How do the tangent lines tell you whether a curve is convex or concave?

If every tangent lies below the curve then it is convex; if every tangent lies above the curve then it is concave. This is a quick check to make against a sketch, since it needs no differentiation at all.

Define point of inflection .

A point at which a curve changes from convex to concave , or from concave to convex. Equivalently, it is a point where the second derivative changes sign .

Does a point of inflection have to be a stationary point?

No. The gradient does not have to be zero at a point of inflection. A stationary point of inflection is simply the special case where the gradient happens to be zero there as well.

What does a curve look like at a non-stationary point of inflection?

It carries on in the same direction, but switches which way it bends , so the tangent line passes through the curve rather than staying on one side of it. That crossing is what makes an inflection visible on a sketch.

Why does it matter which letters in the problem are constants?

Because a constant differentiates to zero and a variable does not, so treating one as the other changes the answer completely. In a cone of fixed shape, for instance, the ratio of radius to height stays constant even though both of them are changing.

Define differential equation .

An equation that contains a derivative , in other words a rate of change. Connected rates of change is how you set one up; solving it comes later, using integration.

A LevelMathsEdexcelPureFlashcardsDifferentiationFurther Applications of Differentiation

Further Applications of Differentiation (Edexcel A Level Maths: Pure): Flashcards

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FrontConcave & Convex Functions
Define a convex function.

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Cards in this collection (16)

Define a convex function.
A function whose graph curves up, so that its tangent lines lie below the graph.
On a given interval, $f (x)$ is convex if and only if $f'' (x) \geq 0$ for every value of $x$ in that interval.
Define a concave function.
A function whose graph curves down, so that its tangent lines lie above the graph.
On a given interval, $f (x)$ is concave if and only if $f'' (x) \leq 0$ for every value of $x$ in that interval.
How do the tangent lines tell you whether a curve is convex or concave?
If every tangent lies below the curve then it is convex; if every tangent lies above the curve then it is concave.
This is a quick check to make against a sketch, since it needs no differentiation at all.
Why is convexity always stated on an interval?
Because one curve is usually convex in some regions and concave in others, so it is rarely one or the other everywhere.
That is why a question asks for the interval on which $f'' (x) \geq 0$ , rather than asking you to classify the whole curve at once.
True or False?
A curve with a positive gradient everywhere is convex.
False.
Convexity depends on the second derivative, not on the first.
The curve $y = \sqrt{x}$ has a positive gradient everywhere it is defined, and yet it is concave throughout.
Define point of inflection.
A point at which a curve changes from convex to concave, or from concave to convex.
Equivalently, it is a point where the second derivative changes sign.
Does a point of inflection have to be a stationary point?
No. The gradient does not have to be zero at a point of inflection.
A stationary point of inflection is simply the special case where the gradient happens to be zero there as well.
How do you find the points of inflection of $y = f (x)$ ?
Differentiate twice, solve $f'' (x) = 0$ , and then check that $f''$ changes sign across each solution.
Substitute the surviving $x$ -values into $f (x)$ to get the $y$ -coordinates.
Why is solving $f'' (x) = 0$ on its own not enough to locate a point of inflection?
Because the second derivative has to change sign there, not merely reach zero.
For $y = x^{4}$ the second derivative is $12 x^{2}$ , which is zero at $x = 0$ but positive on both sides, so the curve stays convex and there is no point of inflection.
What does a curve look like at a non-stationary point of inflection?
It carries on in the same direction, but switches which way it bends, so the tangent line passes through the curve rather than staying on one side of it.
That crossing is what makes an inflection visible on a sketch.
True or False?
Every cubic curve has exactly one point of inflection.
True.
Differentiating a cubic twice leaves a linear expression, which is zero at exactly one value of $x$ and changes sign as it passes through it.
That point is also the cubic's centre of symmetry.
What does the chain rule let you do when three quantities are connected?
Link their rates of change into a single equation, such as $\frac{d V}{d t} = \frac{d V}{d r} \times \frac{d r}{d t}$ .
That lets you work out a rate you cannot measure directly from two that you can.
A question gives you $\frac{d V}{d t}$ and asks for $\frac{d r}{d t}$ . What do you do?
Find a formula connecting $V$ and $r$ , differentiate it to get $\frac{d V}{d r}$ , then substitute into $\frac{d V}{d t} = \frac{d V}{d r} \times \frac{d r}{d t}$ and rearrange.
The formula you have to differentiate is usually a standard volume or surface area result.
Why does it matter which letters in the problem are constants?
Because a constant differentiates to zero and a variable does not, so treating one as the other changes the answer completely.
In a cone of fixed shape, for instance, the ratio of radius to height stays constant even though both of them are changing.
Define differential equation.
An equation that contains a derivative, in other words a rate of change.
Connected rates of change is how you set one up; solving it comes later, using integration.
True or False?
$\frac{d r}{d t}$ can be found by taking the reciprocal of $\frac{d t}{d r}$ .
True.
The reciprocal form of the chain rule holds for any pair of connected variables.
So an awkward rate can often be reached by writing the relationship the other way round and differentiating that instead.

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