Home  18.013A  Chapter 2  Section 2.1 


State and prove these fundamental properties ie, expressions for $\mathrm{exp}(x+r)$ and for $\mathrm{exp}rx$ . (Hint: what value do they have at $x=0$ ? What are their derivatives? Deduce their series from these statements and identify them.)
Solution:
$\mathrm{exp}(x+r)$ has value $\mathrm{exp}r$ at argument 0 and is its own derivative by the chain rule. By the logic used to get the series for $\mathrm{exp}x$ we obtain: ${a}_{0}=\mathrm{exp}r$ , and the relation between the $a$ 's is exactly as before. We can deduce that each $a$ is $\mathrm{exp}r$ multiplied by its value in the previous case, which gives us
Notice that $\mathrm{exp}rx$ has value 1 when $x=0$ , and has derivative $r\mathrm{exp}rx$ . Also notice that, for values of $r$ for which it is defined, ${(\mathrm{exp}x)}^{r}$ has the same value at $x=0$ 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 $r$ for which we have a definition for the latter expression. This allows us to define the second expression to be the first everywhere else.
