Keith's copy of Maria's copy of Mario's copy of Interactive Newton-Raphson method

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Keith Tarnowski 5 years, 9 months ago

Created this as a copy of Maria's copy of Mario's copy of Interactive Newton-Raphson method.

Name	Status	Author	Last Modified
Interactive Newton-Raphson method	Ready to use	Bill Foster	19/04/2017 11:03
Interactive Newton-Raphson method	Should not be used	Christian Lawson-Perfect	14/02/2020 12:57
Senida's copy of Interactive Newton-Raphson method	draft	Senida Krcic	15/11/2018 03:27
Mario's copy of Interactive Newton-Raphson method	draft	Mario Stevanovski	15/11/2018 03:44
Maria's copy of Mario's copy of Interactive Newton-Raphson method	draft	Maria Aneiros	27/05/2019 05:57
Keith's copy of Maria's copy of Mario's copy of Interactive Newton-Raphson method	draft	Keith Tarnowski	03/06/2019 14:24
Anna's copy of Maria's copy of Mario's copy of Interactive Newton-Raphson method	Doesn't work	Anna Strzelecka	16/08/2019 15:12
Min_max_v1	draft	Anna Strzelecka	10/01/2020 14:29
Min_max_v2	draft	Anna Strzelecka	16/08/2019 14:51
Maths_and_Stats's copy of Interactive Newton-Raphson method	draft	Maths_and_Stats Advice	19/09/2019 13:49

There are 8 other versions that do you not have access to.

Question statement

Consider the following equation.

\[\simplify[std]{e^({m}x)+{b}x-{a}=0}\]

Find the approximate solution in the range $0 \le x \le 1$ by using the Newton-Raphson method.

The following diagram demonstrates the method.

$x_0$ is the starting value, you can slide it along the x-axis to see the effect of changing it.

{test(m,b,a,maxy)}

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	12
maxy	integer	58
ans1	number	0.7690
m	number	1.9
results	list	List of 7 items
b	integer	10
tol	integer	0
ans	number	1.4449
tans	number	1.4448547865

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(8..15)

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`

→ Used by:

b
maxy
results
tans

This variable doesn't seem to be used anywhere.

Parts

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

Please note that if you enter a function of the form $xe^{ax}$, then you must input it as $x*e^{ax}$.

$p(x_n)=\;\;$Gap 0. In your answer use $x$ instead of $x_n$.

$g'(x_n)=\;\;$Gap 1. In your answer use $x$ instead of $x_n$.

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

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:
\[\simplify[std]{g(x) = Exp({m} * x) + {b} * x + { -a}} \Rightarrow \simplify[std]{g'(x) = {m}*Exp({m} * x) + {b}}\]
Substituting these expressions into the formula we have:
\[x_{n+1} =\simplify[std]{ x_n -((Exp({m} * x_n) + {b} * x_n + { -a}) / ({m} * Exp({m} * x_n) + {b}))}\]

which can be rearranged to give:
\[x_{n + 1} = \simplify[std]{(({m} * x_n -1) * Exp({m} * x_n) + {a}) / ({m} * Exp({m} * x_n) + {b})}\]

(In your answers you would input $x$ rather than $x_n$.)

In the following let $\displaystyle f(x)=\simplify[std]{ (({m} * x -1) * Exp({m} * x ) + {a}) / ({m} * Exp({m} * x ) + {b})}$

If $x_0=2$ then $x_1$ is simply given by:
\[\simplify[std]{x_1 = (({2*m} -1) * Exp({2*m}) + {a}) / ({m} * Exp({2*m}) + {b})}\]

which to 4 decimal places is: $\;\;x_1= \var{ans}$

We find on running the iteration that the first six values are:

\[\begin{align}x_1&=f(x_0)=f(2)&=\var{results[1]}\\x_2&=f(x_1)=f(\var{results[1]})&=\var{results[2]}\\x_3&=f(x_2)=f(\var{results[2]})&=\var{results[3]}\\x_4&=f(x_3)=f(\var{results[3]})&=\var{results[4]}\\x_5&=f(x_4)=f(\var{results[4]})&=\var{results[5]}\\x_6&=f(x_5)=f(\var{results[5]})&=\var{results[6]}\end{align}\]

So the solution to the equation to four decimal places for $0 \le x \le 1$ is $x=\var{precround(ans1,4)}$

Here we see the graph of $\simplify{e^({m}*x)+{b}*x-{a}}$ and the first four successive approximations to the root:

{test(m,b,a,maxy)}

Note that you can slide the first approximation $x_0$ along the x-axis to see the effect of changing the starting value.

Give a worked solution to the whole question.

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Keith's copy of Maria's copy of Mario's copy of Interactive Newton-Raphson method

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Keith Tarnowski 5 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Keith's copy of Maria's copy of Mario's copy of Interactive Newton-Raphson method

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Keith Tarnowski 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`