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chapter2.page17

chapter2.page17 - ∞ The following table organizes our...

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5. PHASE AND VECTOR FIELD DIAGRAMS 17 Step one : Set f ( x ) = 0 solve the equation. The solution is called the critical number of the equation x 0 = f ( x ) . Step two : Plot the solution one a horizontal (or vertical)number line. The solutions will divide the line into intervals (segments), pick any number from each interval (segment) and determine the sign of the value of f ( x ) at the picked value. Step three : Using the sign in Step two to draw arrow on each segment, a right (up) arrow for positive sign and a left (down) arrow for negative sign. Example 5.1 . Draw phase diagram for x 0 = ( x - 1)( x + 2)( x - 3) . Solution Step one : Here f ( x ) = ( x - 1)( x + 2)( x - 3) so f ( x ) = 0 gives ( x - 1)( x + 2)( x - 3) = 0 , and it has three solutions, x = - 2 , 1 , 3 Step two: Plot the three solution on the number line, Figure 9. Number Line we see that we have four intervals: (- , -2), (-2, 1), (1, 3), and (3,
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Unformatted text preview: ∞ ). The following table organizes our computation, Interval (-∞ , -2) (-2, 1) (1, 3) (3, ∞ ) x-value picked-3 2 4 Value of f(x)= (x-1)(x+2)(x-3) f(-3)=(-3-1) (-3+2)(-3-3) =-24 f(0)=(0-1)(0+2) (0-3)=6 f(2)=(2-1)(2+2) (2-3)=-4 f(4)=(4-1)(4+2) (4-3)=24 Sign of f ( x )-+-+ Step three: Draw the arrows on the number line. A right arrow for positive sign and a left arrow for negative sign. a Figure 10. Phase Diagram If c is a critical number of x = f ( x ), i.e. f ( c ) = 0 , • c is a sink if arrows on both side of it point toward it. • c is a source if arrows on both side of it point away from it. • c is a saddle if one arrow points away from and another points toward it....
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