Question

For the table of values f(x) =xe^x given by
x f(x)
1.8 10.8894
1.9 12.7032
2.0 14.7781
2.1 17.1489
2.2 19.8550
Find f"(2.0) using the central difference formula of 0(h^2) fir h=0.1 and h=0.2. Calculate T.E. and actual error.

Answer 1

Answer on Question #75527 – Math – Quantitative Methods

Question

For the table of values $f(x) = x e^x$ given by

x f(x)

1.8 10.8894

1.9 12.7032

2.0 14.7781

2.1 17.1489

2.2 19.8550

Find $f''(2.0)$ using the central difference formula of $0(h^2)$ for $h=0.1$ and $h=0.2$. Calculate T.E. and actual error.

Solution

Exact value:

$$f(x) = x e^x, \quad f'(x) = (x + 1) e^x, \quad f''(x) = (x + 2) e^x,$$ $$f''(2.0) = 4 e^2 \approx 29.5562.$$

Central difference formula.

$$f''(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}.$$

Using $h = 0.1$:

$$f''(2.0) \approx \frac{f(2.1) - 2f(2.0) + f(1.9)}{0.1^2} = \frac{17.1489 - 2 \times 14.7781 + 12.7032}{0.01} = 29.59.$$

Total error $\varepsilon = 29.5562 - 29.59 = -0.0338$.

Actual error $|\varepsilon| = 0.0338$.

Using $h = 0.2$:

$$f''(2.0) \approx \frac{f(2.2) - 2f(2.0) + f(1.8)}{0.2^2} = \frac{19.8550 - 2 \times 14.7781 + 10.8834}{0.04} = 29.555.$$

Total error $\varepsilon = 29.5562 - 29.555 = -0.0012$. Actual error $|\varepsilon| = 0.0012$.

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