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3.2-bo'lim : Hosilaning talqini

1 – 3 masalalar uchun \(f\left( x \right)\) funksiya grafigidan foydalaning, \(a\) ning berilgan qiymatlari uchun \(f'\left( a \right)\) qiymatini baholang.

    1. \(a = - 5\)
    2. \(a = 1\)
    This graph is a parabola that opens downward with vertex somewhere between x=-1 and x=0. To the left of the vertex the graph goes through the points (-5,-2) and (-2,4). To the right of the vertex the graph goes through the points (1,4) and (2, 3.75).
    1. \(a = - 2\)
    2. \(a = 3\)
    This is a graph that starts at approximately (-4,6) and decreases until it hits (-2,4) where it goes through this point perfectly horizontal and then continues to decrease going through the points (1, 3.25) and (3,2).
    1. \(a = - 3\)
    2. \(a = 4\)
    This is a graph that starts at approximately (-3.8,5) and through the points (-3,3) and (0,1). It then increases until it reaches a peak somewhere between x=2 and x=3 and then decreases through the points (3,1) and (4,-2).

4 – 6 masalalar uchun berilgan shartlarni qanoatlantiradigan funksiya grafigini chizing.

  1. \(f\left( { - 7} \right) = 5\), \(f'\left( { - 7} \right) = - 3\), \(f\left( 4 \right) = - 1\), \(f'\left( 4 \right) = 1\)
  2. \(f\left( 1 \right) = 2\), \(f'\left( 1 \right) = 4\), \(f\left( 6 \right) = 2\), \(f'\left( 6 \right) = 3\)
  3. \(f\left( { - 1} \right) = - 9\), \(f'\left( { - 1} \right) = 0\), \(f\left( 2 \right) = - 1\), \(f'\left( 2 \right) = 3\), \(f\left( 5 \right) = 4\), \(f'\left( 5 \right) = - 1\)

7 – 9 masalalar uchun \(f\left( x \right)\) funksiya grafigi berilgan. Bundan \(f'\left( x \right)\) hosilasining grafigini chizish uchun foydalaning.

  1. This is a graph that starts at approximately (-2, -0.8) and increases until it reaches a peak at x=-1 and approximately y=1.8. It then decreases until it reaches a valley at (3,9) and then increases until it ends at approximately (4, -6.8).
  2. This is a graph that starts at approximately (-2.8, 5) and increases until it reaches a peak at x=-2 and approximately y=4.2. It then decreases going through the origin perfectly flat and continuing to decrease until it reaches a valley at x=2 and approximately y=-4.2. It then increases until it ends at approximately (2.8, 5).
  3. This graph is a series of four line segments. The first line segment starts at (-6,3) and ends at (-3,-2). The second line segment starts at (-3,-2) and ends at (-1,5). The third line segment starts at (-1,5) and ends at (4,5). The final line segment starts at (4,5) and ends at (7,1).
  4. \(g\left( z \right) = 1 + 10z - 7{z^2}\) funksiyasi haqida quyidagi savollarga javob bering.
    1. \(z = 0\) da funktsiya ortib bormoqdami yoki kamayyaptimi?
    2. \(z = 2\) da funktsiya ortib bormoqdami yoki kamayyaptimi?
    3. Funktsiya hech qachon o'zgarishni to'xtatadimi? Ha bo'lsa, \(z\) ning qaysi qiymati(lar)ida funktsiya o'zgarishni to'xtatadi?
  5. \(x = 1\) da \(f\left( x \right) = 5x - {x^3}\) ga teguvchi chiziq tenglamasi qanday?
  6. Ob'ektning istalgan vaqtda \(t\) pozitsiyasi \(s\left( t \right) = 2{t^2} - 8t + 10\) tomonidan beriladi.
    1. Ob'ektning istalgan vaqtda tezligini aniqlang \(t\).
    2. Ob'ekt \(t = 1\) da o'ngga yoki chapga siljiyaptimi?
    3. Ob'ekt \(t = 4\) da o'ngga yoki chapga siljiyaptimi?
    4. Ob'ekt hech qachon harakatni to'xtatadimi? Agar shunday bo'lsa, ob'ekt qaysi vaqtda (lar)da harakatni to'xtatadi?
  7. \(R\left( w \right) = {w^2} - 8w + 20\) funktsiyasi hech qachon o'zgarishni to'xtatadimi? Ha bo'lsa, \(w\) ning qaysi qiymati(lar)ida funktsiya o'zgarishni to'xtatadi?
  8. Faraz qilaylik, \(0 \le t \le 6\) uchun shardagi havo hajmi \(V\left( t \right) = 6t - {t^2}\) tomonidan berilgan.
    1. \(t = 2\) da havo hajmi ortib bormoqdami yoki kamayyaptimi?
    2. \(t = 5\) da havo hajmi ortib bormoqdami yoki kamayyaptimi?
    3. Havo hajmi hech qachon o'zgarishni to'xtatadimi? Ha bo'lsa, ovoz balandligi qaysi paytlarda o'zgarishni to'xtatadi?
  9. \(x = - 4\) da \(f\left( x \right) = 5x + 7\) ga teguvchi chiziq tenglamasi qanday?
  10. \(Z\left( x \right) = 2{x^3} - {x^2} - x\) funksiyasi haqida quyidagi savollarga javob bering.
    1. \(x = - 1\) da funktsiya ortib bormoqdami yoki kamayyaptimi?
    2. \(x = 2\) da funktsiya ortib bormoqdami yoki kamayyaptimi?
    3. Funktsiya hech qachon o'zgarishni to'xtatadimi? Ha bo'lsa, \(x\) ning qaysi qiymati(lar)ida funktsiya o'zgarishni to'xtatadi?
  11. \(V\left( t \right) = \sqrt {14 + 3t} \) funksiyasi berilgan nuqtalarda ortib yoki kamayganligini aniqlang.
    1. \(t = 0\)
    2. \(t = 5\)
    3. \(t = 100\)
  12. Faraz qilaylik, \(t \ge 0\) uchun idishdagi suv hajmi \(\displaystyle Q\left( t \right) = \frac{{{t^2}}}{{t + 2}}\) tomonidan berilgan.
    1. \(t = 0\) da suv hajmi ortib bormoqdami yoki kamayyaptimi?
    2. \(t = 3\) da suv hajmi ortib bormoqdami yoki kamayyaptimi?
    3. Suv hajmi hech qachon o'zgarmaydimi? Agar shunday bo'lsa, ovoz balandligi qaysi vaqtlarda o'zgarishni to'xtatadi?
  13. \(x = 16\) da \(g\left( x \right) = 10\) ga teguvchi chiziqning tenglamasi qanday?
  14. Ob'ektning istalgan vaqtda \(t\) pozitsiyasi \(Q\left( t \right) = \sqrt {1 + 4t} \) tomonidan beriladi.
    1. Ob'ektning istalgan vaqtda tezligini aniqlang \(t\).
    2. Ob'ekt hech qachon harakatni to'xtatadimi? Agar shunday bo'lsa, ob'ekt qaysi vaqtda (lar)da harakatni to'xtatadi?
  15. \(Y\left( t \right) = 2{t^3} + 9t + 5\) funktsiyasi hech qachon o'zgarishni to'xtatadimi? Ha bo'lsa, \(t\) ning qaysi qiymati(lar)ida funktsiya o'zgarishni to'xtatadi?