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Unformatted text preview: n x 2 sin
then drag each of the expressions
different colors. x2 and x2 1
x on the interval . 2, . 2 and into your plot. Revise the plot and give the components 0.04 0.02 0.2 0 0.1 0.1
x 0.2 0.02 0.04 b. Prove that the function f is differentiable on R but that the function f is not continuous at the number 0.
For each t 0 we have
t 2 sin 1
ft f0
t
t sin 1
t
t
t0
and, since
ft f0
t sin 1
t 
t
t0
for each t 0 we can deduce from the sandwich theorem that
ft f0
0.
lim
t0
t0
Now given any x 0 we have
f x 2x sin 1 cos 1
x
x
and the latter expression does not approach a limit as x 0. Therefore f is not continuous at
0.
5. This exercise concerns the function f defined by the equation
fx x 3 sin
0 239 1
x if x 0 if x 0 . a. Ask Scientific Notebook to make a 2D plot of the expression x 3 sin
and then drag each of the expressions
different colors.x 3 sin 1
x x3 and x3 1
x on the interval . 05, . 05 into your plot. Revise the plot and give the components 0.0001
5e05 0.04 0 0.02 0.02 x 0.04 5e05
0.0001 b. Prove that the function f is continuous at the number 0 but does not have a derivative there.
The fact that f 0 0 follows in exactly the same way as Exercise 4. Now given any x 0 we
have
f x 3x 2 sin 1 x cos 1
x
x
and since
f x  3x 2 x 
for x 0 we have
lim f x 0 f 0 . To see why f has no derivative at 0 we can argue as in Exercise 4.
x0 6. Suppose that f is a function defined on an open interval a, b and that x a, b . a. Prove that if f x exists then
f x lim
h0 Solution: We assume that f f xh
h fx . x exists. Thus
f x lim ft tx t fx
.
x To show that
f xh f x
,
h
0. Choose a number 0 such that the inequality
ft fx
fx
tx
f x lim
h0 suppose that holds whenever t
a, b and t x and t x  . Then whenever h 0 and h  and
h
a x, b x we deduce from the fact that x h
a, b and  x h x  that
f xh f x
f xh f x
fx
fx .
h
xh x
b. Prove that if the limit
f xh
h
exists then f x exists and is equal to this limit.
lim
h0 c. Prove that if f x exists then 240 fx . f x lim f xh Hint: Use the fact that whenever h
f xh fx
2h h fx h 2h h0 0 and is sufficiently small we have
f xh f x
fx h fx
2h
1 f xh f x 1 f x h
2
2
h
h fx d. Prove that if f x exists then
f x lim lim
ux ft tx t fu
u Since f x exists, the function f must be continuous at x. Therefore the right side is
ft fu
ft fx
lim
f x.
lim lim
tu
tx
tx ux
tx Exercises on the Mean Value Theorem
1. a. Given that f is a function defined on an interval S and that f x 0 for every x S, prove that f must be
constant on S.
Suppose that a and b are numbers in the interval S and a b. We shall show that f a f b .
Applying the mean value theorem to the function f on the interval a, b we choose a number c
between a and b such that
fb fa
fc
.
ba
Since f c 0 we deduce that f b f a 0 which gives us f a f b .
b. Given that f is a function defined on an interval S and that f x 0 for every x
strictly increasing on S. S, prove that f must...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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