Question

given uncertainties in x and y, what is the uncertainty in $d^{2}$? use the addition rule on the results of the power rule.
addition rule
form $r = x + y$
rule $delta r=delta x+delta y$
a) $delta d^{2}=2(xdelta x + ydelta y)$
b) $delta d^{2}=2(delta x+delta y)(x + y)$
c) $delta d^{2}=delta x^{2}+delta y^{2}$
view hint for question 5
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Explanation:

Step1: Recall power - rule for uncertainty

If $y = x^n$, then $\frac{\Delta y}{y}=n\frac{\Delta x}{x}$, or $\Delta y = n x^{n - 1}\Delta x$. Let $d^{2}=x^{2}+y^{2}$. For $u = x^{2}$, using the power - rule, $\Delta u=2x\Delta x$. For $v = y^{2}$, using the power - rule, $\Delta v = 2y\Delta y$.

Step2: Apply addition rule for uncertainty

Since $d^{2}=x^{2}+y^{2}$, and for $r = u + v$ (where $u=x^{2}$ and $v = y^{2}$), the addition rule for uncertainty $\Delta r=\Delta u+\Delta v$. Substituting $\Delta u = 2x\Delta x$ and $\Delta v=2y\Delta y$ into the addition rule, we get $\Delta d^{2}=2(x\Delta x + y\Delta y)$.

Answer:

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