Return to Main Page
Exercises for This Topic
Index of On-Line Topics
Everything for Calculus
Everything for Finite Math
Everything for Finite Math & Calculus
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 derivatives or, for a more detailed study, the on-line tutorials on derivatives.
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:
For this reason, the linear function whose graph is the tangent line to y = f(x) at a specified point (a, f(a)) is called the linear approximation of f(x) near x = a.
Q What is the formula for the linear approximation?
A All we need is the equation of the tangent line at a specified point (a, f(a)). Since the tangent line at (a, f(a)) has slope f'(a), we can write down its equation using the point-slope formula:
|y||=||y0 + m(x - x0)|
|=||f(a) + f'(a)(x - a)|
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
Let f(x) = x1/2. Find the linear approximation of f near x = 4 (at the point (4, f(4)) = (4, 2) on the graph), and use it to approximate 4.1.
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 Logarithm
Use linear approximation to approximate ln(1.134).
Here, 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):