An Approximation

1. The act, process or result of approximating.
2. (mathematics) An imprecise solution or result that is adequate for a defined purpose.
3. (medicine) The act of bringing together the edges of tissue to be sutured. Synonym: coaptation

\[x <<1\rightarrow(1 + x)^n ≅ 1 + nx\]

Mathematics

Linear Approximation

(also known as tangent line approximation, Lineær tilnærming). It is a calculus method used to estimate the value of a complicated function by replacing it with a simple straight line.

Why this matters: This tool is heavily used in physics and engineering to simplify complex equations. For example, the famous small-angle approximation \(\sin(\theta) \approx \theta\) used in pendulum mechanics is just a linear approximation of the sine wave near zero.

The Core Concept: "Zooming In"

If you take any smooth mathematical curve and zoom in closely enough on a single point, the curve will look almost perfectly straight. That straight line is the tangent line (tangentlinje).

Because calculating points on a curve (like square roots or logarithms) can be difficult, we instead calculate points on that straight tangent line. As long as you stay very close to your original point, the straight line gives you an incredibly accurate estimate of the real curve.

The Formula

The formula for the linear approximation \(L(x)\) near a known point \(a\) is:

\[L(x) = f(a) + f'(a)(x - a)\]

Here is what each part means:

• \(f(a)\): The exact \(y\)-value at your chosen "nice" starting point \(a\).
• \(f'(a)\): The slope of the curve (the derivative, den deriverte) at that exact point.
• \((x - a)\): The small step or distance you are moving away from your starting point.

Final Estimate

Calculating final estimated number, and mathematically proving whether that estimate is slightly too high (overestimate) or slightly too low (underestimate).

To find the complete conclusion, you must look at the second derivative \(f''(x)\) of your original function to see how the curve bends (its concavity).

1. State the Final Estimate

This is the result of plugging the target number into the tangent line formula \(L(x)\). The calculated value \(L(x_0)=l_0\)

2. Check the Second Derivative (Concavity)

Because the tangent line is straight and the real curve is bent, the estimate will always be slightly off. It can be found out how it is off by finding the second derivative \(f''(x)\) at the starting point \(a\).

• If \(f''(a)\) is positive (\(> 0\)): The curve is "concave up". The straight tangent line sits underneath the curve. Conclusion: Your estimate is an underestimate.
• If \(f''(a)\) is negative (\(< 0\)): The curve is "concave down" (bending downwards like a frown). The straight tangent line sits on top of the curve. Conclusion: Your estimate is an overestimate.

Example

\[f(x)=\sin x: x<<\frac {\pi}{12}\]

\[L(x)=f(a)+f'(a)(x-a)\]

\[L(x)=\sin (a)+cos(a)(x-a)\]

The expansion point must be set to \(a = 0\), as the limit evaluates the behavior of the function as \(x\) approaches \(0\).

\[L(x) = \sin(0) + \cos(0)(x - 0)\]

\[L(x) = 0 + 1(x) = x\]

\[\sin (x) \approx L(x)=x\]

\[\sin (x) \approx x\]

\[ \frac {\sin (x)}{x} \approx 1 : x<<\frac {\pi}{12} \iff \lim_{x\to 0} \frac{\sin (x)}{x} = 1\]

Example

binomial approximation

\[f(x)=(1+x)^{\beta}\]

\[L(x)=(1+a)^{\beta}+\beta(1+a)^{\beta-1}(x-a)\]

The expansion point must be set to \(a = 0\), as the limit evaluates the behavior of the function as \(x\) approaches \(0\).

\[L(x)=(1+0)^{\beta}+\beta(1+0)^{\beta-1}(x-0)\]

\[L(x)=1+\beta x\]

\[(1+x)^{\beta}\approx L(x)=1+\beta x\]

\[(1+x)^{\beta}\approx 1+\beta x \ ; x<<1\]

Stirling's approximation

is an asymptotic formula that provides an estimate for the factorial function (n!) when (n) is large. The approximation was named after the Scottish mathematician James Stirling and is expressed as:

\[n! \approx \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n\]

1. The Formula:

- Stirling's approximation provides an asymptotic expansion of the factorial function as (n) approaches infinity.

2. Motivation:

- Stirling's approximation is motivated by the desire to find a simple and accurate expression for (n!) in terms of elementary functions when (n) is large.

3. Comparison with Factorial:

- Stirling's approximation becomes increasingly accurate as (n) becomes larger. For small values of (n), the approximation may not be very precise, but it improves as (n) grows.

4. Applications:

- Stirling's approximation is widely used in asymptotic analysis, probability theory, and statistical physics. It simplifies expressions involving factorials and allows for more straightforward calculations in the limit of large (n).

5. Error Term:

- The approximation can be refined by including an error term. One common refinement is to write Stirling's formula as \(n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n \cdot e^{c/n}\), where (c) is a constant. The error term \(e^{c/n}\) becomes negligible as (n) becomes large.

While Stirling's approximation is a powerful tool for estimating factorials in the limit of large (n), it's important to note that it is an asymptotic formula. This means that, while it becomes increasingly accurate as (n) grows, it may not provide a precise value for finite values of (n). For practical purposes, especially in numerical calculations, Stirling's approximation is often sufficient for sufficiently large (n).