MATH104 December 2010
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q1 (l) • Q1 (m) • Q1 (n) • Q2 (a) • Q2 (b) • Q3 • Q4 • Q5 • Q6 •
Question 01 (g)
You are using the Newton-Raphson Method to approximate a solution of an equation f(x) = x and you make an initial guess to the solution. If the tangent line to y = f(x) at x = 3 has the equation y = 5x - 7, what is the next approximation to the solution?
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
Notice that finding a solution to f(x)=x is the same as finding the root of an equation g(x)=f(x)-x.
How can you use the tangent line y=5x-7 to give you information about and ?
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We want to find a solution to
and so we begin by defining a function
so that we can think instead of finding roots to g(x). Recall that the Newton-Raphson Method for finding roots to a function g(x) gives a future approximation () based on a current approximation () as
Therefore, in our case
since . The task now is to find what g(3) and g'(3) are. Since g(x)=f(x)-x then
so we really only need to determine f(3) and f'(3). To do this we can use the tangent line which is also the linear approximation to f(x). Therefore we have that
Therefore, we have that f(3)=8 and f'(3)=5. Similarly, we then have that,
and so we get that,
Therefore the next approximation to the solution is .
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