Functions and Critical Points

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Andrew Iskauskas

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Ready to use

6 years, 6 months ago

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Andrew Iskauskas 6 years, 6 months ago

Gave some feedback: Ready to use

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Andrew Iskauskas 6 years, 6 months ago

Created this as a copy of Functions and critical points: Cubic.

Name	Status	Author	Last Modified
Functions and critical points: Cubic	Ready to use	Andrew Iskauskas	14/05/2019 15:17
Andrew's copy of Functions and critical points: Cubic	draft	Andrew Iskauskas	12/07/2018 15:13
Functions and Critical Points	Ready to use	Andrew Iskauskas	28/08/2019 16:05
Functions and critical points: Cubic	Should not be used	Hilary Bainbridge	12/02/2019 17:31
Functions and critical points: Cubic	draft	Xiaodan Leng	10/07/2019 21:47
Functions and Critical Points	draft	Xiaodan Leng	10/07/2019 23:10
Functions and Critical Points	Ready to use	Andrew Iskauskas	07/09/2020 23:44

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Question statement

Consider the function $y=\simplify[all,fractionNumbers]{{coeff/3}*x^3-{coeff*(crit1+crit2)/2} *x^2+{coeff*crit1*crit2}*x+{const}}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

crit1	number	-3
crit2	number	-3
crit1func	number	-19
crit2func	number	-19
coeff	integer	2
const	integer	-1
whichpoint	list	[ latex("x_1"), latex("x_2") ]
rand	integer	1
markingmatrix	list	Nested 2×3 list
freevbls	list	[ 2, -1 ]
crits	list	[ -3, -3 ]
tanpoint	integer	2
mini	number	-1
start	integer	0
end	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-3..3#0.5)

JME function reference

Description

A turning point of the function f(x).

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `number`

-3

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part b) Choose one from a list
Part c) Choose one from a list
Part d) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part e) Mathematical expression

Gap-fill

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

Find the values of $x$ at the turning points of the function, where $x_{1}\le x_{2}$.

Point 1: $x_{1}=$Gap 0.

Point 2: $x_{2}=$Gap 1.

Advice

a) Critical points of $f(x)$ are found as solutions to $f'(x)=0$. For a cubic function $f(x)$ the differential is quadratic, so you will find two (not necessarily distinct, in general) roots.

b) The nature of a critical point $x_{*}$ of $f(x)$ is determined by the sign of $f''(x_{*})$. If $f''(x_{*})>0$, $x_{*}$ is a minimum; if $f''(x_{*})=0$, $x_{*}$ is a saddle point; if $f''(x_{*})<0$, $x_{*}$ is a maximum.

c) To match the function to a graph, the elements to consider are:

Roots of the function: $x$ such that $f(x)=0$. This gives the crossings on the $x$-axis;
Critical points: $x$ such that $f'(x)=0$. This gives the turning points on the graph;
Nature of critical points.

d) The gradient of a tangent line at a point $x_{0}$ is simply $f'(x_{0})$. Writing the equation of a line as $y-y_{0}=m(x-x_{0})$, we can see that the equation of the tangent line at $x_{0}$ can be written as $y-f(x_{0})=f'(x_{0})(x-x_{0})$, from which we can find the equation in the form $y=mx+c$. If $m_{t}$ is the gradient of the tangent line, then the gradient of the normal is $m_{n}=-1/m_{t}$, and given that the normal line also passes through the point $(x_{0},f(x_{0}))$, we can find the normal line equation similarly.

e) There are two places where a minimum of a continuous function can occur on a closed interval $[a,b]$: either at an endpoint, giving minimum $f(a)$ or $f(b)$, or at a turning point of the function, provided the turning point is a minimum. So we simply check the value of $f(x)$ at the endpoints $x=a$ and $x=b$, and check $f(x)$ at any critical points (found in (a)), and pick the minimum of those.

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This question is used in the following exam:

Durham Exam Demo by Andrew Iskauskas in Durham Test Questions.

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Andrew Iskauskas 6 years, 6 months ago

Andrew Iskauskas 6 years, 6 months ago

Error in variable testing condition

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Generated value: `number`

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Gap-fill

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Functions and Critical Points

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Andrew Iskauskas 6 years, 6 months ago

Andrew Iskauskas 6 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`