Suppose we want to solve the following equation: x-2x-5=0 1. What does it mean to solve...

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Suppose we want to solve the following equation: x-2x-5=0 1. What does it mean to "solve an equation"? Because r-2x-5 does not factor in an obvious way, this is actually a difficult task! We can use Newton's Method to help us to at least approximate solutions to the equation. 2. Given is the graph of y=x-2r-5. Sketch a line tangent to this curve at the point where x = 2. NORMAL FLOAT AUTO REAL RADIAN MP 1 3. How do the x-intercept of the tangent line, and the z-intercept of y = x- 2x - 5 compare? 4. Find the equation of your tangent line. (Hint: do not use your graph to do this, be- cause if you are not a robot, you may not have drawn your line perfectly.) 5. Find the x-intercept of your tangent line. We will call this value #1. Notice that x, is an approximation to the solution to z-2x-5=0. 6. Given below is the graph of y=x-2x-5, on the window [1.8, 2.2] x[-2,1]. Sketch a line tangent to the curve at r = 2, and a line tangent to the curve at 2 = 11- NORMAL FLOAT AUTO REAL RADIAN MP 7. Which tangent line has an x-intercept closer to the x-intercept of y = x - 2 8. Find the equation of the line tangent to y=x-2r-5 at x = 21- 9. Find the z-intercept of this tangent line (round to 6 decimal places). We will call this 12. Notice that r is an even better approximation to a solution to r-2x-5=0. Let's repeat the process one more time: 10. Find the equation of the line tangent to y=x-2x-5 at the point where x = 12. (Round to 6 decimal places as necessary). 11. Find the x-intercept of this tangent line (round to 6 decimal places). Call this 23. 12. Use your calculator to approximate the solution to r-2x-5=0 (round to 6 decimal places). How does 23 compare to this value? The process that you used above was called Newton's Method (or the Newton-Raphson Method). The idea is to use the x-intercepts of tangent lines in succession to get you closer and closer to the z-intercept of a curve. (xf(x)) (x2f(x2)) 0 1/xz 1x2 x x Our next goal is to come up with a general formula for the nth iteration of the method (that is, what will x, the x-intercept of the nth tangent line, be). 13. Given a function f(z), what is the equation of the line tangent to the function at x = x? 14. What is the z-intercept of this line? Call it . 15. Given a function f(x), what is the equation of the line tangent to the function at x = x? 16. What is the z-intercept of this line? Call it 12. 17. Given a function f(x), what is the equation of the line tangent to the function at x = x2? 18. What is the z-intercept of this line? Call it 3- 19. Using the pattern found above, what is the r-intercept of the nth tangent line, that is, what is z,? Notice that to find F. you do first have to find 1, 2, 3,...-1- Newton's method is an iterative process. This makes it particularly convenient for use with a computer program. As humans, it is less convenient, but still usable! For our last example, we will approximate a solution to the equation cos (2) = 2 20. Rewrite this equation, so that it becomes a problem where we are trying to find a root of an equation. 21. Use 201 and 4 iterations of Newton's Method to approximate the solution to the equation. Round to 6 decimal places wherever necessary. Note: cos(x)=x can also be solved (approximately) by repeatedly taking the cosine of your answer. Start with taking cos (1) on your calculator. Then do cos (ANS) Repeat this process again and again and again until you start to get outputs that are close to your previous ANS. That will be approximately the r value for which cos(x) = x! 22. How does this compare to your approximate solution using Newton's Method? Suppose we want to solve the following equation: x-2x-5=0 1. What does it mean to "solve an equation"? Because r-2x-5 does not factor in an obvious way, this is actually a difficult task! We can use Newton's Method to help us to at least approximate solutions to the equation. 2. Given is the graph of y=x-2r-5. Sketch a line tangent to this curve at the point where x = 2. NORMAL FLOAT AUTO REAL RADIAN MP 1 3. How do the x-intercept of the tangent line, and the z-intercept of y = x- 2x - 5 compare? 4. Find the equation of your tangent line. (Hint: do not use your graph to do this, be- cause if you are not a robot, you may not have drawn your line perfectly.) 5. Find the x-intercept of your tangent line. We will call this value #1. Notice that x, is an approximation to the solution to z-2x-5=0. 6. Given below is the graph of y=x-2x-5, on the window [1.8, 2.2] x[-2,1]. Sketch a line tangent to the curve at r = 2, and a line tangent to the curve at 2 = 11- NORMAL FLOAT AUTO REAL RADIAN MP 7. Which tangent line has an x-intercept closer to the x-intercept of y = x - 2 8. Find the equation of the line tangent to y=x-2r-5 at x = 21- 9. Find the z-intercept of this tangent line (round to 6 decimal places). We will call this 12. Notice that r is an even better approximation to a solution to r-2x-5=0. Let's repeat the process one more time: 10. Find the equation of the line tangent to y=x-2x-5 at the point where x = 12. (Round to 6 decimal places as necessary). 11. Find the x-intercept of this tangent line (round to 6 decimal places). Call this 23. 12. Use your calculator to approximate the solution to r-2x-5=0 (round to 6 decimal places). How does 23 compare to this value? The process that you used above was called Newton's Method (or the Newton-Raphson Method). The idea is to use the x-intercepts of tangent lines in succession to get you closer and closer to the z-intercept of a curve. (xf(x)) (x2f(x2)) 0 1/xz 1x2 x x Our next goal is to come up with a general formula for the nth iteration of the method (that is, what will x, the x-intercept of the nth tangent line, be). 13. Given a function f(z), what is the equation of the line tangent to the function at x = x? 14. What is the z-intercept of this line? Call it . 15. Given a function f(x), what is the equation of the line tangent to the function at x = x? 16. What is the z-intercept of this line? Call it 12. 17. Given a function f(x), what is the equation of the line tangent to the function at x = x2? 18. What is the z-intercept of this line? Call it 3- 19. Using the pattern found above, what is the r-intercept of the nth tangent line, that is, what is z,? Notice that to find F. you do first have to find 1, 2, 3,...-1- Newton's method is an iterative process. This makes it particularly convenient for use with a computer program. As humans, it is less convenient, but still usable! For our last example, we will approximate a solution to the equation cos (2) = 2 20. Rewrite this equation, so that it becomes a problem where we are trying to find a root of an equation. 21. Use 201 and 4 iterations of Newton's Method to approximate the solution to the equation. Round to 6 decimal places wherever necessary. Note: cos(x)=x can also be solved (approximately) by repeatedly taking the cosine of your answer. Start with taking cos (1) on your calculator. Then do cos (ANS) Repeat this process again and again and again until you start to get outputs that are close to your previous ANS. That will be approximately the r value for which cos(x) = x! 22. How does this compare to your approximate solution using Newton's Method?