Parametric 2nd derivative

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Simon Thomas

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Simon Thomas 5 years, 2 months ago

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Simon Thomas 5 years, 2 months ago

Created this as a copy of Parametric Differentiation.

Name	Status	Author	Last Modified
Parametric Differentiation	draft	Simon Thomas	12/02/2020 10:50
Equation of tangent to Parametric curve	draft	Simon Thomas	12/02/2020 10:51
Parametric 2nd derivative	draft	Simon Thomas	12/02/2020 10:52

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Question statement

A curve is defined by the parametric equations $x=\var{a}+\var{b}t^2, y = \var{c}t+\var{d}t^3$

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			Name	Type	Generated Value

a	integer	4
b	integer	4
c	integer	5
d	integer	5
x	integer	-2
t	rational	-3/2

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1. Gap Gap 1 Mathematical expression
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Find $\frac{d^2y}{dx^2}$ in terms of $t$ Gap 1

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We have $x=\var{a}+\var{b}t^2, y = \var{c}t+\var{d}t^3$

So $\frac{dx}{dt} = 2 \times\var{b}t = \var{2b}t$

and $\frac{dy}{dt} = \var{c}+3 \times\var{d}t^2 = \var{c}+\var{3d}t^2$

Since $\frac{dy}{dx}=\frac{dy}{dt}/\frac{dx}{dt}$ we have $\frac{dy}{dx}=\frac{\var{c}+\var{3d}t^2}{\var{2b}t}=\frac{\var{c}}{\var{2b}t}+\frac{\var{3d}t}{\var{2b}}$

To find $\frac{d^2y}{dx^2}$ we use the chain rule to obtain $\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{d}{dt}(\frac{dy}{dx})\times\frac{dt}{dx}=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$

Now $\frac{d}{dt}(\frac{dy}{dx}) = \frac{-\var{c}}{\var{2b}t^2}+\frac{\var{3d}}{\var{2b}}=\frac{-\var{c}+\var{3d}t^2}{\var{2b}t^2}$

Hence $\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}=\frac{-\var{c}+\var{3d}t^2}{\var{2b}t^2\times\var{2b}t}=\frac{\var{3d}t^2-\var{c}}{\var{4b^2}t^3}$

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This question is used in the following exam:

Parametric Differentiation by Simon Thomas in Maths support.

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Simon Thomas 5 years, 2 months ago

Simon Thomas 5 years, 2 months ago

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Generated value: `integer`

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Parametric 2nd derivative

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 5 years, 2 months ago

Simon Thomas 5 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`