arc
. What conclusion can be drawn from your answer about function y ? What conclusion can be drawn about functions arc
and 
? First, differentiate, applying the chain rule to the inverse cotangent function. Then
= 0 .
If y' = 0 for all admissable values of x , then y must be a constant function, i.e.,
for all admissable values of x ,
i.e.,
arc
for all admissable values of x .
In particular, if x = 1 , then
arc
i.e.,
.
Thus, c = 0 and arc
for all admissable values of x . We conclude that
arc
.
Note that this final conclusion follows even more simply and directly from the definitions of these two inverse trigonometric functions.
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SOLUTION 12 : Differentiate 
. Begin by applying the product rule to the first summand and the chain rule to the second summand. Then
.
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SOLUTION 13 : Find an equation of the line tangent to the graph of
at x=2 . If x = 2 , then 
, so that the line passes through the point 
. The slope of the tangent line follows from the derivative
(Recall that when dividing by a fraction, one must invert and multiply by the reciprocal. That is 
.)
.
The slope of the line tangent to the graph at x = 2 is
.
Thus, an equation of the tangent line is
or
or
.
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SOLUTION 14 : Evaluate 
. Since 
and 
, it follows that
takes the indeterminate form `` zero over zero.'' Thus, we can apply
L'H
pital's Rule. Begin by differentiating the numerator and denominator separately. DO NOT apply the quotient rule ! Then
= 
= 
(Recall that when dividing by a fraction, one must invert and multiply by the reciprocal. That is 
.)
= 
= 
= 
.
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SOLUTION 15 : A movie screen on the front wall in your classroom is 16 feet high and positioned 9 feet above your eye-level. How far away from the front of the room should you sit in order to have the ``best" view ? Begin by introducing variables x and 
. (See the diagram below.)
From trigonometry it follows that
,
so that
.
In addition,
so that
.
It follows that
,
that is, angle 
is explicitly represented as a function of distance x . Now find the value of x which maximizes the value of function . Begin by differentiating function and setting the derivative equal to zero. Then
.
.
Now solve this equation for x . Then
iff
iff
iff
iff
iff
feet .
(Use the first or second derivative test (The first derivative test is easier.) to verify that this value of x
determines a maximum value for .)
Thus, the ``best'' view is found x=15 feet from the front of the room.
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