   Chapter 2.R, Problem 12CC

Chapter
Section
Textbook Problem

# (a) Write an expression for the linearization of f at a.(b) If y = f ( x ) , write an expression for the differential dy.(c) If d x = △ x , draw a picture showing the geometric meanings of △ y and dy.

(a)

To determine

To write: Expression for the linearization of f at a.

Explanation

Close to the point of tangency  a curve is nearby to its tangent line. In other words, the graph of a differentiable function looks like its tangent line near the point of tangency. From this we can find the approximate values of functions.

We can calculate a value f(a) of a function, but it might be hard to find the nearby values of f. So we consider the tangent which is a linear function. It is easy to calculate it’s values.

In other words, use tangent line at (a,fa) as an approximation to the curve y=f(x) when x is near a. An equation of tangent line is

y=fa+f'(a)(x-a)

and the approximation

f(x

(b)

To determine

To write: Expression for differential dy.

(c)

To determine

To write: Draw a picture showing the geometrical meaning of y and dy.

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