Quotient rule

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Bill Foster

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Bill Foster 55 years, 2 months ago

Created this.

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Question statement

Let

$\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	5
b	integer	-7
d	integer	-5
s2	integer	-1
s1	integer	-1
det	integer	25
c1	integer	5

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Description

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Generated value: `integer`

→ Used by:

Statement
Advice
"Part a)" → "Gap 0." - Correct answer
"Part a)" → "Unnamed step" - prompt
"Part b)" - prompt

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
1. Step Information only
Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

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

Find numbers $a$ and $b$ such that

$\simplify[std]{f(x) = a + b/({c}x+{d})}$
Enter a and b as integers or fractions, but not as decimals.

$a=\;$ Gap 0.

$b=\;$ Gap 1.

You can click on Show steps to get some help, but you will lose 1 mark if you do so.

Advice

We have $\displaystyle \simplify[std]{{a}x+{b}={a}/{c}*({c}x+{d})+{b}-{a}*{d}/{c}={a}/{c}*({c}x+{d})+{-det}/{c}}$
Hence

$\begin{eqnarray*} \simplify[std]{({a} * x+{b})/({c}x+{d})}&=&\simplify[std]{({a}/{c}*({c}x+{d})+{-det}/{c})/({c}x+{d})}\\ &=&\simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})} \end{eqnarray*}$
Where we have divided out by

$\simplify[std]{{c}x+{d}}$ at the last step.

We have

$\frac{dg}{dx} = \simplify[std]{{-c}/({c}x+{d})^2}$
using standard rules of differentiation.
Since from a),

$f(x) = \simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})}$
we see that

$\begin{eqnarray*}\frac{df}{dx} &=&\simplify[std,fractionNumbers,!unitPower,!zeroFactor,!zeroTerm,!zeroPower]{(-{c})*(({-det}/{c})/({c}x+{d})^2)}\\ &=&\simplify[std]{{det}/({c}x+{d})^2} \end{eqnarray*}$

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Part	Test	Passed?
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Mathematical expression
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Mathematical expression
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Information only
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Mathematical expression
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Mathematical expression
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This question is used in the following exams:

mathcentre: Quotient rule practice by Bill Foster in Bill's workspace.
Bradford: Quotient rule practice by Bill Foster in Bill's workspace.
Maths Support: Quotient rule by Christian Lawson-Perfect in Maths Support Wiki.
Algebraic manipulation: 1 by Aneela Arshad in Aneela's workspace.
Differentiation by Aneela Arshad in Aneela's workspace.
Derivasjon og integrasjon by Tore Gaupseth in Tore's workspace.
Basic Calculus by Bill Foster in Bill's workspace.
Bradford: Basic Calculus by Bill Foster in Bill's workspace.
Stanislav's copy of Bradford: Basic Calculus by Stanislav Senveter in Stanislav's workspace.
Calculous: 1 by Aneela Arshad in Aneela's workspace.
Differentiation by Bill Foster in Bill's workspace.
Quotient rule by Marlon Arcila in Derivadas.

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Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

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Error

Quotient rule

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`