Let f(x) = 6 sin(x – }) + 3. (a) Use the Intermediate Value Theorem to show that f(x) = 2 for some value of r between a = 0 and r = 6. (b) Estimate where f(r) = 2. Address each of the following in your solution. Start Newton's method at r = 5. This is your first approximation for the solution to f(x) = 2. • Choose a function g(x) for which you want to solve g(x) = 0. Hint: should this be f? a function related to f? Compute g'(x). Write an equation for the tangent line to the graph of g(x) at r = 5. • Find the x-intercept of this line. This is your second approximation for the solution to f(r) = 2. • Draw a graph to illustrate how you obtained your second approximation from the first approximation. • Is you second estimate an underestimate or an overestimate of the solution to f(x) = 2. Explain using your graph. How would Newton's Method help you estimate where f(r) = 2? Write the Newton iteration. That means that you should give a formula r„41 in terms of r.. • What technology will you use for this problem? • Continue using Newton's method to estimate where f(x) = 2 to the maximum number of digits available with your technology. Record the approximations neatly in a table. • Check that your final approximation actually has f(r) = 2.
Let f(x) = 6 sin(x – }) + 3. (a) Use the Intermediate Value Theorem to show that f(x) = 2 for some value of r between a = 0 and r = 6. (b) Estimate where f(r) = 2. Address each of the following in your solution. Start Newton's method at r = 5. This is your first approximation for the solution to f(x) = 2. • Choose a function g(x) for which you want to solve g(x) = 0. Hint: should this be f? a function related to f? Compute g'(x). Write an equation for the tangent line to the graph of g(x) at r = 5. • Find the x-intercept of this line. This is your second approximation for the solution to f(r) = 2. • Draw a graph to illustrate how you obtained your second approximation from the first approximation. • Is you second estimate an underestimate or an overestimate of the solution to f(x) = 2. Explain using your graph. How would Newton's Method help you estimate where f(r) = 2? Write the Newton iteration. That means that you should give a formula r„41 in terms of r.. • What technology will you use for this problem? • Continue using Newton's method to estimate where f(x) = 2 to the maximum number of digits available with your technology. Record the approximations neatly in a table. • Check that your final approximation actually has f(r) = 2.
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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Transcribed Image Text:2. Let f(x) = 6 sin(r – ) + 3.
(a) Use the Intermediate Value Theorem to show that f(r) = 2 for some value of x
between r = 0 and r = 6.
(b) Estimate where f(x) = 2. Address each of the following in your solution.
Start Newton's method at r = 5. This is your first approximation for the
solution to f(x) = 2.
• Choose a function g(x) for which you want to solve g(x)
this be f? a function related to f?
• Compute g'(r).
Write an equation for the tangent line to the graph of g(x) at r = 5.
Find the x-intercept of this line. This is your second approximation for the
solution to f(x) = 2.
Draw a graph to illustrate how you obtained your second approximation from
the first approximation.
• Is you second estimate an underestimate or an overestimate of the solution
to f(r) = 2. Explain using your graph.
How would Newton's Method help you estimate where f(x) = 2?
• Write the Newton iteration. That means that you should give a formula rn+1
in terms of rn.
What technology will you use for this problem?
• Continue using Newton's method to estimate where f(r) = 2 to the maximum
number of digits available with your technology.
0. Hint: should
%3D
Record the approximations neatly in a table.
• Check that your final approximation actually has f (x) × 2.
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