Linear Approximation and Differentials
Read this section to learn how linear approximation and differentials are connected. Work through practice problems 1-10.
Linear Approximation
Since this section uses tangent lines frequently, it is worthwhile to recall how we find the equation of the line tangent to at a point . The line tangent to at goes through the point and has slope so, using the point-slope form for linear equations, we have
This final result is the equation of the line tangent to at .
Example 1: Find the equation of the line which is tangent to the graph of at the point . Evaluate and to approximate and .
. If is close to , then the value of is a good approximation of the value of (Fig. 2). The number is close to 9 so
Similarly, In fact, , so our estimate, using , is within of the exact answer. (accurate to 6 decimal places) and our estimate is within of the exact answer.
In each example, we got a good estimate of a square root with very little work. The graph indicates the tangent line is slightly above , and each estimate is slightly larger than the exact value.
Practice 1: Find the equation of the line tangent to the graph of at the point (Fig. 3). Evaluate and to approximate and . Are your approximations using larger or smaller than the exact values of the square roots?
Practice 2: Find the equation of the line tangent to the graph of at the point and use to approximate and . Do you think your approximations using are larger or smaller than the exact values?
The process we have used to approximate square roots and cubics can be used to approximate any differentiable function, and the main result about the linear approximation follows from the two statements in the boxes. Putting these two statements together, we have the process for Linear Approximation.
If is differentiable at and is close to ,
(algebraically) the values of the tangent line function
Example 2: Use the linear approximation process to approximate the value of .
Solution: so We need to pick a value a near for which we know the exact value of and , and is the obvious choice. Then
This approximation is within of the exact value of
Practice 3: Approximate the value of , the amount becomes after 4 years in a bank which pays interest compounded annually. (Take and ).
Practice 4: Use the linear approximation process and the values in the table to estimate the value of when and .
We can also approximate functions as well as numbers.
Example 3: Find a linear approximation formula for when is small. Use your result to approximate and .
Solution: so . Since "x is small", we know that is close to , and we can pick Then and so
If is small, then and . Use your calculator to determine by how much each estimate differs from the true value.