Linear and Quadratic Approximations

\(f(x) \approx f(x_0) + f'(x_0)(x-x_0)\)

Tangent line is near the function curve at values close to \(x_0\) so it serves as an approximation there.

Think back to \(\lim_{x\rightarrow0} \frac{f(x_0+\Delta x) - f(x_0)}{x-x_0}\) and realize that the approximation \(\frac{\Delta f}{\Delta x} \approx f'(x_0)\) arises. This is the same relationship as the first equation!

Why?

• Manipulate second approximation to yield the relation \(\Delta f \approx f'(x_0)\Delta x\).
  
• Substitute to get \(f(x)-f(x_0) \approx f'(x_0)(x-x_0)\).
  
• Put constant on the other side to get the original equation.
  \(f(x) \approx f(x_0) + f'(x_0)(x-x_0)\)
  

To simplify, base point (\(x_0\)) will be 0.
Formula becomes \(f(x) \approx f(0) + f'(0)x\) with \(x_0 = 0\).

NOTE: This formula and the examples based off of it only approximate for values of \(x\) near 0.

For \(x\approx 0\):

• \(\sin{x}\): \(f(x) \approx f(0) + f'(0)x\) so \(\sin{x} \approx x\)
  
• \(\cos{x}\): \(f(x) \approx f(0) + f'(0)x\) so \(\cos{x} \approx 1\)
  
• \(e^{x}\): \(f(x) \approx f(0) + f'(0)x\) so \(e^{x} \approx 1+x\)
  
• \(\ln{(1+x)}\): \(f(x) \approx f(0) + f'(0)x\) so \(\ln{(x+1)} \approx x\)
  
• \((1+x)^{r}\): \(f(x) \approx f(0) + f'(0)x\) so \((1+x)^{r} \approx 1+rx\)
  

Linear approximations greatly simplify what can sometimes be more complicated functions (i.e. logarithms which would require a calculator, while 1+x is much simpler).

Example 3 For \(x\approx0\), find a linear approximation of \(\frac{e^{-3x}}{\sqrt{1+x}}\).

• Remember the earlier approximations of \(e^x\) and \((1+x)^r\).
  
• Rewrite as product: \(e^{-3x}(1+x)^{-1/2} \approx (1-3x)(1-\frac{1}{2}x)\).
  
• Expand: \(e^{-3x}(1+x)^{-1/2} \approx 1-3x-\frac{1}{2}x+\frac{3}{2}x^2\).
  
• Goal is a linear approximation so loosely approximate to \(e^{-3x}(1+x)^{-1/2} \approx 1-\frac{7}{2}x\) (sum up coefficients).
  
  • Drop the \(x^2\) and higher terms as they get small as \(x\) is near zero.
    

Serves as an extension of the linear approximation formula.

\(f(x) \approx f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2}(x-x_0)^2\)

Example 2 Compute \(\ln{1.1}\) with a quadratic approximation.

• Algebra yields \(\ln{1+x} \approx x-\frac{x^2}{2}\).
  
• Plug in: \(\ln(1.1) \approx \frac{1}{10} - \frac{1}{2}\big(\frac{1}{10}\big)^2\) or 0.95
  

Quadratic is not always more helpful, as in approximations like that of \(\sin{x}\), quadratic term vanishes.