Question

When one knows the true values x1 and x2 and has approximations X1 and X2 at hand, one can see where errors may arise. By viewing error as something to be added to an approximation to attain a true value, it follows that the error ei is related to Xi and xi as xi 5 Xi 1 ei (a) Show that the error in a sum X1 1 X2 is (x1 1 x2) 2 (X1 1 X2) 5 e1 1 e2 (b) Show that the error in a difference X1 2 X2 is (x1 2 x2) 2 (X1 2 X2) 5 e1 2 e2 (c) Show that the error in a product X1X2 is x1x2 2 X1X2 < X1X2 a e1 X1 1 e2 X2 b (d) Show that in a quotient X1yX2 the error is x1 x2 2 X1 X2 < X1 X2 a e1 X1 2 e2 X2 b

Answer 1

(a) For the sum X1 + X2, we have:

X1 + X2 = (x1 + e1) + (x2 + e2)

= x1 + x2 + (e1 + e2)

The error in the sum is given by:

e1 + e2 = (x1 + e1) + (x2 + e2) - (x1 + x2)

= (x1 + x2) + (e1 + e2) - (x1 + x2)

= e1 + e2

Therefore, the error in the sum is e1 + e2, as required.

(b) For the difference X1 - X2, we have:

X1 - X2 = (x1 + e1) - (x2 + e2)

= x1 - x2 + (e1 - e2)

The error in the difference is given by:

e1 - e2 = (x1 + e1) - (x2 + e2) - (x1 - x2)

= (x1 - x2) + (e1 - e2) - (x1 + x2)

= e1 - e2

Therefore, the error in the difference is e1 - e2, as required.

(c) Show that the error in a product X1X2 is:

x1x2 - X1X2 ≈ (X1 * e2) + (X2 * e1)

Proof:

We start with the equation:

X1X2 = (x1 + e1)(x2 + e2)

Expanding the right side of the equation, we get:

X1X2 = x1x2 + x1e2 + x2e1 + e1e2

Subtracting x1x2 from both sides, we get:

x1x2 - X1X2 = x1e2 + x2e1 + e1e2

Since e1 and e2 are small compared to x1 and x2, we can ignore the e1e2 term. Therefore, we can approximate the error as:

x1x2 - X1X2 ≈ (X1 * e2) + (X2 * e1)

(d) Show that in a quotient X1 / X2, the error is:

(x1 / x2) - (X1 / X2) ≈ ((e1 * X2) - (e2 * X1)) / (X2)^2

Proof:

We start with the equation:

X1 / X2 = (x1 + e1) / (x2 + e2)

Expanding the right side of the equation, we get:

X1 / X2 = (x1 / x2) + (x1 * e2 - x2 * e1) / (x2)^2 + e1 / x2 - e2 * x1 / (x2)^2

Subtracting (x1 / x2) from both sides, we get:

(x1 / x2) - (X1 / X2) = (x1 * e2 - x2 * e1) / (x2)^2 + e1 / x2 - e2 * x1 / (x2)^2

Simplifying the expression, we get:

(x1 / x2) - (X1 / X2) ≈ ((e1 * X2) - (e2 * X1)) / (X2)^2

This is the error in the quotient.

Step-by-step explanation: