. Show your work.1. 136.151: Test #1
20 minutes
Name: _____________________ Student Number: _______
1. Find an equation of the line passing through a point (2,3) and perpendicular to the line . Show your work.
Solution:
The line through (2,3) and with slope m has an equation 
.
So, we need to find m. But we know that the required line is perpendicular
to the line ; consequently their
slopes are related with the formula 
where
is the slope of the line Now
note that the later equation is equivalent to 
;
so the slope of that line is 
.
Thereby the slope m that we need is 
.
So the required equation is 
.
2. Find the center and the radius of the circle 
.
Solution.
is equivalent to (means the same as ) 
(to check that simply expand the square in the last equation). The last equation is obviously equivalent to 
. We now see that the center of the circle is at (-3,0) and that its radius is 2.
3. Find the domain of the function 
.
Solution.
Since the denominator should not be equal to 0, we must have 
and 
. Since the usual 
is defined only for number that are 
, we must have 
, i.e., 
. We can now summarize: the domain of the function 
consistes of all numbers x (strictly) larger than 2 except 
.
4. Show that if the function is odd and if the function 
is even then their product 
is an odd function.
Solution.
We assume here that is odd and that is even. This means that 
and that 
for every x. We want to show that the function 
is odd, i.e. we want to show that 
. Let's see:
,
where in the second step we have used that is odd and that is even.