121616949-math.141 - it crosses the x-axis in other words...

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6.3 Newton’s Method 127 way that the center of the scissors at A is fixed, and the paper is also fixed. As the blades are closed (i.e., the angle θ in the diagram is decreased), the distance x between A and C increases, cutting the paper. a. Express x in terms of a , θ , and β . b. Express dx/dt in terms of a , θ , β , and dθ/dt . c. Suppose that the distance a is 20 cm, and the angle β is 5 . Further suppose that θ is decreasing at 50 deg/sec. At the instant when θ = 30 , find the rate (in cm/sec) at which the paper is being cut. ....................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .... .. . . . . . . . . . . . . . . . . . . . . . . . . .... .... .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... .. ...... .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... .. ...... .. .. .. .. .. .. .. .. .. .. .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .............. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .............. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. B A C θ Figure 6.16 Scissors. 6.3 Newton's Method Suppose you have a function f ( x ), and you want to find as accurately as possible where
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Unformatted text preview: it crosses the x-axis; in other words, you want to solve f ( x ) = 0. Suppose you know of no way to find an exact solution by any algebraic procedure, but you are able to use an approximation, provided it can be made quite close to the true value. Newton’s method is a way to find a solution to the equation to as many decimal places as you want. It is what is called an “iterative procedure,” meaning that it can be repeated again and again to get an answer of greater and greater accuracy. Iterative procedures like Newton’s method are well suited to programming for a computer. Newton’s method uses the fact that the tangent line to a curve is a good approximation to the curve near the point of tangency. EXAMPLE 6.19 Approximate √ 3. Since √ 3 is a solution to x 2 = 3 or x 2-3 = 0, we use f ( x ) = x 2-3. We start by guessing something reasonably close to the true value; this...
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