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Due on Apr. 30, 2008
Asked on Apr 28, 2008 at 10:58:40PM

Q:
Determine if each function f is differentiable at the points. If it is, find its derivative at that point and if not explain why

a)f(x)=3x+1 if x<0
=x^2+3x+1 if x>0
at x=0

b)f(x)=x^2+1 if x<1
=2x if x >= 1
at x=1

c)f(x)=x^2+2 if x <= 1
=3x if x>1
at x=1

d)f(x)=x^2 if x is rational
=0 if x is irrational
at x=0

e)f(x)=x if x is rational
=0 if x is irrational
at x=0

f)f(x)=sin 1/x if x does not = 0
=0 if x=0
at x=0

g)f(x)=sin x at x=0

h)f(x)=cos x at x=0

i)f(x)=x^2+x if x=1/n, n belonging in N
=0 if x=0
at x=0,1

j)f(x)=x^2 sin 1/x if x is rational
=0 if x is irrational
at x=0

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Posted on Apr 30, 2008 at 4:24:38PM

A:
Preview: ... in part (d), we have that the derivative at x=0 is = lim ( f(h) / h) h->0 The expression f(h)/h = h / h = 1 if h is rational, and = 0 if h is irrational. f(h)/h does not converge (rather, it oscillates between 0 and 1) when h->0. Hence f is NOT differentiable. f) For any h>0 (no matter how small it is), there exists 0 < k < h such that f(k) = sin 1/k = 1 (namely, pick k = 2/((4n+1)*pi) for large enough integer n. then sin 1/k = sin (4n+1)pi/2 = sin (2n*pi+ pi/2) = sin pi/2 = 1). f is not continuou ...

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Ledswimmer08
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Posted on Apr 30, 2008 at 4:28:56PM

A:
Preview: ... ous at x=0. b)Same deal as before: take derivatives of both parts. Plugging in x=1 you get 2 for both therefore it is continuous at x=1 and the derivatives value is 2. c) ...

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