3.10 Linear Approximations and Differentials Understand the concept of a tangent line approximation Compare the value of the differential, dy, with the actual change in y, ∆y

Consider the equation for the tangent line for a differentiable function at the point (c, f(c)) slope or This tangent line is called the linearization of f(x) at c. By choosing values of x sufficiently close to c, the values of y can be used as approximations for values of f. In other words, as x→c, the limit of y is f(c).

Differentials When ∆x is small, ∆y = f(c + ∆x) – f(c) is x – c is called the change in x and is denoted delta x. When delta x is small, the change in y, denoted delta y, can be approximated by When ∆x is small, ∆y = f(c + ∆x) – f(c) is ≈ f ‘(c) ∆x which we’ll call dy

Find the linearization L(x) of the function at a.

Differential form of derivatives:

For such an approximation, the quantity ∆x = dx, and is called the differential of x. Then ∆y ≈ dy as defined below. In many types of applications, the differential of y can be used as an approximation of the change in y. That is, ∆y ≈ dy, or ∆y ≈ f ‘(x) dx

y = f (x + x) – f (x) dy = f (x) dx

Error Propagation y = f (x + x) – f (x) Physicists and engineers make liberal use of approximation dy to replace ∆y. In practice this is in estimation of errors propagated by measuring devices. If x represents the measured value of variable, and x + ∆x is the exact value, then ∆x is the error in measurement. If the measured value x is used to compute another value f(x), the difference between f(x + ∆x) and f(x) is the propagated error. y = f (x + x) – f (x)

The answer is best given in relative terms by comparing dV with V or dA with A Maximum possible error dV relative error dV/V percent error - relative error written as percent

In summary, these are the most useful ideas:   y = f (x + x) – f (x) dy = f (x) dx Tangent line! relative error, which is computed by dividing the error by the total thing you are figuring: for example dV/V or dA/A Percent error, which is relative error written as a percent.