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Ches Beatez Name MATH 152 Solving Equations nates the solution to an equation. Consider t o J()0is the -value where the graph crosses the z-axis. Starting with a first guess o e approximate the graph with a straight line, using the derivative at ro to the line. The value at which the line crosses the z-axis is our next approximation x1, See the his exercise, we nre going to produce an ordered list of numbers aro 띠,xa that approxi- he equation /(z) = 0 for some function f. On a graph of the function, the solution form the slope of picture: T T2 TO 1. The tangent line at the point (o, f (o)) goes through that point and has slope equal to f (ro). Make sure you can find an equation of this line Answer: y = f(xo) + f, (xo)(x-m) 2. Find the point z1 at w Answer: x1 = 20-fro hich this line crosses the x-axis Now find the tangent line to the graph at the point (xi, f(i)) and get a formula for 2, where this line crosses the r-axis 3.

4. Find a formula for r+ n terms of rn, f (r), and f(r). This is what we call a recursive formula that defines each number in our list using the previous nuber 5. Use this method to find a solution to 2 0 correct to three decimal places. 6. Show that when applying this method to equations of the form r2- B 0, the recursive formula simplifies to the Babylonian algorithm 7. Use the simplified formula above to compute the square root of 23. 8. Find the list of numbers starting with xo = 0 to the equation e, 0, computing by hand. Make sure your list makes sense by drawing the graph of the function f(x)e and drawing tangent lines. 9. Explain how to find the solution to cos(x) r correct to two decimal places

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