7.4 Application of Binomial Theorem in Approximate Calculation 二项式定理在近似值计算中的应用 - 独中课程 | 线上补习

高二数学 | 高级数学

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7.4 Application of Binomial Theorem in Approximate Calculation 二项式定理在近似值计算中的应用

[1]

Obtain the first four terms in the expansion of (1 + p) $^6$ in ascending powers of p. By using p = x + x $^2$ , expand (1 + x + x $^2$ ) $^6$ as far as the term containing x $^2$ . Then find the value of (1.0101) $^6$ , giving your answer correct to 3 decimal places.

以p的升幂式表示(1 + p) $^6$ 的首四项. 用p = x + x $^2$ 展开(1 + x + x

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) $^6$ 到有 x $^2$ 的那一项. 跟着求(1.0101) $^6$ , 答案准确至小数三位数.

Answer:

1.062

[2]

Use the binomial theorem to find the values of the following :
用二项式定理求以下的值

(a)
$\sqrt{1.001}$ , correct to six places of decimal.

$\sqrt{1.001}$ , 准确至小数六位数.

(b)
$\frac{1}{(1.02)^2}$ , correct to four places of decimals.

$\frac{1}{(1.02)^2}$ ,准确至小数四位数.

Answer:

(a) 1.000500

(b) 0.9612

[3]

Find the first four terms of the expansion of (1 – 8x) $^\frac{1}{2}$ in ascending powers of x. Substitute x = $\frac{1}{100}$ and obtain the value of $\sqrt{23}$ correct to five significant figures.

按x的升幂展开(1 – 8x) $^\frac{1}{2}$ 的首四项. 代入x = $\frac{1}{100}$ , 求 $\sqrt{23}$ 之值准确至五位有效数字.

Answer:

1 – 4x – 8x $^2$ – 32x $^3$ – …,

4.7958

[4]

If x is so small that x $^3$ and higher powers can be neglected, prove that $\left(\frac{1-x}{1+x}\right)^p$ = $1-2px+2p^2x^2$ . Hence, find an approximation of $\sqrt[5]{\frac{99}{101}}$ to five decimal places.

若x非常小使到x $^3$ 与更高次幂均可略去不计, 试证 $\left(\frac{1-x}{1+x}\right)^p$ = $1-2px+2p^2x^2$ . 据此, 试求 $\sqrt[5]{\frac{99}{101}}$ 的近似值, 取至小数点五位.

Answer:

0.99601