Define stationary point .

A point on a curve where the gradient is zero . It may be a local minimum, a local maximum, or a point of inflection.

What is the difference between a stationary point and a turning point?

Every turning point is stationary, but a point of inflection is stationary without being a turning point. A turning point is one where the curve actually changes direction, from rising to falling or the other way round.

How do you use the first derivative to determine the nature of a stationary point?

Check its sign just to the left and just to the right of the point. Negative then positive is a minimum, positive then negative is a maximum, and the same sign on both sides is a point of inflection.

Which test for the nature of a stationary point should you reach for first?

The second derivative , because it is usually much quicker. Fall back on the first-derivative test only when the second derivative turns out to be zero.

What has to happen before you can optimise a quantity by differentiating?

It has to be written as a formula in a single variable. These problems usually give two variables plus a constraint linking them, and the constraint is what lets you eliminate one.

Why might a stationary point of a model not be a sensible answer?

Because the context may restrict the variable: a length cannot be negative, and a number of items has to be a whole number. A mathematically valid stationary point can fall outside the range the situation allows.

International A Level (IAL)MathsEdexcelPure 2FlashcardsDifferentiationApplications of Differentiation

Applications of Differentiation (Edexcel International A Level (IAL) Maths: Pure 2): Flashcards

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Define stationary point.

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Define stationary point.
A point on a curve where the gradient is zero.
It may be a local minimum, a local maximum, or a point of inflection.
How do you find the stationary points of $f (x)$ ?
Differentiate, solve $f' (x) = 0$ for the $x$ -coordinates, then substitute those back into $f (x)$ for the $y$ -coordinates.
The derivative gives the positions; the original function gives the heights.
What is the difference between a stationary point and a turning point?
Every turning point is stationary, but a point of inflection is stationary without being a turning point.
A turning point is one where the curve actually changes direction, from rising to falling or the other way round.
Complete the second derivative test for the nature of a stationary point:
$f'' (x) > 0 \Rightarrow local______$
$f'' (x) < 0 \Rightarrow local______$
A positive second derivative means a local minimum, and a negative one means a local maximum.
Substitute the stationary point's $x$ -coordinate into $f'' (x)$ to find out which.
How do you use the first derivative to determine the nature of a stationary point?
Check its sign just to the left and just to the right of the point.
Negative then positive is a minimum, positive then negative is a maximum, and the same sign on both sides is a point of inflection.
True or False?
If $f'' (x) = 0$ at a stationary point, the point is a point of inflection.
False.
A zero second derivative tells you nothing: the point could still be a maximum or a minimum.
The first-derivative test has to be used instead, and that one always works.
Which test for the nature of a stationary point should you reach for first?
The second derivative, because it is usually much quicker.
Fall back on the first-derivative test only when the second derivative turns out to be zero.
How many stationary points does a quadratic have, and what kind?
Exactly one, and it is always a turning point rather than a point of inflection.
Its $y$ -value is therefore the minimum or maximum value the whole quadratic can take.
What does a stationary point on $y = f (x)$ become on the graph of $y = f' (x)$ ?
A point where the gradient graph meets the $x$ -axis, since the gradient there is zero.
The stationary points of $f$ are exactly the roots of $f'$ .
Where $f (x)$ is increasing, $f' (x)$ lies $______$ the $x$ -axis; where $f (x)$ is decreasing, it lies $______$ it.
Where $f$ is increasing, $f'$ lies above the axis; where $f$ is decreasing, it lies below it.
That is simply because the gradient is positive on a rising section and negative on a falling one.
True or False?
Where $f (x)$ cuts the $x$ -axis, $f' (x)$ does something notable.
False.
Where $f$ crosses the axis tells you nothing at all about $f'$ .
Only the gradient of $f$ matters, and a curve can cross the axis at any gradient whatever.
If $f (x)$ is a smooth curve, what does that tell you about $f' (x)$ ?
It will be a smooth curve as well.
So once the key features have been marked, the rest of the gradient graph is drawn by joining them smoothly.
What can you not work out about $f' (x)$ from the graph of $f (x)$ alone?
The exact coordinates of its $y$ -intercept, or of its own stationary points.
A sketch of $f$ gives you the shape of $f'$ , not its precise values.
Can you differentiate variables other than $y$ and $x$ ?
Yes: a derivative can be taken with respect to any variable.
$\frac{d V}{d r}$ , for instance, gives the rate of change of a volume with respect to a radius.
What has to happen before you can optimise a quantity by differentiating?
It has to be written as a formula in a single variable.
These problems usually give two variables plus a constraint linking them, and the constraint is what lets you eliminate one.
Why is the answer to an optimisation question often not just the value of $x$ ?
Because the question usually asks for the maximum or minimum value itself, not for where it occurs.
Substituting the stationary point's $x$ back into the original formula is what gives that value.
True or False?
A derivative in a modelling question always represents a gradient on a graph.
False.
It represents a rate of change of one quantity with respect to another, whatever those quantities happen to be.
The gradient of a graph is simply the case where they are $y$ and $x$ .
Why might a stationary point of a model not be a sensible answer?
Because the context may restrict the variable: a length cannot be negative, and a number of items has to be a whole number.
A mathematically valid stationary point can fall outside the range the situation allows.

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