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7 lip 2017 · Feynman ostensibly used the trivial linear approximation of ex: ex ≈ 1 + x. which works well for small values of x. Thus: e3.3 =e1 ⋅e2.3026−0.0026 ≈ e ⋅ 10 ⋅e−0.0026 ≈ 10e(1 − 0.0026) = 27.1121 …. The correction adds two more correct decimal places and is quite easy to compute by hand.
The first function approximations is: $$ 1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) \approx \frac{y^2}{2x^2 (1-p^2)}$$ ,when $y \ll x$. Note that I tried using the approximation $e^x \approx 1+x$, when x is small, but all I got was the conclusion that $1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y ...
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The simplest way to approximate a function f(x) for values of x near a is to use a linear function. The linear function we shall use is the one whose graph is the tangent line to f(x) at x = a. This makes sense because the tangent line at (a, f(a)) gives a good approximation to the graph of f(x), f(a) f(x) if x is close to a. That is, for x ≈ a,
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28 maj 2023 · Equation 3.4.6 Quadratic approximation. \begin {gather*} f (x)\approx f (a)+f' (a) (x-a)+\frac {1} {2} f'' (a) (x-a)^2 \end {gather*} Here is a figure showing the graphs of a typical \ (f (x)\) and approximating function \ (F (x)\text {.}\) This new approximation looks better than both the first and second.
