Linearization and Differentials
- Calculus 3.2
Linearization
If $f$ is a function that is differentiable at $x = a$, then the linearization of $f$ at $a$ is given by:
\[ L(x) = f(a) + f'(a)(x - a) \]
This linear function $L(x)$ approximates the value of $f(x)$ near $x = a$. The point $x = a$ is called the center of the approximation.
Differentials
The differential of a function $y=f(x)$ at a point $x$ is defined as:
\[ \dd{y} = f'(x) \, \dd{x} \]
Here, $\dd{y}$ represents the change in $y$ corresponding to a small change $\dd{x}$ in $x$. The differential $\dd{y}$ can be thought of as the linear approximation of the change in $y$ when $x$ changes by a small amount.
Estimating Changes
To estimate the change in $y$ when $x$ changes from $a$ to $a + \Delta x$, we can use the differential:
\[ \Delta y = f(a+\Delta x) -f(a) \approx \dd{y} = f'(a) \, \Delta x \]
This approximation is valid when $\Delta x$ is small, allowing us to use the linearization of $f$ at $a$.
Error in Differential Approximation
Supposing that $\dd{x} = \Delta x$, the approximation error is calculated as:
\[ \begin{align*} \text{Error} &= \Delta y - \dd{y} \nl &= f(a + \Delta x) - f(a) - f'(a) \, \dd{x} \nl &= \left( \frac{f(a + \Delta x) - f(a)}{\Delta x} - f'(a) \right) \Delta x \nl &= \varepsilon \Delta x \end{align*} \]
where $\varepsilon$ is the error term, which approaches zero as $\Delta x$ approaches zero.
\[ \lim_{\Delta x \to 0} \varepsilon = f'(a) - f'(a) = 0 \]
Thus, the error in the differential approximation becomes negligible as $\Delta x$ becomes very small. In summary,
\[ \Delta y = f'(a) \, \Delta x + \varepsilon \Delta x \]
