Equation of a Chord

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Created this.

Name	Status	Author	Last Modified
Equation of a Chord	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
George's copy of Equation of a Chord	draft	George Loughlin	06/07/2017 15:09

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Question statement

Let $f(x)=\simplify[std]{(x+{b})^{n}}$ . What are the gradient and equation of the chord between $(\var{a},f(\var{a}))$ and $(\simplify[std]{{a}+{h}},f(\simplify[std]{{a}+{h}}))$ ?

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			Name	Type	Generated Value

a	integer	1
b	integer	-3
d	number	8.05000
val	number	-4.050
h	number	-0.05
n	integer	2
s	integer	-1
b1	integer	-3
tol	number	0.001
val1	number	8.050
d1	number	-4.05000

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Description

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Generated value: `integer`

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1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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1. Step Mathematical expression

Gap-fill

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The gradient $m =$ Gap 0. (input your answer to 3 decimal places).

The equation of the chord is $y=ax+b$ where:

$a= \;$ Gap 1. and $b=\;$ Gap 2.

Enter both values $a$ and $b$ correct to 3 decimal places.

Advice

Given two points $(a,f(a))$ and $(a+h,f(a+h))$ on the graph of the function $y=f(x)$ .
Then the chord is the straight line between these two points and has the equation

$y-f(a)=m(x-a)$ where

$m$ is the gradient of the chord.
The gradient is given by dividing the change in

$y$ by the change in

$x$ .
Hence for this example

$m = \frac{f(a+h)-f(a)}{h} = \frac{f(\var{a+h})-f(\var{a})}{\var{h}} = \var{d1} = \var{val}$ to 3 decimal places.
Hence the equation of the chord is of the form

$y=\var{d1}x+b$ for some constant

$b$ .
But we know that when

$x=\var{a}$ then

$y=f(\var{a}) = \var{a+b}^\var{n}=\var{(a+b)^n}$
So

$b=\var{(a+b)^n}-\var{d1}\times\var{a} = \var{d}=\var{val1}$ to 3 decimal places

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Equation of a Chord

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`