Question

given uncertainties in x and y, what is the uncertainty in d²? use the addition rule on the results of the power rule.
addition rule
form r = x + y
rule δr=δx + δy
a) δd²=2(xδx + yδy)
b) δd²=2(δx + δy)(x + y)
c) δd²=δx² + δy²
hide hint for question 5
first, express δd² in terms of the uncertainties in x² and y² using the addition rule, then reduce it further by using the power rule to express δx² and δy² in terms of the uncertainties in x and y. problems with more than one algebraic operation are always approached step - by - step, keeping in mind order of operations.

Explanation:

Step1: Apply power - rule for uncertainty

The power - rule for uncertainty states that if $u = x^n$, then $\Delta u=n x^{n - 1}\Delta x$. For $u = x^2$, $\Delta x^2 = 2x\Delta x$ and for $u = y^2$, $\Delta y^2=2y\Delta y$.

Step2: Apply addition rule for uncertainty

We know that if $d^2=x^2 + y^2$, then by the addition rule for uncertainty $\Delta d^2=\Delta x^2+\Delta y^2$. Substituting $\Delta x^2 = 2x\Delta x$ and $\Delta y^2=2y\Delta y$ into the equation, we get $\Delta d^2=2(x\Delta x + y\Delta y)$.

Answer:

A. $\Delta d^2=2(x\Delta x + y\Delta y)$