Bill's copy of Newton-Raphson 1.2

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Bill Foster 7 years, 11 months ago

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Bill Foster 11 years, 4 months ago

Created this as a copy of Newton-Raphson 1.2.

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Question statement

Consider the following equation:

\[\simplify[std]{g(x) = x ^ 3 + {d} * x ^ 2 + ({q * (c * q ^ 2 -(p ^ 2 * d)) -(p ^ 3)} / {p * q ^ 2}) * x + { -c} = 0}\]

Give any introductory information the student needs.

			Name	Type	Generated Value

c	integer	4
d	integer	2
p	integer	2
ans1	decimal	dec("0.6667")
results	list	List of 5 items
q	integer	3
troot	rational	2/3
ans	decimal	dec("0.7129")
root	decimal	dec("0.67")
tans	decimal	dec("7.128712871287128430546024899519666116986e-1")

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
if(d=2,random(3..7),if(d=3,random(4..7),if(d=4,random(5..7),if(d=5,random(6..7),7))))

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

← Depends on:

→ Used by:

results
tans

Statement
Advice
"Part a)" → "Gap 0." - Correct answer
"Part a)" → "Gap 1." - Correct answer

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
1. Step Mathematical expression
Part b) Gap-fill
1. Gap Number entry
  Add
Part c) Gap-fill
1. Gap Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

This equation has a root in the range $0 \lt x \lt 1$.
Using the Newton-Raphson formula, if $x_n$ is the $n$th estimate for this root, show that the next estimate can be written in the form \[x_{n+1}= \frac{p(x_n)}{g'(x_n)}\]
Enter $p(x_n)$ and $g'(x_n)$ in the boxes below.

$p(x_n)=\;\;$Gap 0.
In your answer simply use $x$ instead of $x_n$.
Do not enter decimals in this part of the question; enter fractions instead

$g'(x_n)=\;\;$Gap 1.
In your answer simply use $x$ instead of $x_n$.
Do not enter decimals in this part of the question; enter fractions instead

If you have forgotten the Newton-Raphson formula you can click on Steps to see it. You will not lose any marks in doing so.

Advice

a)
Recall that the Newton-Raphson method is defined by:
\[x_{n+1}=x_n-\frac{g(x_n)}{g'(x_n)}\]
where we would like to find the root of the equation $g(x)=0$.

In this question we have:
\[\simplify[std]{g(x) = x ^ 3 + {d} * x ^ 2 + ({q * (c * q ^ 2 -(p ^ 2 * d)) -(p ^ 3)} / {p * q ^ 2}) * x + { -c}} \Rightarrow \simplify[std]{g'(x) = 3 * x ^ 2 + {2 * d} * x + {q * (c * q ^ 2 -(p ^ 2 * d)) -(p ^ 3)} / {p * q ^ 2}}\]

Hence in substituting into the formula we get:

\[\begin{eqnarray*}x_{n + 1} &=&\simplify[std]{ x_n - ((x_n^3)+ {d} * (x_n^2)+ ({q * (c * q ^ 2 -(p ^ 2 * d)) -(p ^ 3)} / {p * q ^ 2}) * x_n + { -c}) / (3 * (x_n^2) + {2 * d} * x_n + {q * (c * q ^ 2 -(p ^ 2 * d)) -(p ^ 3)} / {p * q ^ 2})}\\ &=&\simplify{(2 * (x_n^3) + {d} * (x_n^2) + {c}) / (3 * (x_n^2)+ {2 * d} * x_n + {q * (c * q ^ 2 -(p ^ 2 * d)) -(p ^ 3)} / {p * q ^ 2})} \end{eqnarray*} \]
You input your answers using $x$ rather than $x_n$.

In the following let $ \displaystyle f(x)=\simplify{(2 * (x^3) + {d} * (x^2) + {c}) / (3 * (x^2) + {2 * d} * x + {q * (c * q ^ 2 -(p ^ 2 * d)) -(p ^ 3)} / {p * q ^ 2})}$

b)
If $x_0=1$ then we find:

\[x_1= \simplify[std]{(2 + {d} + {c}) / (3 + {2 * d} + {q * (c * q ^ 2 -(p ^ 2 * d)) -(p ^ 3)} / {p * q ^ 2})} = \var{ans}\]
to 4 decimal places.

c)
We find on running the iteration that the first three values are:

\[\begin{align}x_1&=f(x_0)=f(1)&=\var{results[1]}\\x_2&=f(x_1)=f(\var{results[1]})&=\var{results[2]}\\x_3&=f(x_2)=f(\var{results[2]})&=\var{results[3]}\end{align}\]

and then to 4 d.p.s

\[x=\var{ans1}\]

Here is a graph of the function {funfornr(-5,100,5,-100)}

Looking in more detail between 0 and 1 we see where it crosses the x-axis:

{funfornr(ans1-0.1,2,ans1+0.1,-2)}

Give a worked solution to the whole question.

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Part	Test	Passed?
Gap-fill
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Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
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Mathematical expression
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Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet

This question is used in the following exams:

Isaac's copy of Review Questions by Isaac Mulolani in Isaac's workspace.
Review Questions by Isaac Mulolani in Isaac's workspace.

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Taxonomy: mathcentre

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Bill Foster 7 years, 11 months ago

Bill Foster 7 years, 11 months ago

Bill Foster 11 years, 4 months ago

Error in variable testing condition

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Generated value: `integer`

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Bill's copy of Newton-Raphson 1.2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 7 years, 11 months ago

Bill Foster 7 years, 11 months ago

Bill Foster 11 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`