A normal line is a line that is perpendicular to another line, curve or surface. Give an everyday example of a normal line to ea
ch of the following: another line, a circle, a plane.
2 answers:
Answer:
a) A cross b) A stool c) One foot table
Step-by-step explanation:
Hi there!
Check the pictures below for better visualizing.
a) Cross
The figure of a cross shows clearly one normal line perpendicular to another perpendicular one. They cross each other.
b) Circle
A stool has a normal line perpendicular to the ground.
c) One foot table
In this Table, we have a normal line perpendicular to the ground and its top.
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Answer:
Step-by-step explanation:
subtract 1 from both sides
-6x<-17-1
simplify
-6x<-18
divide by 6 on both sides
x>3
The half-range sine series is the expansion for 
with the assumption that is considered to be an odd function over its full range, 
. So for (a), you're essentially finding the full range expansion of the function
with period 2 so that
for 
and integers 
.
Now, since is odd, there is no cosine series (you find the cosine series coefficients would vanish), leaving you with
where
In this case,
, so
The half-range sine series expansion for is then
which can be further simplified by considering the even/odd cases of, but there's no need for that here.
The half-range cosine series is computed similarly, this time assuming is even/symmetric across its full range. In other words, you are finding the full range series expansion for
Now the sine series expansion vanishes, leaving you with
where
for
. Again, . You should find that
Here, splitting into even/odd cases actually reduces this further. Notice that when is even, the expression above simplifies to
while for odd, you have
So the half-range cosine series expansion would be
Attached are plots of the first few terms of each series overlaid onto plots of. In the half-range sine series (right), I use 
terms, and in the half-range cosine series (left), I use 
or 
terms. (It's a bit more difficult to distinguish from the latter because the cosine series converges so much faster.)
with period 2 so that
Now, since
where
In this case,
The half-range sine series expansion for
which can be further simplified by considering the even/odd cases of
The half-range cosine series is computed similarly, this time assuming
Now the sine series expansion vanishes, leaving you with
where
for
Here, splitting into even/odd cases actually reduces this further. Notice that when
while for odd
So the half-range cosine series expansion would be
Attached are plots of the first few terms of each series overlaid onto plots of
Answer at very bottom of picture above
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