Asosiy tarkibga o'tish

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

3. \(W\left( x \right) = \ln \left( {1 + {x^4}} \right)\) funksiyasi va \(x = 1\) 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. 1.5
    2. 1.1
    3. 1.01
    4. 1.001
    5. 1.0001
    1. 0.5
    2. 0.9
    3. 0.99
    4. 0.999
    5. 0.9999
  2. \(x = 1\) da \(W\left( x \right)\) 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. 1.5
  2. 1.1
  3. 1.01
  4. 1.001
  5. 1.0001
  1. 0.5
  2. 0.9
  3. 0.99
  4. 0.999
  5. 0.9999
Yechim

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

\[{m_{PQ}} = \frac{{W\left( x \right) - W\left( 1 \right)}}{{x - 1}} = \frac{{\ln \left( {1 + {x^4}} \right) - \ln \left( 2 \right)}}{{x - 1}}\]

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

\(x\) \({m_{PQ}}\) \(x\) \({m_{PQ}}\)
1.5 2.21795015 0.5 1.26504512
1.1 2.08679449 0.9 1.88681740
1.01 2.00986668 0.99 1.98986668
1.001 2.00099867 0.999 1.99899867
1.0001 2.00009999 0.9999 1.99989999


b \(x = 1\) da \(W\left( x \right)\) ga tangens chiziqning qiyaligini taxmin qilish uchun (a) dagi ma'lumotlardan foydalaning va tangens chiziq tenglamasini yozing. Yechim

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

Tangens chiziq tenglamasi u holda,

\[y = W\left( 1 \right) + m\left( {x - 1} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\ln \left( 2 \right) + 2\left( {x - 1} \right)}}\]

Bu funksiya va tangens chiziqning grafigi.

The graph of \(W\left( x \right) = \ln \left( {1 + {x^4}} \right)\) in red for \(-1 \le x \le 2\), along with its tangent line \(y = \ln \left( 2 \right) + 2\left( {x - 1} \right)\) in blue. The curve decreases from about 0.7 at \(x = -1\), is nearly flat along the \(x\)-axis around \(x = 0\) and then increases to about 2.8 at \(x = 2\). The tangent line touches the curve at the labeled point P = \(\left( {1,\ln \left( 2 \right)} \right)\) and crosses the \(x\)-axis at about \(x = 0.65\).