Ugur's copy of Parameterisation of a curve - tangent and coordinates at given point,

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Created this as a copy of Parameterisation of a curve - tangent and coordinates at given point, .

Name	Status	Author	Last Modified
Parameterisation of a curve - tangent and coordinates at given point,	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Ugur's copy of Parameterisation of a curve - tangent and coordinates at given point,	Ready to use	Ugur Efem	07/06/2023 16:49

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Question statement

You are given the following curve, $t\mapsto\pmatrix{t^\var{a},\simplify{{b}t}}$ , defined with respect to the parameter $t$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	4
c	integer	6
b	number	10
d	integer	5
dxdtc	number	864
dxdtd	number	500
y	number	60
x	number	1296
speed	number	500.10

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xxxxxxxxxx
 
random(2..4)

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Description

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Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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3. Gap Gap 2. Number entry
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4. Gap Gap 3. Number entry
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Part c) 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.

Find the coordinates $\pmatrix{x,y}$ of the point corresponding to $t=\var{c}$ .

$\pmatrix{x,y}=($ Gap 0. $,$ Gap 1. $)$ .

Advice

To find the coordinates of the point corresponding to $t=\var{c}$ , substitute $t=\var{c}$ into the expression for the curve, i.e.

$\pmatrix{x,y}=\pmatrix{\var{c}^\var{a},\var{b}\times\var{c}}=\pmatrix{\var{x},\var{y}}.$

Differentiate each component of the vector in part a) to find the tangent vector $\boldsymbol{u}$ , i.e.

$\boldsymbol{u}=\pmatrix{\frac{\mathrm{d}}{\mathrm{d}t}t^\var{a},\frac{\mathrm{d}}{\mathrm{d}t}\simplify{{b}t}}=\pmatrix{\var{a}t^\var{a-1},\var{b}}.$

The tangent vector at $t=\var{c}$ is found by substituting $t=\var{c}$ into the tangent vector $\boldsymbol{u}$ , i.e.

$\boldsymbol{u}\vert_{t=\var{c}}=\pmatrix{\var{a}\times\var{c}^\var{a-1},\var{b}}=\pmatrix{\var{dxdtc},\var{b}}.$

The velocity $u$ is given by $u=\lvert\boldsymbol{u}\rvert=\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2}$ . We must calculate the speed at $t=\var{d}$ , however, therefore

$u\vert_{t=\var{d}}=\sqrt{\left(\var{a}\times\var{d}^\var{a-1}\right)^2+\var{b}^2}=\sqrt{\var{dxdtd^2}+\var{b^2}}=\var{speed} \; \text{to 2d.p.}$

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This question is used in the following exam:

Vector Calculus by Ugur Efem in Ugur's workspace.

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Ugur's copy of Parameterisation of a curve - tangent and coordinates at given point,

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Ugur Efem 1 year, 10 months ago

Ugur Efem 1 year, 10 months ago

Ugur Efem 1 year, 10 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Ugur's copy of Parameterisation of a curve - tangent and coordinates at given point,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ugur Efem 1 year, 10 months ago

Ugur Efem 1 year, 10 months ago

Ugur Efem 1 year, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`