Asosiy tarkibga o'tish

2.1-bo'lim : Tangent chiziqlar va o'zgarish tezligi

2. \(g\left( x \right) = \sqrt {4x + 8} \) funksiyasi va \(x = 2\) tomonidan berilgan \(P\) nuqtasi uchun quyidagi savollarning har biriga javob bering.

  1. \(x\) ning quyidagi qiymatlari bilan berilgan \(Q\) nuqtalari uchun \(P\) va \(Q\) nuqtalari orqali ajratilgan chiziqning qiyaligini, \({m_{PQ}}\) (kamida 8 kasrgacha aniq) hisoblang.
    1. 2.5
    2. 2.1
    3. 2.01
    4. 2.001
    5. 2.0001
    1. 1.5
    2. 1.9
    3. 1.99
    4. 1.999
    5. 1.9999
  2. \(x = 2\) da \(g\left( x \right)\) ga tangens chiziqning qiyaligini baholash uchun (a) dagi ma’lumotlardan foydalaning va tangens chiziq tenglamasini yozing.
a \(x\) ning quyidagi qiymatlari bilan berilgan \(Q\) nuqtalari uchun \(P\) va \(Q\) nuqtalari orqali ajratilgan chiziqning qiyaligini (kamida 8 kasrgacha aniq) hisoblang, \({m_{PQ}}\).
  1. 2.5
  2. 2.1
  3. 2.01
  4. 2.001
  5. 2.0001
  1. 1.5
  2. 1.9
  3. 1.99
  4. 1.999
  5. 1.9999
Yechim

Biz qilishimiz kerak bo'lgan birinchi narsa - sekant chiziqlarning qiyalik formulasini o'rnatish. Ushbu bo'limda muhokama qilinganidek, bu tomonidan berilgan:

\[{m_{PQ}} = \frac{{g\left( x \right) - g\left( 2 \right)}}{{x - 2}} = \frac{{\sqrt {4x + 8} - 4}}{{x - 2}}\]

Endi biz faqat \(x\) ning berilgan qiymatlari uchun \({m_{PQ}}\) qiymatlari jadvalini tuzishimiz kerak. Quyidagi jadvaldagi barcha qiymatlar 8 kasrgacha aniq.

\(x\) \({m_{PQ}}\) \(x\) \({m_{PQ}}\)
2.5 0.48528137 1.5 0.51668523
2.1 0.49691346 1.9 0.50316468
2.01 0.49968789 1.99 0.50031289
2.001 0.49996875 1.999 0.50003125
2.0001 0.49999688 1.9999 0.50000313


b \(x = 2\) da \(g\left( x \right)\) ga tangens chiziqning qiyaligini taxmin qilish uchun (a) dagi ma'lumotlardan foydalaning va tangens chiziq tenglamasini yozing. Yechim

Yuqoridagi qiymatlar jadvalidan biz \(x = 2\) ning har ikki tomonidan ajratilgan chiziqlarning qiyaligi 0,5 qiymatiga qarab harakat qilayotganini ko'rishimiz mumkin va shuning uchun biz tangens chizig'ining qiyaligi : \(\require{bbox} \bbox[2pt,border:1px solid black]{{m = 0.5 = \frac{1}{2}}}\) ekanligini taxmin qilishimiz mumkin.

Tangens chiziq tenglamasi u holda,

\[y = g\left( 2 \right) + m\left( {x - 2} \right) = 4 + \frac{1}{2}\left( {x - 2} \right)\hspace{0.5in} \Rightarrow \hspace{0.5in}\,\require{bbox} \bbox[2pt,border:1px solid black]{{y = \frac{1}{2}x + 3}}\]

Bu funksiya va tangens chiziqning grafigi.

The graph of \(g\left( x \right) = \sqrt {4x + 8} \) in red, which starts at (-2,0) and increases while bending downward, along with its tangent line \(y = \frac{1}{2}x + 3\) in blue. The tangent line touches the curve at the labeled point P = (2,4) and lies just above the curve on either side of that point.