3 Newton's method 3. For this problem, we define the function g(x) = x³ +2 - 1. We want to find a number such that g(x) = 0. In other words, we want to solve the equation r³3+x-1=0. You may use a calculator for this question. (a) Calculate g(0) and g(1). This guarantees there has to be some number 0<<1 such that g(x) = 0. Why? Hint: which theorem we can imply here. (b) We are going to make a bunch of successive guesses for the solution to the equation. None of them will be exact, but each one will be better than the previous one. Our first guess is going to be ₁ = 1. Write the equation of the line tangent to y = g(x) at the point with x-coordinate 21. Draw this line. We will call it L₁. (c) We are looking for the point of the graph y= g(x) that intersects the x-axis. Since this point is not too far from (1,9 (x1)), we can look for the point where the line L₁ intersects the x-axis instead. (Convince yourself that this makes sense!) Calculate this point. Call its x-coordinate 22. This is our second guess. Calculuate g(x2). Notice that g(x2) is not zero yet but it is closer to zero that g(x1). We are improving!
3 Newton's method 3. For this problem, we define the function g(x) = x³ +2 - 1. We want to find a number such that g(x) = 0. In other words, we want to solve the equation r³3+x-1=0. You may use a calculator for this question. (a) Calculate g(0) and g(1). This guarantees there has to be some number 0<<1 such that g(x) = 0. Why? Hint: which theorem we can imply here. (b) We are going to make a bunch of successive guesses for the solution to the equation. None of them will be exact, but each one will be better than the previous one. Our first guess is going to be ₁ = 1. Write the equation of the line tangent to y = g(x) at the point with x-coordinate 21. Draw this line. We will call it L₁. (c) We are looking for the point of the graph y= g(x) that intersects the x-axis. Since this point is not too far from (1,9 (x1)), we can look for the point where the line L₁ intersects the x-axis instead. (Convince yourself that this makes sense!) Calculate this point. Call its x-coordinate 22. This is our second guess. Calculuate g(x2). Notice that g(x2) is not zero yet but it is closer to zero that g(x1). We are improving!
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer part d of this question, thank you
Transcribed Image Text:3 Newton's method
3. For this problem, we define the function g(x) = x³ +2 - 1. We want to
find a number such that g(x) = 0. In other words, we want to solve the
equation r³3+x-1=0. You may use a calculator for this question.
(a) Calculate g(0) and g(1). This guarantees there has to be some number
0<<1 such that g(x) = 0. Why? Hint: which theorem we can imply here.
(b) We are going to make a bunch of successive guesses for the solution to
the equation. None of them will be exact, but each one will be better than the
previous one. Our first guess is going to be ₁ = 1. Write the equation of the
line tangent to y = g(x) at the point with x-coordinate 21. Draw this line. We
will call it L₁.
(c) We are looking for the point of the graph y= g(x) that intersects the
x-axis. Since this point is not too far from (1,9 (x1)), we can look for the point
where the line L₁ intersects the x-axis instead. (Convince yourself that this
makes sense!) Calculate this point. Call its x-coordinate 22. This is our second
guess.
Calculuate g(x2). Notice that g(x2) is not zero yet but it is closer to
zero that g(x1). We are improving!
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