Calculus With Analytic Geometry
Hakan TOR
March 4,2010
1. The graph of
g
is given.
(a) At what numbers is
g
discontinuous? Why?
(b) At what numbers is
g
not diﬀerentiable? Why?
Solution:
(a) Although lim
x
→
2

g
(
x
) = lim
x
→
2
+
g
(
x
), the function
g
(
x
) is discontinuous at the
point

2 because lim
x
→
2
g
(
x
)
6
=
g
(

2).
Since
g
(5) = lim
x
→
5

g
(
x
)
6
= lim
x
→
5
+
g
(
x
),
g
(
x
) is discontinuous at the point 5.
Remark
Although
∞
= lim
x
→
2

g
(
x
)
6
= lim
x
→
2
+
g
(
x
) =
∞
, we can not say any thing
about continuity of
g
, because the point 0 is not in the domain of
g
.
(b) Since
g
is discontinuous at the points 2 and 5,
g
is not diﬀerentiable at that points. The
function
g
is also not diﬀerentiable at the point 2.
2. The normal line to a curve C at a point P is, by deﬁnition, the line that passes through P
and is perpendicular to the tangent line to C at P. Find an equation of the normal line to the
parabola
y
= 1

x
2
at the point (2
,

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 Spring '10
 Tor
 Calculus, Geometry, Derivative, Mathematical analysis, Continuous function

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