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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). fx = cos x If f is an odd function, then it’s graph is symmetric with respect to the origin and f(-x)= -f(x). fx = sinx Evaluate f(-x) and determine if each function is even, odd or neither. 1. f ( x) x cos x 2sin 3 x 3. f ( x) 2 x 2 2. f ( x) x sin x 4. f ( x) sin x x 2 2 Recall: The Intermediate Value Theorem (IVT) p.206 in Pre-Calc Text If a and b are real numbers with a b and if f is continuous on the interval a, b , then f takes on every value between f (a) and f (b). In other words, if y0 is between f (a ) and f (b), then y0 =f (c) for some number c in a, b . In particular, if f (a) and f (b) have opposite signs (i.e., one is negative and the other is positive), then f (c) 0 for some number c in a, b . Note: The Intermediate Value Theorem is an existence theorem. It indicates whether at least one c exists, but does not give a method for finding c. Making Sense of the IVT Think 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 1. f ( x) sin(2 x) on , 6 3 2 2. f ( x) 2 cos ( x) sin x on , 3 3 2 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 f ( x) 2sin 2 x 1 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. Example 3: Answer the following questions about f ( x) 2sin 2 12 x sin( 12 x) on [0, 4π]. What are the zeros of f ? Do a number line sign study for f . Identify the intervals for which f < 0. 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!!