= 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'Hpital'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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Tue Sep 16 16:10:59 PDT 1997