Let g be a function that is differentiable throughout an
Chapter , Problem 6AAE(choose chapter or problem)
Let \(g\) be a function that is differentiable throughout an open interval containing the origin. Suppose \(g\) has the following properties:
i. \(g(x+y)=\frac{g(x)+g(y)}{1-g(x) g(v)}\) for all real numbers \(x, y\), and \(x + y\) in the domain \(g\).
ii. \(\lim _{h \rightarrow 0} g(h)=0\)
iii. \(\lim \limits_{h \rightarrow 0} \frac{g(h)}{h}=1\)
a. Show that \(g(0)=0\) .
b. Show that \(g^{\prime}(\mathrm{x})=1+[g(\mathrm{x})]^{2}\).
c. Find \(g(\mathrm{x})\) by solving the differential equation in part (b).
Equation Transcription:
Text Transcription:
g(x+y) = g(x) + g(y) / 1 - g(x)g(y)
x, y and x+y
g
lim_h right arrow 0 g(h) = 0
lim_h right arrow 0 g(h) /h = 1
g(0) = 0
g^prime (x) = 1 + [g (x)]^2
g (x)
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