Linear Approximation and Differentials
Read this section to learn how linear approximation and differentials are connected. Work through practice problems 1-10.
Linear Approximation
The Differential of f
In Fig. 7, the change in value of the function near the point is and the change along the tangent line is . If is small, then we have used the approximation that . This leads to the definition of a new quantity, , called the differential of .
The differential of represents the change in , as changes from to , along the tangent line to the graph of at the point . If we take to be the number , then the differential is an approximation of .
Example 7: Determine the differential df of each of , and . Solution: , and .
Practice 9: Determine the differentials of , and .
We will do little with differentials for a while, but are used extensively in integral calculus.