Lens Formula and Derivative for Error Analysis:
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Taking the derivative of both sides with respect to \(u\) and \(v\), we get:
\[ -\frac{df}{f^2} = -\frac{dv}{v^2} + \frac{du}{u^2} \]
Rearranging for \(df\):
\[ df = f^2 \left( \frac{dv}{v^2} + \frac{du}{u^2} \right) \]
Error in Measurement of Focal Length:
Since \(dv\) and \(du\) represent the measurement errors in \(v\) and \(u\),
respectively, we can substitute \(dv = \Delta v\) and \(du = \Delta u\):
\[ \Delta f = f^2 \left[ \frac{\Delta v}{v^2} + \frac{\Delta u}{u^2} \right] \]
Conclusion:
The error in the measurement of the focal length \(f\) is:
\[ \Delta f = f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right] \]
Let \[ I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}} (x+15)^{\frac{15}{13}}} \] If \[ I(37) - I(24) = \frac{1}{4} \left( b^{\frac{1}{13}} - c^{\frac{1}{13}} \right) \] where \( b, c \in \mathbb{N} \), then \[ 3(b + c) \] is equal to:
For the thermal decomposition of \( N_2O_5(g) \) at constant volume, the following table can be formed, for the reaction mentioned below: \[ 2 N_2O_5(g) \rightarrow 2 N_2O_4(g) + O_2(g) \] Given: Rate constant for the reaction is \( 4.606 \times 10^{-2} \text{ s}^{-1} \).