View
228
Download
0
Category
Preview:
Citation preview
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
1/13
SOLUTIONS TO DIFFERENTIATION OF INVERSE TRIGONOMETRIC
FUNCTIONS
SOLUTION 1:Differentiate . Apply the product rule. Then
(Factor an x from each term.)
.
SOLUTION 2 : Differentiate . Apply the quotient rule. Then
.
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
2/13
SOLUTION 3 : Differentiate arc arc . Apply the product rule. Then
arc arc arc arc
arc arc
= ( arc arc .
SOLUTION 4 : Let arc . Solve f'(x) = 0 forx . Begin by
differentiating f. Then
(Get a common denominator and subtract fractions.)
.
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
3/13
(It is a fact that if , then A = 0 .) Thus,
2(x - 2)(x+2) = 0 .
(It is a fact that ifAB = 0 , then A = 0 orB=0 .) It follows that
x-2 = 0 orx+2 = 0 ,
that is, the only solutions to f'(x) = 0 are
x = 2 orx = -2 .
SOLUTION 5 : Let . Show that f'(x) = 0 . Conclude
that . Begin by differentiating f. Then
.
Iff'(x) = 0 for all admissable values ofx , then fmust be a constant function, i.e.,
for all admissable values ofx ,
i.e.,
for all admissable values ofx .
In particular, ifx = 0 , then
i.e.,
.
Thus, and for all admissable values ofx .
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
4/13
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
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
5/13
.
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.)
.
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
6/13
SOLUTION 8 : Differentiate . Apply the chain rule twice. Then
(Recall that .)
.
SOLUTION 9 : Differentiate . Apply the chain rule twice. Then
(Recall that .)
.
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
7/13
SOLUTION 10 : Determine the equation of the line tangent to the graph
of at x = e . Ifx = 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
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
8/13
= 0 .
Ify' = 0 for all admissable values ofx , then y must be a constant function, i.e.,
for all admissable values ofx ,
i.e.,
arc for all admissable values of x .
In particular, ifx = 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
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
9/13
.
SOLUTION 13 : Find an equation of the line tangent to the graph
of at x=2 . Ifx = 2 ,
then , so that the line passes through thepoint . 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 .)
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
10/13
.
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 `` 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 .)
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
11/13
=
=
= .
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 introducingvariables x and . (See the diagram below.)
From trigonometry it follows that
,
so that
.
In addition,
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
12/13
so that
.
It follows that
,
that is, angle is explicitly represented as a function of distance x . Now find the
value ofx which maximizes the value of function . Begin by differentiating
function and setting the derivative equal to zero. Then
.
.
Now solve this equation forx . Then
if
if
if
8/7/2019 Solutions to Differentiation of Inverse Trigonometric Functions
13/13
if
if
feet .
(Use the first or second derivative test (The first derivative test is easier.) to verify
that this value ofx determines a maximum value for .)
Thus, the ``best'' view is found x=15 feet from the front of the room.
Recommended