.125truein ^^^^''0 0^''
SOLUTION 1: Differentiate . Apply the product rule.
Then
(Factor an x from each term.)
.
SOLUTION 2: Differentiate . Apply the quotient rule. Then
.
SOLUTION 3: Differentiate arc
arc
.
Apply the product rule. Then
arc
arc
arc
arc
arc
arc
= ( arcarc
.
SOLUTION 4: Let arc
. Solve f'(x) = 0 for x . Begin by differentiating f . Then
(Get a common denominator and subtract fractions.)
.
(It is a fact that if , then A = 0 .) Thus,
2(x - 2)(x+2) = 0 .
(It is a fact that if AB = 0 , then A = 0 or B=0 .) It follows that
x-2 = 0 or x+2 = 0 ,
that is, the only solutions to f'(x) = 0 are
x = 2 or x = -2 .
SOLUTION 5: Let . Show that f'(x) = 0 . Conclude that
. Begin by differentiating f . Then
.
If f'(x) = 0 for all admissable values of x , then f must be a constant function, i.e.,
for all admissable values of x ,
i.e.,
for all admissable values of x .
In particular, if x = 0 , then
i.e.,
.
Thus, and
for all admissable values of x .
SOLUTION 6: Evaluate . It may not be obvious, but this problem can be viewed as a derivative problem. Recall that
(Since h approaches 0 from either side of 0, h can be either a positve or a negative number. In addition,
is equivalent to
. This explains the following
equivalent variations in the limit definition of the derivative.)
.
If , then
, and letting
, it follows that
.
The following problems require use of the chain rule.
SOLUTION 7: Differentiate . Use the product rule first. Then
(Apply the chain rule in the first summand.)
(Factor out . Then get a common denominator and add.)
.
SOLUTION 8: Differentiate . Apply the chain rule twice. Then
(Recall that .)
.
SOLUTION 9: Differentiate . Apply the chain rule twice. Then
(Recall that .)
.
SOLUTION 10: Determine the equation of the line tangent to the graph of
at x = e . If x = e , then
, so that the line passes through the point
. The slope of the tangent line follows from the derivative (Apply the chain rule.)
.
The slope of the line tangent to the graph at x = e is
.
Thus, an equation of the tangent line is
.
SOLUTION 11: Differentiate 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.
SOLUTION 12: Differentiate . Begin by applying the product rule to the first summand and the chain rule to the second summand. Then
.
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
.
SOLUTION 14: Evaluate . Since
and
, it follows that
takes the indeterminate form . 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 .)
=
=
= .
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 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 .