What is the value of the 4th iterate of [tex]f(x)=\frac{3 x-1}{x^2}[/tex] if [tex]x_0=-2[/tex]? Round to three decimal places.
A. [tex]x_4=-1.659[/tex]
B. [tex]x_4=-2.041[/tex]
C. [tex]x_4=-1.75[/tex]
D. [tex]x_4=-2.096[/tex]
Answer (2)
Calculate x 1 = f ( x 0 ) = f ( − 2 ) = − 1.75 .
Calculate x 2 = f ( x 1 ) = f ( − 1.75 ) ≈ − 2.041 .
Calculate x 3 = f ( x 2 ) = f ( − 2.041 ) ≈ − 1.710 .
Calculate x 4 = f ( x 3 ) = f ( − 1.710 ) ≈ − 2.096 . The final answer is − 2.096 .
Explanation
Understanding the Problem We are given the function f ( x ) = f r a c 3 x − 1 x 2 and the initial value x 0 = − 2 . We need to find the 4th iterate of the function, which means we need to calculate x 1 = f ( x 0 ) , x 2 = f ( x 1 ) , x 3 = f ( x 2 ) , and x 4 = f ( x 3 ) . Finally, we need to round the value of x 4 to three decimal places.
Calculating x_1 First, we calculate x 1 = f ( x 0 ) = f ( − 2 ) .
x 1 = f ( − 2 ) = ( − 2 ) 2 3 ( − 2 ) − 1 = 4 − 6 − 1 = 4 − 7 = − 1.75
Calculating x_2 Next, we calculate x 2 = f ( x 1 ) = f ( − 1.75 ) .
x 2 = f ( − 1.75 ) = ( − 1.75 ) 2 3 ( − 1.75 ) − 1 = 3.0625 − 5.25 − 1 = 3.0625 − 6.25 = − 2.04081632653 ≈ − 2.041
Calculating x_3 Now, we calculate x 3 = f ( x 2 ) = f ( − 2.041 ) .
x 3 = f ( − 2.041 ) = ( − 2.041 ) 2 3 ( − 2.041 ) − 1 = 4.165681 − 6.123 − 1 = 4.165681 − 7.123 = − 1.709955 ≈ − 1.710
Calculating x_4 Finally, we calculate x 4 = f ( x 3 ) = f ( − 1.710 ) .
x 4 = f ( − 1.710 ) = ( − 1.710 ) 2 3 ( − 1.710 ) − 1 = 2.9241 − 5.13 − 1 = 2.9241 − 6.13 = − 2.09634 ≈ − 2.096
Final Answer Therefore, the value of the 4th iterate of f ( x ) when x 0 = − 2 , rounded to three decimal places, is − 2.096 .
Examples
In weather forecasting, complex models use iterative functions to predict future weather conditions based on current data. Each iteration refines the prediction, providing a more accurate forecast. Similarly, in financial modeling, iterative functions can be used to predict stock prices or investment returns, where each iteration incorporates new market data to adjust the predictions. These models help in making informed decisions in various real-world scenarios.
The 4th iterate of the function f ( x ) = x 2 3 x − 1 starting from x 0 = − 2 is − 2.096 . This value is obtained by calculating successive iterations of the function. Therefore, the correct answer is option D.
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