(Solved): Where do we go to see how errors propagate through a calculation? CALCULUS If \( F \) is a function ...

Where do we go to see how errors propagate through a calculation? CALCULUS If \( F \) is a function of \( x \) and \( y \), How does uncertainty in \( \mathrm{x} \) and \( \mathrm{y} \) translate into uncertainty in \( \mathrm{F} \) ? Same as asking... How do small CHANGES in \( \mathrm{x} \) and \( \mathrm{y}(\Delta \mathrm{x}, \Delta \mathrm{y}) \) lead to changes in \( \mathrm{F}(=\Delta \mathrm{F}) \) \[ \begin{aligned} F &=F(x, y) \\ d F &=\left(\frac{\partial F}{\partial x}\right) d x+\left(\frac{\partial F}{\partial y}\right) d y \\ \Delta F & \approx\left(\frac{\partial F}{\partial x}\right) \Delta x+\left(\frac{\partial F}{\partial y}\right) \Delta y \end{aligned} \] If \( F(x, y)=x+y \quad \) How does \( \mathrm{dF} \) relate to \( \mathrm{dx} \) and \( \mathrm{dy} \) ? \[ \begin{array}{l} d F=\left(\frac{\partial F}{\partial x}\right) d x+\left(\frac{\partial F}{\partial y}\right) d y \\ \left(\frac{\partial F}{\partial x}\right)=? \\ \left(\frac{\partial F}{\partial y}\right)=? \\ d F=? \end{array} \] How would \( \Delta \mathrm{F} \) relate to \( \Delta \mathrm{x} \) and \( \Delta \mathrm{y} \) (at least approximately)? \[ \Delta F=\left(\frac{\partial F}{\partial x}\right) \Delta x+\left(\frac{\partial F}{\partial y}\right) \Delta y \] If \( \mathrm{F}(\mathrm{x}, \mathrm{y})=\mathrm{axy} \), where \( \mathrm{a} \) is a constant How does \( \mathrm{dF} \) relate to \( \mathrm{dx} \) and \( \mathrm{dy} \) ? What is \( \left(\frac{\partial F}{\partial x}\right) \) What is \( \left(\frac{\partial F}{\partial y}\right) \) ? Put it together to find \( \mathrm{dF} \) using: \( d F=\left(\frac{\partial F}{\partial x}\right) d x+\left(\frac{\partial F}{\partial y}\right) d y \) Then, then find \( \Delta \mathrm{F} \) (at least approximately) using \( \Delta F \approx\left(\frac{\partial F}{\partial x}\right) \Delta x+\left(\frac{\partial F}{\partial y}\right) \Delta y \) Now, divide both sides by \( \mathrm{F} \) - what do you get?