1. 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

2. 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).

4. 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

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

6. Differential form of derivatives:

7. 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

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

10. 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)

11. 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

13. 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.