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Chapter 4 | Applications of Derivatives
That is,
f ( a + dx )− f ( a ) ≈ L ( a + dx )− f ( a ) = f ′( a ) dx . In other words, the actual change in the function f if x increases from a to a + dx is approximately the difference between L ( a + dx ) and f ( a ), where L ( x ) is the linear approximation of f at a . By definition of L ( x ), this difference is equal to f ′( a ) dx . In summary, Δ y = f ( a + dx )− f ( a ) ≈ L ( a + dx )− f ( a ) = f ′( a ) dx = dy . Therefore, we can use the differential dy = f ′( a ) dx to approximate the change in y if x increases from x = a to x = a + dx . We can see this in the following graph.
Figure 4.11 The differential dy = f ′( a ) dx is used to approximate the actual change in y if x increases from a to a + dx .
We now take a look at how to use differentials to approximate the change in the value of the function that results from a small change in the value of the input. Note the calculation with differentials is much simpler than calculating actual values of functions and the result is very close to what we would obtain with the more exact calculation. Example 4.9 Approximating Change with Differentials
Let y = x 2 +2 x . Compute Δ y and dy at x =3 if dx =0.1.
Solution The actual change in y if x changes from x =3 to x =3.1 is given by
Δ y = f (3.1)− f (3) = [(3.1) 2 + 2(3.1)] − [3 2 + 2(3)] = 0.81. The approximate change in y is given by dy = f ′(3) dx . Since f ′( x ) =2 x +2, we have dy = f ′(3) dx = (2(3) + 2)(0.1) = 0.8.
For y = x 2 +2 x , find Δ y and dy at x =3 if dx =0.2.
4.9
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