7/27/2019 Calculus I Notes, Section 2-9

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7/27/2019 Calculus I Notes, Section 2-9

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Here is a view of the cube root function along

with the linear approximation over the interval

from (1,15).

The graph of both the cube root function (red)

and the linear approximation (green) show that if

close enough {this time the interval is (7,9)}, a

line can be an excellent approximation tool.

It should be noted that this approximation technique is only good "near" to the value for x which you use to calculate the slope of the

tangent line (calculate the derivative).

The basic idea here is that a straight line is a decent approximation to any curve for "small" enough values of x. That is, if you "zoom" in

on any graph, you will see the graph get more and more linear in appearance. (See the last line of the above chart)

Check Concepts

#1: Linear approximations can be especially useful in ___________.

#2: True or False: When you calculate a value of a function using alinear approximation, the proximity of point of the function that youuse and the point you are trying to estimate is extremely important.

#3: Another name for a linear approximation is linearization. Yetanother is _________________.

#4: In this expression, what type of notation is used?:

#5 True or False: Any curve, if examined with a small enough intervaltends toward a linear function.

Calculus I Notes, Section 2-9 http://www.blc.edu/fac/rbuelow/calc/nt2-9.html

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