Linear approximation is a fundamental application of differentiation, used to approximate functions by linear ones near a given point. This page presents detailed examples with solutions on how to find linear approximations to functions.
Linear Approximation Formula
A linear approximation \( f_l \) to a function \( f \) at \( x = a \) is obtained from the equation of the tangent line to the graph of \( f \) at \( x = a \):
For values of \( x \) close to \( a \), the values of \( f(x) \) and \( f_l(x) \) are close. Since \( f_l(x) \) is a linear function, it serves as a linear approximation to \( f \).
This technique is especially useful to linearize non-algebraic functions such as sine, cosine, logarithm, and exponential functions to simplify their computations. Examples follow below.
Step-by-Step Examples
Example 1: Linear approximation of \( f(x) = \tan x \) near \( x = 0 \)
First, compute the derivative:
\[ f'(x) = \sec^2 x \]Evaluate at \( x = 0 \):
\[ f'(0) = \sec^2(0) = 1 \]Thus, the linear approximation \( f_l(x) \) at \( x = 0 \) is:
\[ f_l(x) = f(0) + f'(0)(x - 0) = 0 + 1 \cdot x = x \]This means \(\tan x \approx x\) for \( x \) close to 0, where \( x \) is measured in radians.
Try calculating \(\tan x\) and compare it with \(x\) for values: \[ x = 0, 0.001, 0.01, 0.1, 0.2, 0.3, 0.5 \] Use your calculator set in radians mode and observe the approximation.
Example 2: Linear approximation of \( f(x) = \ln x \) near \( x = 1 \)
Compute the derivative:
\[ f'(x) = \dfrac{1}{x} \]Evaluate at \( x = 1 \):
\[ f'(1) = 1 \]The linear approximation \( f_l(x) \) at \( x = 1 \) is:
\[ f_l(x) = \ln 1 + 1 \cdot (x - 1) = 0 + (x - 1) = x - 1 \]So, \[ \ln x \approx x - 1 \] for \( x \) near 1.
Compare \(\ln x\) and \(x - 1\) for values: \[ x = 1, 1.001, 1.01, 1.1, 1.5 \] using your calculator.
Example 3: Linear approximation of \( f(x) = e^x \) near \( x = 0 \)
The derivative is:
\[ f'(x) = e^x \]Evaluate at \( x = 0 \):
\[ f'(0) = 1 \]The linear approximation \( f_l(x) \) at \( x = 0 \) is:
\[ f_l(x) = e^0 + 1 \cdot (x - 0) = 1 + x \]Compare \( e^x \) and \( 1 + x \) for values: \[ x = 0, 0.001, 0.01, 0.1, 0.5 \] using your calculator.
Conclusion & Further Study
Linear approximation is one of the simplest ways to approximate transcendental functions that cannot be expressed algebraically. There exist more advanced techniques for better approximations.
For further study, see more applications of differentiation.