求表达式$((\frac{\sqrt{3}}{5})^3-2)^5$的绝对误差限及近似值场景下误差限

Let's work through this problem step by step. We'll cover two scenarios: first deriving the general absolute error bound for the expression, then computing the bound when we use the approximate value $(\frac{\sqrt{3}}{5})^* = 0.34$.

1. General Absolute Error Bound for the Exact Expression #

First, let's define the function we're working with: let $f(x) = (x^3 - 2)^5$, where $x = \frac{\sqrt{3}}{5}$. To estimate absolute error bounds, we'll use the differential approximation method—a standard go-to for error propagation in functions.

The absolute error bound $\Delta f$ for $f(x)$ is approximated by:
$$\Delta f \approx |f'(x)| \cdot \Delta x$$
Here, $\Delta x$ is the absolute error bound of the input $x$, and $f'(x)$ is the derivative of $f(x)$ with respect to $x$.

First, compute the derivative of $f(x)$:
$$f'(x) = 5(x^3 - 2)^4 \cdot 3x^2 = 15x^2(x3 - 2)^4$$

Now plug in the exact value of $x = \frac{\sqrt{3}}{5}$ to calculate $|f'(x)|$:

• Calculate $x^2 = \left(\frac{\sqrt{3}}{5}\right)^2 = \frac{3}{25} = 0.12$
• Calculate $x^3 = \left(\frac{\sqrt{3}}{5}\right)^3 = \frac{3\sqrt{3}}{125} \approx 0.04157$
• Calculate $x^3 - 2 = 0.04157 - 2 = -1.95843$, so $(x^3 - 2)^4 \approx (-1.95843)^4 \approx 14.707$
• Multiply it all together: $|f'(x)| = 15 \times 0.12 \times 14.707 \approx 26.47$

So the general absolute error bound for the expression is:
$$\Delta f \approx 26.47 \cdot \Delta x$$
(Note: $\Delta x$ here is the absolute error bound of $\frac{\sqrt{3}}{5}$. If you have an error bound for $\sqrt{3}$ itself, you can derive $\Delta x$ from that by scaling by $\frac{1}{5}$.)

2. Absolute Error Bound When Using $(\frac{\sqrt{3}}{5})^* = 0.34$ #

First, let's find the absolute error bound of the approximate $x^* = 0.34$ relative to the exact $x$:

• Exact $x = \frac{\sqrt{3}}{5} \approx \frac{1.73205}{5} \approx 0.34641$
• Absolute error $\Delta x = |x - x^*| = |0.34641 - 0.34| = 0.00641$

Now plug this $\Delta x$ into our earlier error bound formula:
$$\Delta f \approx 26.47 \times 0.00641 \approx 0.17$$

To verify, we can compute the exact and approximate values of the expression directly:

• Exact expression value: $((0.34641)^3 - 2)^5 \approx (-1.95843)^5 \approx -28.80$
• Approximate expression value: $((0.34)^3 - 2)^5 = ((0.039304) - 2)^5 \approx (-1.9607)^5 \approx -28.97$
• The absolute difference is $|-28.80 - (-28.97)| = 0.17$, which matches our error bound estimate perfectly.

内容的提问来源于stack exchange，提问作者peterxd4