The sensitivity S(x)S(x)S(x) of a precision electronic sensor is modeled by the function S(x)=(1+4x)−12S(x) = (1 + 4x)^{-\frac{1}{2}}S(x)=(1+4x)−21, where xxx is a small dimensionless parameter.
Find the first three terms, in ascending powers of xxx, of the binomial expansion of S(x)S(x)S(x).
An engineer attempts to use this expansion with x=0.3x = 0.3x=0.3 to calibrate the sensor. Explain why this approach is invalid.
By substituting x=−0.05x = -0.05x=−0.05 into your expansion, find an approximation for 5\sqrt{5}5. Give your answer to three significant figures.
