On-line Math 21
On-line Math 21
4.5 Curve Sketching
Example 8
Solution
Asymptotes
As x® 0 , this function gets a little complicated. For x > 0 ,
yet x near 0, this one-sided limit
is an <a href="hospital.html»indeterminate form</a>, of type 0·¥.
In order to convert this to something that l'Hôpital's rule can handle,
it needs to be switched to either ¥/¥ or 0/0 . Since
I don't want to invert the exponential term, we'll convert to ¥/¥:
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lim
x® 0+
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e1/x(-1/x2) -1/x2
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so from the right side, the line x = 0 is a vertical asymptote. However,
if you approach from the other side,
is not an indeterminate form at all, since
because 1/x is headed to negative infinity, and so
as well. So, the line x = 0 is a vertical asymptote on one side only,
and the graph approaches the origin from the other side (although (0,0)
is of course not on the graph).
As x® ±¥, then e1/x® e0 = 1 ,
and so xe1/x approaches ±¥ as x does. But,
for x large, since that exponential term is nearly 1, the graph ought
to look like a line, and in fact
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lim
x® ¥
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xe1/x-(x+1) = |
lim
x® ¥
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x( e1/x-(1+1/x)) , |
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which is an indeterminate form of type ¥·0 , so converting
to 0/0 ,
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lim
x® ¥
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( e1/x-(1+1/x)) 1/x
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lim
x® ¥
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e1/x(-1/x2)-(-1/x2) -1/x2
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, using l¢Hopital¢s rule |
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so that the graph has a slant-asymptote y = x+1 .
Intercepts
Since 0 is not in the domain, there can be no y -intercept, even
though there is a one-sided limit towards a point on the y -axis. For
x ¹ 0 , though, there is no solution to f(x) = 0 , so that there
are no x -intercepts, either.
Increasing/decreasing, and critical points
so there is a critical point at x = 1 only, with critical value f(1) = e ,
and the curve is increasing ( f¢(x) > 0 ) for x > 1 , and for x < 0 .
The curve is decreasing, f¢(x) < 0 , only when 0 < x < 1 .
Concavity, and inflection points
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æ ç
è
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e1/x |
æ ç
è
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1- |
1 x
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ö ÷
ø
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ö ÷
ø
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¢ |
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e1/x(-1/x2) |
æ ç
è
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1- |
1 x
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ö ÷
ø
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+e1/x(+1/x2) |
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e1/x x2
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æ ç
è
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æ ç
è
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1 x
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-1 |
ö ÷
ø
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+1 |
ö ÷
ø
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so there are no inflection points, the graph is concave up for x > 0 and
concave down for x < 0 .
Draw the graph
Use this information to draw a fair representation of the graph.
- The first information you plot are the asymptotes and ``tails'' [Link
to ex8-1.gif]
- Then find intercepts and endpoints. But there are none for this graph.
- Then plot the critical points (and values). I always mark it with a horizontal
line to remind myself that it is a critical point. [Link to ex8-2.gif]
- Plot the inflection points. None of these, at least
- Then fill in the graph, connecting the dots and making sure that the graph you
draw is increasing/decreasing where the number line indicates it should be,
and concave-up or concave-down where the sign of the second derivative indicates.
[Link to ex8-3.gif]
[Each of these items should trigger the appearance of a new drawing with that
information added.]
Copyright (c) 2000 by David L. Johnson.
File translated from
TEX
by
TTH,
version 2.61.
On 21 Dec 2000, 00:46.