Why does a parametric model always state a range for the parameter?

Because it fixes how much of the curve actually belongs to the model. Outside that range the equations still produce points, but they describe nothing real, such as a time before the motion began.

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Calculus & Modelling with Parametric Equations (Edexcel A Level Maths: Pure): Flashcards

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FrontParametric Differentiation
Complete the rule for differentiating parametric equations:
$\frac{d y}{d x} = \frac{d y}{d t} \div______$

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Complete the rule for differentiating parametric equations:
$\frac{d y}{d x} = \frac{d y}{d t} \div______$
The completed rule is:
$\frac{d y}{d x} = \frac{d y}{d t} \div \frac{d x}{d t}$
It follows from the chain rule with the reciprocal property, since $\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x}$ .
Why is it acceptable for $\frac{d y}{d x}$ to come out in terms of $t$ ?
Because every point on the curve is identified by its parameter value rather than by its $x$ coordinate.
To get a numerical gradient you find the value of $t$ at the point you want and substitute that, which is one step earlier than in ordinary differentiation.
How do you find the gradient of a parametric curve at a given point?
Differentiate both equations to get $\frac{d y}{d t}$ and $\frac{d x}{d t}$ , then divide to get $\frac{d y}{d x}$ in terms of $t$ .
Find the value of $t$ at that point and substitute it in.
What condition gives a stationary point on a parametric curve?
$\frac{d y}{d x} = 0$ , which happens exactly when $\frac{d y}{d t} = 0$ .
It is the numerator of the quotient that has to vanish, so it is the $y$ equation you differentiate and set to zero.
What happens where $\frac{d x}{d t} = 0$ ?
The gradient $\frac{d y}{d x}$ is undefined, since the quotient would be dividing by zero.
Geometrically the tangent there is vertical, which is something an ordinary $y = f (x)$ curve can never show.
True or False?
To find $\frac{d y}{d x}$ you must first eliminate the parameter.
False.
Differentiating each equation with respect to $t$ and dividing is quicker, and it still works when the parameter cannot be eliminated neatly.
Eliminating first is extra work that usually makes the differentiation harder rather than easier.
Complete the rule for the area under a parametrically defined curve:
$\int y d x = \int y______d t$
The completed rule is:
$\int y d x = \int y \frac{d x}{d t} d t$
The $d x$ is replaced by $\frac{d x}{d t} d t$ , and $y$ has to be written in terms of $t$ as well.
What is the key thing to change when integrating parametrically?
The limits, which must be converted from $x$ values into the matching values of the parameter.
Leaving $x$ limits on an integral that is now with respect to $t$ is the commonest error in the topic.
An area is required from $x = 0$ to $x = 4$ , and $x = t^{2}$ . What are the limits in $t$ ?
$t = 0$ and $t = 2$ , found by solving $t^{2} = 0$ and $t^{2} = 4$ .
Where solving gives two possible values, the stated range of the parameter decides which one belongs to the region you want.
Why must $y$ be rewritten in terms of $t$ before integrating?
Because the integral is now with respect to $t$ , so nothing in it may still be written in terms of $x$ .
That is the same requirement as in any substitution: every part of the integral changes together or none of it does.
True or False?
Parametric integration is a substitution in disguise.
True.
The parameter plays exactly the part that $u$ plays in an ordinary substitution: replace $d x$ , rewrite the integrand, convert the limits.
Seeing that means there is no separate method to learn here, only a familiar one to recognise.
What kinds of question can a parametric motion model be asked?
Where the object is at a given time, when it reaches a given position, and when it crosses an axis.
Each one becomes a substitution into, or an equation in, the parameter, rather than anything done with $x$ and $y$ directly.
A model gives $x = 2 \cos t$ and $y = 3 \sin 2 t$ . How do you find when the object is at a given $x$ ?
Solve $2 \cos t = x$ for $t$ , keeping only the solutions that lie inside the stated range of the parameter.
The range is part of the model, so a mathematically valid value outside it is not an answer to the question.
How do you find the greatest height reached in a parametric motion model?
Maximise $y$ as a function of the parameter, by solving $\frac{d y}{d t} = 0$ .
The $x$ equation plays no part in it, since the height depends on $y$ alone.
Why does a parametric model always state a range for the parameter?
Because it fixes how much of the curve actually belongs to the model.
Outside that range the equations still produce points, but they describe nothing real, such as a time before the motion began.
True or False?
Two different values of the parameter can give the same point on the curve.
True.
A path that crosses itself, or that repeats, returns to the same $(x, y)$ at more than one time.
That is why solving for $t$ can produce several answers, and why the stated range of the parameter matters so much.

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