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Home | 18.013A | Chapter 2 | Section 2.1 |
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State and prove these fundamental properties ie, expressions for and for . (Hint: what value do they have at ? What are their derivatives? Deduce their series from these statements and identify them.)
Solution:
has value at argument 0 and is its own derivative by the chain rule. By the logic used to get the series for we obtain: , and the relation between the 's is exactly as before. We can deduce that each is multiplied by its value in the previous case, which gives us
Notice that has value 1 when , and has derivative . Also notice that, for values of for which it is defined, has the same value at and the same derivative (use the power rule and the chain rule). (We conclude that these functions are the same thing: their difference is 0 and has 0 derivative everywhere, which means it never changes from 0)
So we have
for all values of for which we have a definition for the latter expression. This allows us to define the second expression to be the first everywhere else.
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