3.6: Summary and Further Problems
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Lumping turns calculus on its head. Whereas calculus analyzes a changing process by dividing it into ever finer intervals, lumping simplifies a changing process by combining it into one unchanging process. It turns curves into straight lines, difficult integrals into multiplication, and mildly nonlinear differential equations into linear differential equations.
Problem 3.36 FWHM for another decaying function
Use the FWHM heuristic to estimate
\[\int_{-\infty}^{\infty} \frac{dx}{1 + x^{4}}. \label{3.56} \]
Then compare the estimate with the exact value of \(π/ 2\). For an enjoyable additional problem, derive the exact value.
Problem 3.37 Hypothetical pendulum equation
Suppose the pendulum equation had been
\[\frac{d^{2}θ}{dθ^{2}} + \frac{g}{l}\tan θ = 0. \label {3.57} \]
How would the period \(T\) depend on amplitude \(θ_{0}\)? In particular, as \(θ_{0}\) increases, would \(T\) decrease, remain constant, or increase? What is the slope \(dT/dθ_{0}\) at zero amplitude? Compare your results with the results of Problem 3.33.
For small but nonzero \(θ_0\), find an approximate expression for the dimensionless period \(h(θ_0)\) and use it to check your previous conclusions.
Problem 3.38 Gaussian 1-sigma tail
The Gaussian probability density function with zero mean and unit variance is
\[p(x) = \frac{e^{-x^{2}/2}}{\sqrt{2\pi}}. \label{3.58} \]
The area of its tail is an important quantity in statistics, but it has no closed form. In this problem you estimate the area of the 1-sigma tail
\[\int_{1}^{\infty} \frac{e^{-x^{2}/2}}{\sqrt{2\pi}}dx. \label{3.59} \]
a. Sketch the above Gaussian and shade the 1-sigma tail.
b. Use the 1/e lumping heuristic (Section 3.2.1) to estimate the area.
c. Use the FWHM heuristic to estimate the area.
d. Compare the two lumping estimates with the result of numerical integration:
\[\int_{1}^{\infty} \frac{e^{-x^{2}/2}}{\sqrt{2\pi}}dx = \frac{1 - erf(1 - \sqrt{2})}{2} ≈ 0.159, \label{3.60} \]
where erf(z) is the error function.
Problem 3.39 Distant Gaussian tails
For the canonical probability Gaussian, estimate the area of its n-sigma tail (for large n). In other words, estimate
\[\int_{n}^{\infty} \frac{e^-x^{2}/2}{\sqrt{2\pi}}dx. \label{3.61} \]