Inverse Trig. Functions - PowerPoint PPT Presentation

1 / 55
About This Presentation
Title:

Inverse Trig. Functions

Description:

tan-1 = arctan, has domain and range . INVERSE TANGENT FUNCTIONS. We know that: So, the lines ... the lines y = p/2. and y = -p/2. are horizontal. asymptotes ... – PowerPoint PPT presentation

Number of Views:3332
Avg rating:3.0/5.0
Slides: 56
Provided by: Tra869
Category:
Tags: functions | inverse | lines | tan | trig

less

Transcript and Presenter's Notes

Title: Inverse Trig. Functions


1
Inverse Trig. Functions Differentiation
  • Section 5.8

2
INVERSE TRIGONOMETRIC FUNCTIONS
  • Here, you can see that the sine function y sin
    x is not one-to-one.
  • Use the Horizontal Line Test.

3
INVERSE TRIGONOMETRIC FUNCTIONS
  • However, here, you can see that the function
    f(x) sin x, , is
    one-to-one.

4
INVERSE SINE FUNCTIONS
Equation 1
  • As the definition of an inverse function states
  • that
  • we have
  • Thus, if -1 x 1, sin-1x is the number between
    and whose sine is x.

5
INVERSE SINE FUNCTIONS
Example 1
  • Evaluate
  • a.
  • b.

6
Solve.
7
Solve.
8
Example 1 a
INVERSE SINE FUNCTIONS
  • We have
  • This is because , and
    lies between and .

9
Example 1 b
INVERSE SINE FUNCTIONS
  • Let , so .
  • Then, we can draw a right triangle with angle ?.
  • So, we deduce from the Pythagorean Theorem that
    the third side has length .

10
INVERSE SINE FUNCTIONS
Example 1b
  • This enables us to read from the triangle that

11
INVERSE SINE FUNCTIONS
Equations 2
  • In this case, the cancellation equations
  • for inverse functions become

12
INVERSE SINE FUNCTIONS
  • The graph is obtained from that of
  • the restricted sine function by reflection
  • about the line y x.

13
INVERSE SINE FUNCTIONS
  • We know that
  • The sine function f is continuous, so the
    inverse sine function is also continuous.
  • The sine function is differentiable, so the
    inverse sine function is also differentiable
    (from Section 3.4).

14
INVERSE SINE FUNCTIONS
  • since we know that is sin-1 differentiable, we
    can just as easily calculate it by implicit
    differentiation as follows.

15
INVERSE SINE FUNCTIONS
  • Let y sin-1x.
  • Then, sin y x and p/2 y p/2.
  • Differentiating sin y x implicitly with respect
    to x,we obtain

16
INVERSE SINE FUNCTIONS
Formula 3
  • Now, cos y 0 since p/2 y p/2, so

17
INVERSE SINE FUNCTIONS
Example 2
  • If f(x) sin-1(x2 1), find
  • the domain of f.
  • f (x).

18
INVERSE SINE FUNCTIONS
Example 2 a
  • Since the domain of the inverse sine function is
    -1, 1, the domain of f is

19
Example 2 b
INVERSE SINE FUNCTIONS
  • Combining Formula 3 with the Chain Rule, we have

20
INVERSE COSINE FUNCTIONS
Equation 4
  • The inverse cosine function is handled similarly.
  • The restricted cosine function f(x) cos x, 0
    x p, is one-to-one.
  • So, it has an inverse function denoted by cos-1
    or arccos.

21
INVERSE COSINE FUNCTIONS
Equation 5
  • The cancellation equations are

22
INVERSE COSINE FUNCTIONS
  • The inverse cosine function,cos-1, has domain
    -1, 1 and range , and is a
    continuous function.

23
INVERSE COSINE FUNCTIONS
Formula 6
  • Its derivative is given by
  • The formula can be proved by the same method as
    for Formula 3.

24
INVERSE TANGENT FUNCTIONS
  • The inverse tangent function, tan-1 arctan,
    has domain and range .

25
INVERSE TANGENT FUNCTIONS
  • We know that
  • So, the lines are vertical asymptotes of the
    graph of tan.

26
INVERSE TANGENT FUNCTIONS
  • The graph of tan-1 is obtained by reflecting the
    graph of the restricted tangent function about
    the line y x.
  • It follows that the lines y p/2 and y -p/2
    are horizontal asymptotes of the graph of
    tan-1.

27
Inverse Trig. Functions
  • None of the 6 basic trig. functions has an
    inverse unless you restrict their domains.

28
  • Function Domain Range
  • y arcsin x -1
  • y arccos x -1
  • y arctan x
  • y arccot x
  • y arcsec x I II
  • y arccsc I IV

29
The Inverse Trigonometric Functions
Graphs of six inverse trigonometric functions
30
The Inverse Trigonometric Functions
Graphs of six inverse trigonometric functions
31
Inverse Properties
  • f (f 1(x)) x and f 1(f (x)) x
  • Remember that the trig. functions have inverses
    only in restricted domains.

32
Table 11
DERIVATIVES

33
Derivatives of Inverse Trig. Functions
  • Let u be a differentiable function of x.

34

35
DERIVATIVES
  • Each of these formulas can be combined with the
    Chain Rule.
  • For instance, if u is a differentiable function
    of x, then

36
DERIVATIVES
Example 5
  • Differentiate

37
DERIVATIVES
Example 5 a

38
DERIVATIVES
Example 5 b

39
Find each derivative with respect to x.
40
Find each derivative with respect to x.
41
Find each derivative with respect to the given
variable.
42
Find each derivative with respect to the given
variable.
43
Example 1
44
Example 2
45
Example 2
46
Example 2
47
Example 2
48
Example 2
49
Example 2
50
Example 2
51
Some homework examples
Write the expression in algebraic form
Let then
Solution Use the right triangle
Now using the triangle we can find the hyp.
3x
y
1
52
Some homework examples
Find the derivative of
Let u
53
Example
54
Example
55
Find an equation for the line tangent to the
graph of at x -1
Slope of tangent line
When x -1, y
At x -1
Write a Comment
User Comments (0)
About PowerShow.com