4.8 symmetry, ivt and number line sign studies for composite trig functions
TRANSCRIPT
4.8 Symmetry, IVT and Number line sign studies for composite trig functions
Recall the definitions of even/odd functions: If f is an even function, then it’s graph is
symmetric with respect to the y-axis and f(-x)=f(x).
If f is an odd function, then it’s graph is symmetric with respect to the origin and f(-x)= -f(x).
f x = cos x
f x = sin x
Evaluate f(-x) and determine if each function is even, odd or neither.
22
2
2sin 31. ( ) cos 3. ( )
2
2. ( ) sin 4. ( ) sin
xf x x x f x
x
f x x x f x x x
Recall: The Intermediate Value Theorem (IVT) p.206 in Pre-Calc Text
0
0
If a and b are real numbers with and if is continuous on the interval , ,
then takes on every value between ( ) and ( ). In other words, if is between
( ) and ( ), then = ( ) for
a b f a b
f f a f b y
f a f b y f c
some number in , .
In particular, if ( ) and ( ) have opposite signs (i.e., one is negative and the
other is positive), then ( ) 0 for some number in , .
Note: The Intermediate Value Theorem i
c a b
f a f b
f c c a b
s an
existence theorem. It indicates whether at least
one c exists, but does not give a method for
finding c.
Making Sense of the IVTThink of the Intermediate Value Theorem as “crossing a river.” In the picture below, if you are walking on a continuous path from f(a) to f(b), and there is a river across your path at the horizontal line y=y0 , then you would have to cross the river to reach your destination.
River
Use the Intermediate value Theorem to determine if a zero must exist on the interval:
2
21. ( ) sin(2 ) on ,
6 3
22. ( ) 2cos ( )sin on ,
3 3
f x x
f x x x
Note: the fact that the IVT does not guarantee a zero does not mean that one does not exist in the interval. For instance, check f(π/2) in number 2.
Example 1: Answer the following questions about on [0, 2π].
What are the zeros of f ?
Describe the symmetry of f.
Do a number line sign study for f and use interval notation to identify where f > 0.
2( ) 2sin 1f x x
Example 3: Answer the following questions about on [0, 4π]. What are the zeros of f ?
Do a number line sign study for f .
Identify the intervals for which f < 0.
2 1 12 2( ) 2sin sin( )f x x x
Assignment
A4.8, Sections I, II and III to be completed by Monday
Test #11 will be at the end of this week and includes Polar Equations and Complex Numbers.
See you Tmrrw!!