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	9
maxy	integer	71
ans1	number	0.5806
m	number	2
results	list	List of 7 items
b	integer	10
tol	integer	0
ans	number	1.4497
tans	number	1.4496628671

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(8..15)

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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`