Difference tables 2

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Frank Doheny

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Frank Doheny

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8 years ago

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Frank Doheny 7 years, 1 month ago

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Frank Doheny 8 years ago

Created this as a copy of Difference tables.

Name	Status	Author	Last Modified
Difference tables	Ready to use	Frank Doheny	14/03/2018 21:18
Difference tables 2	Ready to use	Frank Doheny	14/03/2018 21:20

Question statement

The following set of data was generated by a polynomial.

$\begin{matrix} {x}&{f(x)} \\ \var{x0}&\var{y0}\\\var{x1}&\var{y1}\\\var{x2}&\var{y2}\\\var{x3}&\var{y3}\\\var{x4}&\var{y4}\\\var{x5}&\var{y5}\\\var{x6}&\var{y6}\\\var{x7}&\var{y7}\\\end{matrix}$

Give any introductory information the student needs.

			Name	Type	Generated Value

x0	integer	1
stp	number	0.5
x1	number	1.5
x2	number	2
x3	number	2.5
x4	number	3
x5	number	3.5
x6	number	4
x7	number	4.5
a	integer	5
b	integer	9
c	integer	4
d	integer	24
y0	number	42
y1	number	67.125
y2	number	108
y3	number	168.375
y4	number	252
y5	number	362.625
y6	number	504
y7	number	679.875
d10	number	25.125
d11	number	40.875
d12	number	60.375
d13	number	83.625
d14	number	110.625
d15	number	141.375
d16	number	175.875
d20	number	15.75
d21	number	19.5
d22	number	23.25
d23	number	27
d24	number	30.75
d25	number	34.5
d30	number	3.75
d31	number	3.75
d32	number	3.75
y10	number	37
y11	number	50.25
y12	number	68
y13	number	90.25
y14	number	117
y15	number	148.25
y16	number	184
y17	number	224.25
e10	number	13.25
e11	number	17.75
e12	number	22.25
e13	number	26.75
e14	number	31.25
e15	number	35.75
e16	number	40.25
e20	number	4.5
e21	number	4.5
e22	number	4.5
e23	number	4.5
e24	number	4.5
e25	number	4.5
y20	integer	28
y21	number	30
y22	number	32
y23	number	34
y24	number	36
y25	number	38
y26	number	40
y27	number	42
f1	number	2

Automatically regenerate variables?

Name

Data type

Value

A random number between and (inclusive) with step size

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`

This variable doesn't seem to be used anywhere.

Parts

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1. Gap Number entry
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Part b) Gap-fill
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Gap-fill

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

Use a difference table to show it was generated by a degree 3 polynomial $f(x)=ax^3+bx^2+cx+d$ and hence calculate the leading coefficient, $a$ .

$a=$ Unnamed gap

Advice

The difference table should take the form:

$\begin{matrix} {x}&{f(x)}&\delta_1&\delta_2&\delta_3 \\ \var{x0}&\var{y0}&\var{d10}&\var{d20}&\var{d30}\\\var{x1}&\var{y1}&\var{d11}&\var{d21}&\var{d31}\\\var{x2}&\var{y2}&\var{d12}&\var{d22}&\var{d32}\\\var{x3}&\var{y3}&\var{d13}&\var{d23}&\var{d32}\\\var{x4}&\var{y4}&\var{d14}&\var{d24}&\var{d32}\\\var{x5}&\var{y5}&\var{d15}&\var{d25}\\\var{x6}&\var{y6}&\var{d16}\\\var{x7}&\var{y7}\\\end{matrix}$

All the 3rd differences equal the constant $\var{d32}$

The coefficient of $x^3$ is given by the formula: $a=\frac{constant}{3!*(step-size)^3}$

$a=\frac{\var{d32}}{3!*(\var{stp})^3}=\var{a}$

The next difference table should relate the $x$ values to the $f(x)-\var{a}x^3$ values.

Repeat the process:

$\begin{matrix} {x}&{f(x)-\var{a}x^3}&\delta_1&\delta_2 \\ \var{x0}&\var{y10}&\var{e10}&\var{e20}\\\var{x1}&\var{y11}&\var{e11}&\var{e21}\\\var{x2}&\var{y12}&\var{e12}&\var{e22}\\\var{x3}&\var{y13}&\var{e13}&\var{e23}\\\var{x4}&\var{y14}&\var{e14}&\var{e24}\\\var{x5}&\var{y15}&\var{e15}&\var{e25}\\\var{x6}&\var{y16}&\var{e16}\\\var{x7}&\var{y17}\\\end{matrix}$

All the 2nd differences equal the constant $\var{e25}$

The coefficient of $x^2$ is given by: $b=\frac{\var{e25}}{2!*(\var{stp})^2}=\var{b}$

The next difference table should relate the $x$ values to the $f(x)-\var{a}x^3-(\var{b}x^2)$ values.

Repeat the process:

$\begin{matrix} {x}&{f(x)-\var{a}x^3}-(\var{b}x^2)&\delta_1 \\ \var{x0}&\var{y20}&\var{f1}&\\\var{x1}&\var{y21}&\var{f1}\\\var{x2}&\var{y22}&\var{f1}\\\var{x3}&\var{y23}&\var{f1}\\\var{x4}&\var{y24}&\var{f1}\\\var{x5}&\var{y25}&\var{f1}\\\var{x6}&\var{y26}&\var{f1}\\\var{x7}&\var{y27}\\\end{matrix}$

All the 1st differences equal the constant $\var{f1}$

The coefficient of $x$ is given by: $c=\frac{\var{f1}}{1!*(\var{stp})}=\var{c}$

$f(x)=\simplify{{a}x^3+{b}x^2}+\var{c}x+{d}$

From the initial set of data we see that $(\var{x0},\var{y0})$ satisfies the polynomial function.

$\var{y0}=\var{a}(\var{x0})^3+(\var{b})(\var{x0}^2)+\var{c}(\var{x0})+{d}$

${d}=\var{d}$

Therefore $f(x)=\simplify{{a}x^3+{b}x^2}+\var{c}x+\var{d}$

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This question is used in the following exam:

Numerical methods by Frank Doheny in Numerical methods.

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Frank Doheny 7 years, 1 month ago

Frank Doheny 8 years ago

Frank Doheny 8 years ago

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Difference tables 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Frank Doheny 7 years, 1 month ago

Frank Doheny 8 years ago

Frank Doheny 8 years ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`