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Note To understand this topic, you will need to be familiar with derivatives, as discussed in Chapter 3 of Calculus Applied to the Real World. If you like, you can review the topic summary material on techniques of differentiation or, for a more detailed study, the on-line tutorials on derivatives of powers, sums, and constant multipes.
We start with the observation that if you zoom in to a portion of a smooth curve near a specified point, it becomes indistinguishable from the tangent line at that point. In other words:
Q The above argument is based on geometry: the fact that the tangent line is close to the original graph near the point of tangency. Is there an algebriac way of seeing why this is true?
A Yes. This links to an algebraic derivation of the linear approximation.
Linear Approximation of $f(x)$ Near $x = a$
If $x$ is close to a, then
Example 1 Linear Approximation of the Square Root
We can use $L(x)$ to approximate the square root of any number close to $4$ very easily without using a calculator. For example,
Example 2 Linear Approximation of the LogarithmUse linear approximation to approximate $\ln(1.134).$
SolutionHere, we are not given a value for $a.$ The key is to use a value close to $1.134$ whose natural logarithm we know. Since we know that $\ln(1) = 0,$ we take a to be $1.$
Now use the formula for linear approximation:
Substituting and simplifying gives (numerical answers should be accurate to 4 decimal places):