§3.10 Linear Approximations and Differentials Main idea We have seen that a curve lies very close to its tangent line near the point of tangency. So we can use the tangent line to approximate the graph near the point. Topics: Linear approximation Differentials

I. linear approximation The tangent line at (a, f(a)) is y = f(a) + f (a)(x – a) The approximation f(x)  f(a) + f (a)(x – a) is called linear approximation of f at a. The linear function L(x) = f(a) + f (a)(x – a) is called the linearization of f at a.

Differentials Consider y = f(x). Then we have We treat the left side as a ratio of two quantities. Then we have

Definition: In equation (2) on last slide, both dx and dy are called differentials. dy is the dependent variable and dx is the independent variable. dy depends on both dx and x. So dy is a two variable function. Comment: Computing derivatives y' = f ' (x) is equivalent to computing differentials dy = f ' (x)dx .

Increment and differential If x is given an increment (i.e. change x), then y = f(x + x) – f(x) is the increment of y. The differential dy can be used to approximate y when dx = x is small. That is: y  dy f(x + x) – f(x)  f (x)x. Note: Computing y is hard in general. But computing dy is always easy.

Relationship between linear approximation and differential: L(x) = f(a) + f (a)(x – a) = f(a) + dy where dy = f (a) dx and dx = x = x – a. L(x) – f(a) = dy f(x)  L(x) = f(a) + dy  f(x) – f(a)  dy  y  dy where y = f(x) – f(a)