let f be a function that is differentiable throughout its domain and that has the following properties.?
Posted on Mar 24, 2008 under domain news and reports | 210 Commentslet f be a function that is differentiable throughout its domain and that has the following properties.
i.f(x+y)=(f(x)+f(y))/(1-f(x)f(y)), for all real numbers x,y, and x+y in the domain of f.
ii. lim as h–>inf. of f(h)=0
iii. lim as h–>inf. of f(h)/h=1
a. show that f(0)=0
b. use the definition of the derivatvie to show that f'(x)=1+(f(x))^2.
c. find f(x) by solving the differential equation in part (b).
March 24th, 2008 at 7:24 am
I'm pretty sure those limits are supposed to be as h→0, not as h→∞. I say this, because solving the differential equation yields a function for which [h→∞]lim f(h) ≠ 0, and also if [h→∞]lim f(h)/h = 1 as (iii) states, then [h→∞]lim f(h) = [h→∞]lim h*f(h)/h = [h→∞]lim h * [h→∞]lim f(h)/h = ∞*1 = ∞ ≠ 0.
In other words, I think the reason you're having trouble is because you copied the problem wrong. So henceforth I will proceed under the assumption that (ii) reads [h→0]lim f(h) = 0 and (iii) reads [h→0]lim f(h)/h = 1
a) This is easy — we are given that f is differentiable, thus continuous, so f(0) = [h→0]lim f(h) = 0.
b) Taking the derivative:
f'(x) = [h→0]lim (f(x+h) – f(x))/h
Using (i):
= [h→0]lim ((f(x) + f(h))/(1 – f(x)f(h)) – f(x))/h
Extracting a factor of 1/(1-f(x)f(h)) and writing the limit of the product as the product of limits:
= [h→0]lim 1/(1-f(x)f(h)) * (f(x) + f(h) – (1-f(x)f(h))f(x))/h
= ([h→0]lim 1/(1-f(x)f(h))) * ([h→0]lim (f(x) + f(h) – (1-f(x)f(h))f(x))/h)
Evaluating the left-hand limit, using the known value for [h→0]lim f(h):
= 1/(1-f(x)*0) * ([h→0]lim (f(x) + f(h) – (1-f(x)f(h))f(x))/h)
= 1/1 * [h→0]lim (f(x) + f(h) – (1-f(x)f(h))f(x))/h
= [h→0]lim (f(x) + f(h) – (1-f(x)f(h))f(x))/h
Expanding the numerator and simplifying:
= [h→0]lim (f(x) + f(h) – f(x) + f(x)²f(h))/h
= [h→0]lim (f(h) + f(x)²f(h))/h
Extracting the factor of 1+f(x)², which is constant w.r.t. h:
= [h→0]lim (1+f(x)²) * f(h)/h
= (1+f(x)²) * [h→0]lim f(h)/h
Invoking (iii):
= 1+f(x)²
So f'(x) = 1 + f(x)²
c) Solving the differential equation:
f'(x) = 1+f(x)²
f'(x)/(1+f(x)²) = 1
∫f'(x)/(1+f(x)²) dx = ∫1 dx
arctan (f(x)) = x + C
Substituting x=0 to solve for the constant:
arctan (f(0)) = C
C = arctan 0 = 0
So arctan (f(x)) = x, which means f(x) = tan x. And we are done.
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