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Chapter 4: Problem 6

Does the differential \(d y\) represent the change in \(f\) or the change in the linear approximation to \(f\) ? Explain.

Short Answer

Expert verified
Answer: The differential dy represents the change in the linear approximation to the function f, not the actual change in the function f.

Step by step solution

01

Definition of Differential dy

The differential dy is defined as the change in the dependent variable (y) for a given change in the independent variable (x), which is represented by the differential dx. It can be mathematically expressed as: dy = f'(x) * dx, where f'(x) denotes the derivative of the given function f with respect to x.
02

Change in function f

The change in the function f for a given change in the independent variable x (from a to a + dx) can be mathematically expressed as: 螖f = f(a + dx) - f(a). This is the actual change in the value of the function f over the given interval.
03

Linear Approximation to f

The linear approximation of function f at a point x = a is given by the tangent line to the function at that point. The equation for the linear approximation is: L(x) = f(a) + f'(a) * (x - a). The change in the linear approximation for a given change in x (from a to a + dx) is: 螖L = L(a + dx) - L(a).
04

Comparing dy with the change in f and the linear approximation

As mentioned, dy = f'(x) * dx. From the definition of the linear approximation, the change in the linear approximation 螖L can be written as: 螖L = f'(a) * (a + dx - a) = f'(a) * dx. We see that dy and 螖L are similar, however, dy represents the change in y due to the change in x at a given point, while 螖L represents the change in the linear approximation to f. On the other hand, the change in the function f (螖f = f(a + dx) - f(a)) is generally not equal to dy, as it represents the actual change in the value of the function over the given interval.
05

Conclusion

The differential dy represents the change in the linear approximation to the function f and not the actual change in the function f for a given change in the independent variable x. This is because dy equals the change in the linear approximation (螖L) to the function f and is different from the actual change in the function f (螖f).

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