\text{sec}[\/latex].<\/li>\r\n \t
  16. The depth of the water decreases at [latex]\\frac{128}{125\\pi}[\/latex] ft\/min.<\/li>\r\n \t
  17. The volume is decreasing at a rate of [latex]\\frac{(25\\pi )}{16}{\\text{ft}}^{3}\\text{\/min}.[\/latex]<\/li>\r\n \t
  18. The water flows out at rate [latex]\\frac{2\\pi}{5} \\, \\text{m}^3[\/latex]\/min.<\/li>\r\n \t
  19. [latex]\\frac{3}{2}[\/latex] m\/sec<\/li>\r\n \t
  20. The angle decreases at [latex]\\frac{400}{1681}[\/latex] rad\/sec.<\/li>\r\n \t
  21. [latex]100\\pi[\/latex] mi\/min<\/li>\r\n \t
  22. The angle is changing at a rate of [latex]\\frac{11}{25}[\/latex] rad\/sec.<\/li>\r\n \t
  23. The distance is increasing at a rate of [latex]62.50[\/latex] ft\/sec.<\/li>\r\n \t
  24. The distance is decreasing at a rate of [latex]11.99[\/latex] ft\/sec.<\/li>\r\n<\/ol>\r\n

    Linear Approximations and Differentials<\/h2>\r\n
      \r\n \t
    1. <\/li>\r\n \t
    2. [latex]f^{\\prime}(a)=0[\/latex]<\/li>\r\n \t
    3. <\/li>\r\n \t
    4. The linear approximation exact when [latex]y=f(x)[\/latex] is linear or constant.<\/li>\r\n \t
    5. [latex]L(x)=\\frac{1}{2}-\\frac{1}{4}(x-2)[\/latex]<\/li>\r\n \t
    6. [latex]L(x)=1[\/latex]<\/li>\r\n \t
    7. [latex]L(x)=0[\/latex]<\/li>\r\n \t
    8. [latex]0.02[\/latex]<\/li>\r\n \t
    9. [latex]1.9996875[\/latex]<\/li>\r\n \t
    10. [latex]0.001593[\/latex]<\/li>\r\n \t
    11. [latex]1[\/latex]; error, [latex]~0.00005[\/latex]<\/li>\r\n \t
    12. [latex]0.97[\/latex]; error, [latex]~0.0006[\/latex]<\/li>\r\n \t
    13. [latex]3-\\frac{1}{600}[\/latex]; error, [latex]~4.632\\times 10^{-7}[\/latex]<\/li>\r\n \t
    14. [latex]dy=(\\cos x-x \\sin x) \\, dx[\/latex]<\/li>\r\n \t
    15. [latex]dy=(\\frac{x^2-2x-2}{(x-1)^2}) \\, dx[\/latex]<\/li>\r\n \t
    16. [latex]dy=-\\frac{1}{(x+1)^2} \\, dx[\/latex], [latex]-\\frac{1}{16}[\/latex]<\/li>\r\n \t
    17. [latex]dy=\\frac{9x^2+12x-2}{2(x+1)^{3\/2}} \\, dx[\/latex], -[latex]0.1[\/latex]<\/li>\r\n \t
    18. [latex]dy=(3x^2+2-\\frac{1}{x^2}) \\, dx[\/latex], [latex]0.2[\/latex]<\/li>\r\n \t
    19. [latex]12x \\, dx[\/latex]<\/li>\r\n \t
    20. [latex]4\\pi r^2 \\, dr[\/latex]<\/li>\r\n \t
    21. [latex]-1.2\\pi \\, \\text{cm}^3[\/latex]<\/li>\r\n \t
    22. [latex]-100 \\, \\text{ft}^3[\/latex]<\/li>\r\n \t
    23. <\/li>\r\n \t
    24. <\/li>\r\n \t
    25. <\/li>\r\n<\/ol>\r\n

      Maxima and Minima<\/h2>\r\n
        \r\n \t
      1. Answers may vary<\/li>\r\n \t
      2. Answers will vary<\/li>\r\n \t
      3. No; answers will vary<\/li>\r\n \t
      4. Since the absolute maximum is the function (output) value rather than the [latex]x[\/latex] value, the answer is no; answers will vary<\/li>\r\n \t
      5. When [latex]a=0[\/latex]<\/li>\r\n \t
      6. Absolute minimum at [latex]3[\/latex]; Absolute maximum at \u2212[latex]2.2[\/latex]; local minima at \u2212[latex]2[\/latex], [latex]1[\/latex]; local maxima at \u2212[latex]1[\/latex], [latex]2[\/latex]<\/li>\r\n \t
      7. Absolute minima at \u2212[latex]2[\/latex], [latex]2[\/latex]; absolute maxima at \u2212[latex]2.5[\/latex], [latex]2.5[\/latex]; local minimum at [latex]0[\/latex]; local maxima at \u2212[latex]1[\/latex], [latex]1[\/latex]<\/li>\r\n \t
      8. Answers may vary.<\/li>\r\n \t
      9. Answers may vary.<\/li>\r\n \t
      10. [latex]x=1[\/latex]<\/li>\r\n \t
      11. None<\/li>\r\n \t
      12. [latex]x=0[\/latex]<\/li>\r\n \t
      13. None<\/li>\r\n \t
      14. [latex]x=-1,1[\/latex]<\/li>\r\n \t
      15. Absolute maximum: [latex]x=4[\/latex], [latex]y=\\frac{33}{2}[\/latex]; absolute minimum: [latex]x=1[\/latex], [latex]y=3[\/latex]<\/li>\r\n \t
      16. Absolute minimum: [latex]x=\\frac{1}{2}[\/latex], [latex]y=4[\/latex]<\/li>\r\n \t
      17. Absolute maximum: [latex]x=2\\pi [\/latex], [latex]y=2\\pi [\/latex]; absolute minimum: [latex]x=0[\/latex], [latex]y=0[\/latex]<\/li>\r\n \t
      18. Absolute maximum: [latex]x=-3[\/latex]; absolute minimum: [latex]-1\\le x\\le 1[\/latex], [latex]y=2[\/latex]<\/li>\r\n \t
      19. Absolute maximum: [latex]x=\\frac{\\pi}{4}[\/latex], [latex]y=\\sqrt{2}[\/latex]; absolute minimum: [latex]x=\\frac{5\\pi}{4}[\/latex], [latex]y=\u2212\\sqrt{2}[\/latex]<\/li>\r\n \t
      20. Absolute minimum: [latex]x=-2[\/latex], [latex]y=1[\/latex]<\/li>\r\n \t
      21. Absolute minimum: [latex]x=-3[\/latex], [latex]y=-135[\/latex]; local maximum: [latex]x=0[\/latex], [latex]y=0[\/latex]; local minimum: [latex]x=1[\/latex], [latex]y=-7[\/latex]<\/li>\r\n \t
      22. Local maximum: [latex]x=1-2\\sqrt{2}[\/latex], [latex]y=3-4\\sqrt{2}[\/latex]; local minimum: [latex]x=1+2\\sqrt{2}[\/latex], [latex]y=3+4\\sqrt{2}[\/latex]<\/li>\r\n \t
      23. Absolute maximum: [latex]x=\\frac{\\sqrt{2}}{2}[\/latex], [latex]y=\\frac{3}{2}[\/latex]; absolute minimum: [latex]x=-\\frac{\\sqrt{2}}{2}[\/latex], [latex]y=-\\frac{3}{2}[\/latex]<\/li>\r\n \t
      24. Local maximum: [latex]x=-2[\/latex], [latex]y=59[\/latex]; local minimum: [latex]x=1[\/latex], [latex]y=-130[\/latex]<\/li>\r\n \t
      25. Absolute maximum: [latex]x=0[\/latex], [latex]y=1[\/latex]; absolute minimum: [latex]x=-2,2[\/latex], [latex]y=0[\/latex]<\/li>\r\n \t
      26. Absolute minima: [latex]x=0[\/latex], [latex]x=2[\/latex], [latex]y=1[\/latex]; local maximum at [latex]x=1[\/latex], [latex]y=2[\/latex]<\/li>\r\n \t
      27. No maxima\/minima if [latex]a[\/latex] is odd, minimum at [latex]x=1[\/latex] if [latex]a[\/latex] is even<\/li>\r\n<\/ol>\r\n

        The Mean Value Theorem<\/h2>\r\n
          \r\n \t
        1. <\/li>\r\n \t
        2. One example is [latex]f(x)=|x|+3, \\, -2 \\le x \\le 2[\/latex]<\/li>\r\n \t
        3. <\/li>\r\n \t
        4. Yes, but the Mean Value Theorem still does not apply<\/li>\r\n \t
        5. [latex](\u2212\\infty,0), \\, (0,\\infty)[\/latex]<\/li>\r\n \t
        6. [latex](\u2212\\infty,-2), \\, (2,\\infty)[\/latex]<\/li>\r\n \t
        7. 2 points]<\/li>\r\n \t
        8. 5 points<\/li>\r\n \t
        9. [latex]c=\\frac{2\\sqrt{3}}{3}[\/latex]<\/li>\r\n \t
        10. [latex]c=\\frac{1}{2}, \\, 1, \\, \\frac{3}{2}[\/latex]<\/li>\r\n \t
        11. [latex]c=1[\/latex]<\/li>\r\n \t
        12. Not differentiable<\/li>\r\n \t
        13. Not differentiable<\/li>\r\n \t
        14. Yes<\/li>\r\n \t
        15. The Mean Value Theorem does not apply since the function is discontinuous at [latex]x=\\frac{1}{4}, \\, \\frac{3}{4}, \\, \\frac{5}{4}, \\, \\frac{7}{4}[\/latex].<\/li>\r\n \t
        16. Yes<\/li>\r\n \t
        17. The Mean Value Theorem does not apply; discontinuous at [latex]x=0[\/latex].<\/li>\r\n \t
        18. Yes<\/li>\r\n \t
        19. The Mean Value Theorem does not apply; not differentiable at [latex]x=0[\/latex].<\/li>\r\n \t
        20. [latex]c=\\pm \\frac{1}{\\pi} \\cos^{-1}(\\frac{\\sqrt{\\pi}}{2})[\/latex]; [latex]c=\\pm 0.1533[\/latex]<\/li>\r\n \t
        21. The Mean Value Theorem does not apply.<\/li>\r\n \t
        22. [latex]\\frac{1}{2\\sqrt{c+1}}-\\frac{2}{c^3}=\\frac{521}{2880}[\/latex]; [latex]c=3.133,5.867[\/latex]<\/li>\r\n \t
        23. Yes<\/li>\r\n \t
        24. It is constant.<\/li>\r\n<\/ol>\r\n

          Derivatives and the Shape of a Graph<\/h2>\r\n
            \r\n \t
          1. <\/li>\r\n \t
          2. It is not a local maximum\/minimum because [latex]f^{\\prime}[\/latex] does not change sign<\/li>\r\n \t
          3. <\/li>\r\n \t
          4. No<\/li>\r\n \t
          5. <\/li>\r\n \t
          6. False; for example, [latex]y=\\sqrt{x}[\/latex].<\/li>\r\n \t
          7. <\/li>\r\n \t
          8. Increasing for [latex]-2 < x < 1[\/latex] and [latex]x > 2[\/latex]; decreasing for [latex]x < 2[\/latex] and [latex]-1 < x < 2[\/latex]<\/li>\r\n \t
          9. Decreasing for [latex]x < 1[\/latex]; increasing for [latex]x > 1[\/latex]<\/li>\r\n \t
          10. Decreasing for [latex]-2 < x < -1[\/latex] and [latex]1 < x < 2[\/latex]; increasing for [latex]-1 < x < 1[\/latex] and [latex]x < -2[\/latex] and [latex]x > 2[\/latex]<\/li>\r\n \t
          11. \r\n
              \r\n \t
            1. Increasing over [latex]-2 < x < 1, \\, 0 < x < 1, \\, x > 2[\/latex]; decreasing over [latex]x < -2 \\, -1 < x < 0, \\, 1 < x < 2[\/latex]<\/li>\r\n \t
            2. maxima at [latex]x=-1[\/latex] and [latex]x=1[\/latex], minima at [latex]x=-2[\/latex] and [latex]x=0[\/latex] and [latex]x=2[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
            3. \r\n
                \r\n \t
              1. Increasing over [latex]x > 0[\/latex], decreasing over [latex]x < 0[\/latex]<\/li>\r\n \t
              2. Minimum at [latex]x=0[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
              3. Concave up on all [latex]x[\/latex], no inflection points<\/li>\r\n \t
              4. Concave up on all [latex]x[\/latex], no inflection points<\/li>\r\n \t
              5. Concave up for [latex]x < 0[\/latex] and [latex]x > 1[\/latex], concave down for [latex]0 < x < 1[\/latex], inflection points at [latex]x=0[\/latex] and [latex]x=1[\/latex]<\/li>\r\n \t
              6. Answers will vary<\/li>\r\n \t
              7. Answers will vary<\/li>\r\n \t
              8. \r\n
                  \r\n \t
                1. Concave up for [latex]x > \\frac{4}{3}[\/latex], concave down for [latex]x < \\frac{4}{3}[\/latex]<\/li>\r\n \t
                2. Inflection point at [latex]x=\\frac{4}{3}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                3. \r\n
                    \r\n \t
                  1. Increasing over [latex]x < 0[\/latex] and [latex]x > 4[\/latex], decreasing over [latex]0 < x < 4[\/latex]<\/li>\r\n \t
                  2. Maximum at [latex]x=0[\/latex], minimum at [latex]x=4[\/latex]<\/li>\r\n \t
                  3. Concave up for [latex]x > 2[\/latex], concave down for [latex]x < 2[\/latex]<\/li>\r\n \t
                  4. Infection point at [latex]x=2[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                  5. \r\n
                      \r\n \t
                    1. Increasing over [latex]x < 0[\/latex] and [latex]x > \\frac{60}{11}[\/latex], decreasing over [latex]0 < x < \\frac{60}{11}[\/latex]<\/li>\r\n \t
                    2. Minimum at [latex]x=\\frac{60}{11}[\/latex]<\/li>\r\n \t
                    3. Concave down for [latex]x < \\frac{54}{11}[\/latex], concave up for [latex]x > \\frac{54}{11}[\/latex]<\/li>\r\n \t
                    4. Inflection point at [latex]x=\\frac{54}{11}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                    5. \r\n
                        \r\n \t
                      1. Increasing over [latex]x > -\\frac{1}{2}[\/latex], decreasing over [latex]x < -\\frac{1}{2}[\/latex]<\/li>\r\n \t
                      2. Minimum at [latex]x=-\\frac{1}{2}[\/latex]<\/li>\r\n \t
                      3. Concave up for all [latex]x[\/latex]<\/li>\r\n \t
                      4. No inflection points<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                      5. \r\n
                          \r\n \t
                        1. Increases over [latex]-\\frac{1}{4} < x < \\frac{3}{4}[\/latex], decreases over [latex]x > \\frac{3}{4}[\/latex] and [latex]x < -\\frac{1}{4}[\/latex]<\/li>\r\n \t
                        2. Minimum at [latex]x=-\\frac{1}{4}[\/latex], maximum at [latex]x=\\frac{3}{4}[\/latex]<\/li>\r\n \t
                        3. Concave up for [latex]-\\frac{3}{4} < x < \\frac{1}{4}[\/latex], concave down for [latex]x < \\frac{3}{4}[\/latex] and [latex] > \\frac{1}{4}[\/latex]<\/li>\r\n \t
                        4. Inflection points at [latex]x=-\\frac{3}{4}, \\, x=\\frac{1}{4}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                        5. \r\n
                            \r\n \t
                          1. Increasing for all [latex]x[\/latex]<\/li>\r\n \t
                          2. No local minimum or maximum<\/li>\r\n \t
                          3. Concave up for [latex]x > 0[\/latex], concave down for [latex]x < 0[\/latex]<\/li>\r\n \t
                          4. Inflection point at [latex]x=0[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                          5. \r\n
                              \r\n \t
                            1. Increasing for all [latex]x[\/latex] where defined<\/li>\r\n \t
                            2. No local minima or maxima<\/li>\r\n \t
                            3. Concave up for [latex]x < 1[\/latex], concave down for [latex]x > 1[\/latex]<\/li>\r\n \t
                            4. No inflection points in domain<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                            5. \r\n
                                \r\n \t
                              1. Increasing over [latex]-\\frac{\\pi }{4} < x < \\frac{3\\pi }{4}[\/latex], decreasing over [latex]x > \\frac{3\\pi }{4}, \\, x < -\\frac{\\pi }{4}[\/latex]<\/li>\r\n \t
                              2. Minimum at [latex]x=-\\frac{\\pi }{4}[\/latex], maximum at [latex]x=\\frac{3\\pi }{4}[\/latex]<\/li>\r\n \t
                              3. Concave up for [latex]-\\frac{\\pi }{2} < x < \\frac{\\pi }{2}[\/latex], concave down for [latex]x < \\frac{\\pi }{2}, \\, x > \\frac{\\pi }{2}[\/latex]<\/li>\r\n \t
                              4. Infection points at [latex]x=\\pm \\frac{\\pi }{2}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                              5. \r\n
                                  \r\n \t
                                1. Increasing over [latex]x > 4[\/latex], decreasing over [latex]0 < x < 4[\/latex]<\/li>\r\n \t
                                2. Minimum at [latex]x=4[\/latex]<\/li>\r\n \t
                                3. Concave up for [latex]0 < x < 8\\sqrt[3]{2}[\/latex], concave down for [latex]x > 8\\sqrt[3]{2}[\/latex]<\/li>\r\n \t
                                4. Inflection point at [latex]x=8\\sqrt[3]{2}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                5. [latex]f > 0, \\, f^{\\prime} > 0, \\, f^{\\prime \\prime} < 0[\/latex]<\/li>\r\n \t
                                6. [latex]f > 0, \\, f^{\\prime} < 0, \\, f^{\\prime \\prime} < 0[\/latex]<\/li>\r\n \t
                                7. [latex]f > 0, \\, f^{\\prime} > 0, \\, f^{\\prime \\prime} > 0[\/latex]<\/li>\r\n \t
                                8. True, by the Mean Value Theorem]<\/li>\r\n \t
                                9. True, examine derivative<\/li>\r\n<\/ol>","rendered":"

                                  Related Rates<\/h2>\n
                                    \n
                                  1. [latex]8[\/latex]<\/li>\n
                                  2. [latex]\\frac{13}{\\sqrt{10}}[\/latex]<\/li>\n
                                  3. [latex]2\\sqrt{3}[\/latex] ft\/sec<\/li>\n
                                  4. The distance is decreasing at [latex]390[\/latex] mi\/h.<\/li>\n
                                  5. The distance between them shrinks at a rate of [latex]\\frac{1320}{13}\\approx 101.5[\/latex] mph.<\/li>\n
                                  6. <\/li>\n
                                  7. [latex]\\frac{9}{2}[\/latex] ft\/sec<\/li>\n
                                  8. <\/li>\n
                                  9. It grows at a rate [latex]\\frac{4}{9}[\/latex] ft\/sec<\/li>\n
                                  10. <\/li>\n
                                  11. The distance is increasing at [latex]\\frac{135\\sqrt{26}}{26}[\/latex] ft\/sec<\/li>\n
                                  12. [latex]-\\frac{5}{6}[\/latex] m\/sec<\/li>\n
                                  13. [latex]240\\pi \\, \\text{m}^2[\/latex]\/sec<\/li>\n
                                  14. [latex]\\frac{1}{2\\sqrt{\\pi}}[\/latex] cm<\/li>\n
                                  15. The area is increasing at a rate [latex]\\frac{(3\\sqrt{3})}{8} \\, \\text{ft}^{2} \/ \\text{sec}[\/latex].<\/li>\n
                                  16. The depth of the water decreases at [latex]\\frac{128}{125\\pi}[\/latex] ft\/min.<\/li>\n
                                  17. The volume is decreasing at a rate of [latex]\\frac{(25\\pi )}{16}{\\text{ft}}^{3}\\text{\/min}.[\/latex]<\/li>\n
                                  18. The water flows out at rate [latex]\\frac{2\\pi}{5} \\, \\text{m}^3[\/latex]\/min.<\/li>\n
                                  19. [latex]\\frac{3}{2}[\/latex] m\/sec<\/li>\n
                                  20. The angle decreases at [latex]\\frac{400}{1681}[\/latex] rad\/sec.<\/li>\n
                                  21. [latex]100\\pi[\/latex] mi\/min<\/li>\n
                                  22. The angle is changing at a rate of [latex]\\frac{11}{25}[\/latex] rad\/sec.<\/li>\n
                                  23. The distance is increasing at a rate of [latex]62.50[\/latex] ft\/sec.<\/li>\n
                                  24. The distance is decreasing at a rate of [latex]11.99[\/latex] ft\/sec.<\/li>\n<\/ol>\n

                                    Linear Approximations and Differentials<\/h2>\n
                                      \n
                                    1. <\/li>\n
                                    2. [latex]f^{\\prime}(a)=0[\/latex]<\/li>\n
                                    3. <\/li>\n
                                    4. The linear approximation exact when [latex]y=f(x)[\/latex] is linear or constant.<\/li>\n
                                    5. [latex]L(x)=\\frac{1}{2}-\\frac{1}{4}(x-2)[\/latex]<\/li>\n
                                    6. [latex]L(x)=1[\/latex]<\/li>\n
                                    7. [latex]L(x)=0[\/latex]<\/li>\n
                                    8. [latex]0.02[\/latex]<\/li>\n
                                    9. [latex]1.9996875[\/latex]<\/li>\n
                                    10. [latex]0.001593[\/latex]<\/li>\n
                                    11. [latex]1[\/latex]; error, [latex]~0.00005[\/latex]<\/li>\n
                                    12. [latex]0.97[\/latex]; error, [latex]~0.0006[\/latex]<\/li>\n
                                    13. [latex]3-\\frac{1}{600}[\/latex]; error, [latex]~4.632\\times 10^{-7}[\/latex]<\/li>\n
                                    14. [latex]dy=(\\cos x-x \\sin x) \\, dx[\/latex]<\/li>\n
                                    15. [latex]dy=(\\frac{x^2-2x-2}{(x-1)^2}) \\, dx[\/latex]<\/li>\n
                                    16. [latex]dy=-\\frac{1}{(x+1)^2} \\, dx[\/latex], [latex]-\\frac{1}{16}[\/latex]<\/li>\n
                                    17. [latex]dy=\\frac{9x^2+12x-2}{2(x+1)^{3\/2}} \\, dx[\/latex], –[latex]0.1[\/latex]<\/li>\n
                                    18. [latex]dy=(3x^2+2-\\frac{1}{x^2}) \\, dx[\/latex], [latex]0.2[\/latex]<\/li>\n
                                    19. [latex]12x \\, dx[\/latex]<\/li>\n
                                    20. [latex]4\\pi r^2 \\, dr[\/latex]<\/li>\n
                                    21. [latex]-1.2\\pi \\, \\text{cm}^3[\/latex]<\/li>\n
                                    22. [latex]-100 \\, \\text{ft}^3[\/latex]<\/li>\n
                                    23. <\/li>\n
                                    24. <\/li>\n
                                    25. <\/li>\n<\/ol>\n

                                      Maxima and Minima<\/h2>\n
                                        \n
                                      1. Answers may vary<\/li>\n
                                      2. Answers will vary<\/li>\n
                                      3. No; answers will vary<\/li>\n
                                      4. Since the absolute maximum is the function (output) value rather than the [latex]x[\/latex] value, the answer is no; answers will vary<\/li>\n
                                      5. When [latex]a=0[\/latex]<\/li>\n
                                      6. Absolute minimum at [latex]3[\/latex]; Absolute maximum at \u2212[latex]2.2[\/latex]; local minima at \u2212[latex]2[\/latex], [latex]1[\/latex]; local maxima at \u2212[latex]1[\/latex], [latex]2[\/latex]<\/li>\n
                                      7. Absolute minima at \u2212[latex]2[\/latex], [latex]2[\/latex]; absolute maxima at \u2212[latex]2.5[\/latex], [latex]2.5[\/latex]; local minimum at [latex]0[\/latex]; local maxima at \u2212[latex]1[\/latex], [latex]1[\/latex]<\/li>\n
                                      8. Answers may vary.<\/li>\n
                                      9. Answers may vary.<\/li>\n
                                      10. [latex]x=1[\/latex]<\/li>\n
                                      11. None<\/li>\n
                                      12. [latex]x=0[\/latex]<\/li>\n
                                      13. None<\/li>\n
                                      14. [latex]x=-1,1[\/latex]<\/li>\n
                                      15. Absolute maximum: [latex]x=4[\/latex], [latex]y=\\frac{33}{2}[\/latex]; absolute minimum: [latex]x=1[\/latex], [latex]y=3[\/latex]<\/li>\n
                                      16. Absolute minimum: [latex]x=\\frac{1}{2}[\/latex], [latex]y=4[\/latex]<\/li>\n
                                      17. Absolute maximum: [latex]x=2\\pi[\/latex], [latex]y=2\\pi[\/latex]; absolute minimum: [latex]x=0[\/latex], [latex]y=0[\/latex]<\/li>\n
                                      18. Absolute maximum: [latex]x=-3[\/latex]; absolute minimum: [latex]-1\\le x\\le 1[\/latex], [latex]y=2[\/latex]<\/li>\n
                                      19. Absolute maximum: [latex]x=\\frac{\\pi}{4}[\/latex], [latex]y=\\sqrt{2}[\/latex]; absolute minimum: [latex]x=\\frac{5\\pi}{4}[\/latex], [latex]y=\u2212\\sqrt{2}[\/latex]<\/li>\n
                                      20. Absolute minimum: [latex]x=-2[\/latex], [latex]y=1[\/latex]<\/li>\n
                                      21. Absolute minimum: [latex]x=-3[\/latex], [latex]y=-135[\/latex]; local maximum: [latex]x=0[\/latex], [latex]y=0[\/latex]; local minimum: [latex]x=1[\/latex], [latex]y=-7[\/latex]<\/li>\n
                                      22. Local maximum: [latex]x=1-2\\sqrt{2}[\/latex], [latex]y=3-4\\sqrt{2}[\/latex]; local minimum: [latex]x=1+2\\sqrt{2}[\/latex], [latex]y=3+4\\sqrt{2}[\/latex]<\/li>\n
                                      23. Absolute maximum: [latex]x=\\frac{\\sqrt{2}}{2}[\/latex], [latex]y=\\frac{3}{2}[\/latex]; absolute minimum: [latex]x=-\\frac{\\sqrt{2}}{2}[\/latex], [latex]y=-\\frac{3}{2}[\/latex]<\/li>\n
                                      24. Local maximum: [latex]x=-2[\/latex], [latex]y=59[\/latex]; local minimum: [latex]x=1[\/latex], [latex]y=-130[\/latex]<\/li>\n
                                      25. Absolute maximum: [latex]x=0[\/latex], [latex]y=1[\/latex]; absolute minimum: [latex]x=-2,2[\/latex], [latex]y=0[\/latex]<\/li>\n
                                      26. Absolute minima: [latex]x=0[\/latex], [latex]x=2[\/latex], [latex]y=1[\/latex]; local maximum at [latex]x=1[\/latex], [latex]y=2[\/latex]<\/li>\n
                                      27. No maxima\/minima if [latex]a[\/latex] is odd, minimum at [latex]x=1[\/latex] if [latex]a[\/latex] is even<\/li>\n<\/ol>\n

                                        The Mean Value Theorem<\/h2>\n
                                          \n
                                        1. <\/li>\n
                                        2. One example is [latex]f(x)=|x|+3, \\, -2 \\le x \\le 2[\/latex]<\/li>\n
                                        3. <\/li>\n
                                        4. Yes, but the Mean Value Theorem still does not apply<\/li>\n
                                        5. [latex](\u2212\\infty,0), \\, (0,\\infty)[\/latex]<\/li>\n
                                        6. [latex](\u2212\\infty,-2), \\, (2,\\infty)[\/latex]<\/li>\n
                                        7. 2 points]<\/li>\n
                                        8. 5 points<\/li>\n
                                        9. [latex]c=\\frac{2\\sqrt{3}}{3}[\/latex]<\/li>\n
                                        10. [latex]c=\\frac{1}{2}, \\, 1, \\, \\frac{3}{2}[\/latex]<\/li>\n
                                        11. [latex]c=1[\/latex]<\/li>\n
                                        12. Not differentiable<\/li>\n
                                        13. Not differentiable<\/li>\n
                                        14. Yes<\/li>\n
                                        15. The Mean Value Theorem does not apply since the function is discontinuous at [latex]x=\\frac{1}{4}, \\, \\frac{3}{4}, \\, \\frac{5}{4}, \\, \\frac{7}{4}[\/latex].<\/li>\n
                                        16. Yes<\/li>\n
                                        17. The Mean Value Theorem does not apply; discontinuous at [latex]x=0[\/latex].<\/li>\n
                                        18. Yes<\/li>\n
                                        19. The Mean Value Theorem does not apply; not differentiable at [latex]x=0[\/latex].<\/li>\n
                                        20. [latex]c=\\pm \\frac{1}{\\pi} \\cos^{-1}(\\frac{\\sqrt{\\pi}}{2})[\/latex]; [latex]c=\\pm 0.1533[\/latex]<\/li>\n
                                        21. The Mean Value Theorem does not apply.<\/li>\n
                                        22. [latex]\\frac{1}{2\\sqrt{c+1}}-\\frac{2}{c^3}=\\frac{521}{2880}[\/latex]; [latex]c=3.133,5.867[\/latex]<\/li>\n
                                        23. Yes<\/li>\n
                                        24. It is constant.<\/li>\n<\/ol>\n

                                          Derivatives and the Shape of a Graph<\/h2>\n
                                            \n
                                          1. <\/li>\n
                                          2. It is not a local maximum\/minimum because [latex]f^{\\prime}[\/latex] does not change sign<\/li>\n
                                          3. <\/li>\n
                                          4. No<\/li>\n
                                          5. <\/li>\n
                                          6. False; for example, [latex]y=\\sqrt{x}[\/latex].<\/li>\n
                                          7. <\/li>\n
                                          8. Increasing for [latex]-2 < x < 1[\/latex] and [latex]x > 2[\/latex]; decreasing for [latex]x < 2[\/latex] and [latex]-1 < x < 2[\/latex]<\/li>\n
                                          9. Decreasing for [latex]x < 1[\/latex]; increasing for [latex]x > 1[\/latex]<\/li>\n
                                          10. Decreasing for [latex]-2 < x < -1[\/latex] and [latex]1 < x < 2[\/latex]; increasing for [latex]-1 < x < 1[\/latex] and [latex]x < -2[\/latex] and [latex]x > 2[\/latex]<\/li>\n
                                          11. \n
                                              \n
                                            1. Increasing over [latex]-2 < x < 1, \\, 0 < x < 1, \\, x > 2[\/latex]; decreasing over [latex]x < -2 \\, -1 < x < 0, \\, 1 < x < 2[\/latex]<\/li>\n
                                            2. maxima at [latex]x=-1[\/latex] and [latex]x=1[\/latex], minima at [latex]x=-2[\/latex] and [latex]x=0[\/latex] and [latex]x=2[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                            3. \n
                                                \n
                                              1. Increasing over [latex]x > 0[\/latex], decreasing over [latex]x < 0[\/latex]<\/li>\n
                                              2. Minimum at [latex]x=0[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                              3. Concave up on all [latex]x[\/latex], no inflection points<\/li>\n
                                              4. Concave up on all [latex]x[\/latex], no inflection points<\/li>\n
                                              5. Concave up for [latex]x < 0[\/latex] and [latex]x > 1[\/latex], concave down for [latex]0 < x < 1[\/latex], inflection points at [latex]x=0[\/latex] and [latex]x=1[\/latex]<\/li>\n
                                              6. Answers will vary<\/li>\n
                                              7. Answers will vary<\/li>\n
                                              8. \n
                                                  \n
                                                1. Concave up for [latex]x > \\frac{4}{3}[\/latex], concave down for [latex]x < \\frac{4}{3}[\/latex]<\/li>\n
                                                2. Inflection point at [latex]x=\\frac{4}{3}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                3. \n
                                                    \n
                                                  1. Increasing over [latex]x < 0[\/latex] and [latex]x > 4[\/latex], decreasing over [latex]0 < x < 4[\/latex]<\/li>\n
                                                  2. Maximum at [latex]x=0[\/latex], minimum at [latex]x=4[\/latex]<\/li>\n
                                                  3. Concave up for [latex]x > 2[\/latex], concave down for [latex]x < 2[\/latex]<\/li>\n
                                                  4. Infection point at [latex]x=2[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                  5. \n
                                                      \n
                                                    1. Increasing over [latex]x < 0[\/latex] and [latex]x > \\frac{60}{11}[\/latex], decreasing over [latex]0 < x < \\frac{60}{11}[\/latex]<\/li>\n
                                                    2. Minimum at [latex]x=\\frac{60}{11}[\/latex]<\/li>\n
                                                    3. Concave down for [latex]x < \\frac{54}{11}[\/latex], concave up for [latex]x > \\frac{54}{11}[\/latex]<\/li>\n
                                                    4. Inflection point at [latex]x=\\frac{54}{11}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                    5. \n
                                                        \n
                                                      1. Increasing over [latex]x > -\\frac{1}{2}[\/latex], decreasing over [latex]x < -\\frac{1}{2}[\/latex]<\/li>\n
                                                      2. Minimum at [latex]x=-\\frac{1}{2}[\/latex]<\/li>\n
                                                      3. Concave up for all [latex]x[\/latex]<\/li>\n
                                                      4. No inflection points<\/li>\n<\/ol>\n<\/li>\n
                                                      5. \n
                                                          \n
                                                        1. Increases over [latex]-\\frac{1}{4} < x < \\frac{3}{4}[\/latex], decreases over [latex]x > \\frac{3}{4}[\/latex] and [latex]x < -\\frac{1}{4}[\/latex]<\/li>\n
                                                        2. Minimum at [latex]x=-\\frac{1}{4}[\/latex], maximum at [latex]x=\\frac{3}{4}[\/latex]<\/li>\n
                                                        3. Concave up for [latex]-\\frac{3}{4} < x < \\frac{1}{4}[\/latex], concave down for [latex]x < \\frac{3}{4}[\/latex] and [latex]> \\frac{1}{4}[\/latex]<\/li>\n
                                                        4. Inflection points at [latex]x=-\\frac{3}{4}, \\, x=\\frac{1}{4}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                        5. \n
                                                            \n
                                                          1. Increasing for all [latex]x[\/latex]<\/li>\n
                                                          2. No local minimum or maximum<\/li>\n
                                                          3. Concave up for [latex]x > 0[\/latex], concave down for [latex]x < 0[\/latex]<\/li>\n
                                                          4. Inflection point at [latex]x=0[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                          5. \n
                                                              \n
                                                            1. Increasing for all [latex]x[\/latex] where defined<\/li>\n
                                                            2. No local minima or maxima<\/li>\n
                                                            3. Concave up for [latex]x < 1[\/latex], concave down for [latex]x > 1[\/latex]<\/li>\n
                                                            4. No inflection points in domain<\/li>\n<\/ol>\n<\/li>\n
                                                            5. \n
                                                                \n
                                                              1. Increasing over [latex]-\\frac{\\pi }{4} < x < \\frac{3\\pi }{4}[\/latex], decreasing over [latex]x > \\frac{3\\pi }{4}, \\, x < -\\frac{\\pi }{4}[\/latex]<\/li>\n
                                                              2. Minimum at [latex]x=-\\frac{\\pi }{4}[\/latex], maximum at [latex]x=\\frac{3\\pi }{4}[\/latex]<\/li>\n
                                                              3. Concave up for [latex]-\\frac{\\pi }{2} < x < \\frac{\\pi }{2}[\/latex], concave down for [latex]x < \\frac{\\pi }{2}, \\, x > \\frac{\\pi }{2}[\/latex]<\/li>\n
                                                              4. Infection points at [latex]x=\\pm \\frac{\\pi }{2}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                              5. \n
                                                                  \n
                                                                1. Increasing over [latex]x > 4[\/latex], decreasing over [latex]0 < x < 4[\/latex]<\/li>\n
                                                                2. Minimum at [latex]x=4[\/latex]<\/li>\n
                                                                3. Concave up for [latex]0 < x < 8\\sqrt[3]{2}[\/latex], concave down for [latex]x > 8\\sqrt[3]{2}[\/latex]<\/li>\n
                                                                4. Inflection point at [latex]x=8\\sqrt[3]{2}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                5. [latex]f > 0, \\, f^{\\prime} > 0, \\, f^{\\prime \\prime} < 0[\/latex]<\/li>\n
                                                                6. [latex]f > 0, \\, f^{\\prime} < 0, \\, f^{\\prime \\prime} < 0[\/latex]<\/li>\n
                                                                7. [latex]f > 0, \\, f^{\\prime} > 0, \\, f^{\\prime \\prime} > 0[\/latex]<\/li>\n
                                                                8. True, by the Mean Value Theorem]<\/li>\n
                                                                9. True, examine derivative<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":108,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/154"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/154\/revisions"}],"predecessor-version":[{"id":167,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/154\/revisions\/167"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/108"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/154\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=154"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=154"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=154"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}