Finding the Linearization of f(x): Find the linear approximation L0(x) of the fu

Question

Finding the Linearization of f(x): Find the linear approximation L0(x) of the function f(x) = x/ROOT x+1 at x = 0. Use

this linearization to approximate f(0.03).

1a) In Maple, create a text region and enter the following sentence: Question 1. Insert an execution group (command

prompt) after the the text region.

b) In Maple, create the function f(x) = x/ ROOT x+1.

c) Using Maple, calculate f(0). ANSWER:__________________________________________

d) Using Maple, calculate f '(x) and f 0(0) . ANSWER:____________________________________________

e) What is L'(x)? (Determine this by hand.)

ANSWER:___________________________________________________________________________

f) Using your answer from part (e), enter L'(x) in Maple? (Note: Use

Explanation / Answer

Operations such as differentiation are treated differently from those of ordinary algebraic expressions.

For example, to differentiate the expression d:=x^2*y+z*sin(y*z), we can use diff as follows:

which gives us

However if we had the function

then using diff(f,z) gives us zero. In order to differentiate the function, we have to specify the arguments since Maple would treat f as a symbol in and of itself and return 0.

Thus the proper expression would be

A common problem among many users who use Maple to do differentiation of mathematical functions is in the evaluation of the derivatives of functions at points.

Say for example, we wished to evaluate the derivative of f at the points (x1,y1,z1). If we tried to define a new function in terms of the diff function, viz

we would get the result

and evaluating at the point (Pi,Pi,Pi) we would get

which is clearly wrong. This is due to the fact that in defining r as above results in delayed evaluation of the expression and substitues the values of the args before differentiation which results in Maple treating Pi as a symbolic variable with which to differentiate r with respect to.

The way to solve this would be to use theunapplyoperator, viz

and evaulating r(Pi,Pi,Pi), we obtain

which is indeed correct.

A few notes on unapply, adapted (and corrected) from the ?unapply help page:

although this depends somewhat on the evaluation of f(x). A safer statement would be:

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