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MAT 125A Homework 6
M.Fukuda
Please submit your answers at the discussion session on November 29th Thursday.
You can use theorems in the lecuture unless otherwise stated. When you do so write those
statements clearly instead of quoting them by numbers.
1. Use the de±nition of derivative to calculate the derivatives of the following functions at
the indicated points.
i)
f
(
x
) =
x
2
cos
x
at
x
= 0.
ii)
g
(
x
) =
3
x
+4
2
x
−
1
at
x
= 1.
2. Let
f
be
f
(
x
) =
x
2
x
∈
Q
0
x
∈
R
\
Q
.
i) Show that
f
is continuous at
x
= 0.
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Unformatted text preview: ii) Show that f is discontinuous at x n = 0. iii) Show that f is di²erentiable at x = 0. 3. Prove (iii) of Corollary 29.7. 4. Let f : R → R and di²erentiable on R . Suppose sup x ∈ R { f ′ ( x ) } = M < 1. Take s ∈ R and de±ne a sequence s n +1 = f ( s n ) n = 0 , 1 , 2 , . . .. Then, show that { s n } is a Cauchy sequence. Hint: Show ±rst that  s n +1 − s n  ≤ M  s n − s n − 1  . 1...
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This homework help was uploaded on 04/08/2008 for the course MATH 125A taught by Professor Fukuda during the Fall '07 term at UC Davis.
 Fall '07
 Fukuda

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