9709. Pure 1. Applied Differentiation

Increasing and Decreasing Functions

There are two ways we can assess if functions are increasing or decreasing:

The function f(x) is:
- increasing on the interval [a,b] if f'(x) ≥ 0 for all values of x such that a < x < b.
- decreasing on the interval [a,b] if f'(x) ≤ 0 for all values of x such that a < x < b.

The function f(x) is:
- increasing on the interval [a,b] if f(a) ≤ f(b) for all values of x such that a < x < b.
- decreasing on the interval [a,b] if f(a) ≥ f(b) for all values of x such that a < x < b.

Worked Examples

Show that the function f(x) = x³ + 12x²+ 48x + 13 is increasing for all real values of x.
Find the interval on which the function f(x) = x³ + 3x² – 9x is decreasing.
Find the values of x for which f(x) is a decreasing function, given that f(x) equals:
1. x²-9x
2. 5x-x²
3. 4-2x-x²
4. 2x³-3x²-12x

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Stationary points

Any point on the curve y = f(x) where f'(x) = 0 is called a stationary point.

Stationary points can be a local maximum, a local minimum or a point of inflection. We can classify them as follows (for small positive value h):

If the function is easy to differentiate, it is often easier to identify the nature of stationary points by calculating the second derivative, f”(x).

If a function f(x) has a stationary point when x = a, then:

If f”(a) > 0, the point is a local minimum;
If f”(a) < 0, the point is a local maximum.

If f”(a) = 0, the point could be a local minimum, a local maximum, or a point of inflection. You will need to inspect points on either side to determine its nature.

Worked Example

Find the coordinates of the stationary points on the curve with equation y=2x³-15x²+24x+6
For (q1) find $\frac {dy}{dx}$ and use it to determine the nature of the stationary points.
The curve with equation $y=\frac{1}{x} + 27x^3$ has stationary points at x = ±a. Find the value of a.
Sketch the graph of $y=\frac{1}{x} + 27x^3$ .
By considering the gradient on either side of the stationary point on the curve y=x³-3x²+3x, show that this is a point of inflection.
Sketch the curve y=x³-3x²+3x

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Practical Applications of Maxima and Minima

In many disciplines, people are interested in achieving maximum amounts (e.g. profit, volume, effectiveness of drug) or minimum amounts (e.g. cost, pollution, noise) for given constraints on variables as represented by equations.

Worked Example

A closed box with a square base has a total surface area of 36m². Find the maximum possible volume of the box:

Worked Example

A farmer has a rectangular piece of land for pigs. One of the sides of the rectangle is a wall. The other three sides have fencing. The fencing is 80m in length. Find the maximum possible area of this rectangular piece of land.

Given that x+y=3, find the least possible value of x²+14y.

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Rates of change

We can consider $\frac{dx}{dt}$ as the rate of change of variable x over time.

In some problems, the form of the chain rule $(\frac{dy}{dx})(\frac{dx}{dt})=\frac{dy}{dx}$ will be useful to us (✤when there are two variables, both y and x varying over time).

Worked Example

The radius, r, of a circle, is increasing at the rate of $\frac{2}{r^2}ms^{-1}$ . Find the rate at which the area, A, is increasing when r = 8.

2. A spherical balloon is blown up so that its volume increases at a rate of 3cm³s^-1. Find the rate of increase of the radius of the balloon when the volume of the balloon is 60cm³.

3. The surface area, A, of a cube is increasing at the rate of 12cm²s^-1. Find the rate of increase of the volume, V, of the cube when each edge of the cube is 10cm.

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Practical Uses of Rates of Change

Worked Examples

1.) A cuboid has a square base. The height of the cuboid is twice the length of the side of the base. The surface area of the cuboid is increasing at a rate of 10cm³s^-1. Find the rate of increase of the volume of the cuboid when the height of the cuboid is 12cm.

2.) Paint is poured onto a table, forming a circle which increases at a rate of 2.5cm²s^-1. Find the rate the radius is increasing when the area of the circle is 20π cm².

3.) The surface area of a cube is increasing at 0.3m²s^-1. Find the rate of increase of the volume of the cube when the length of the side is 5m.

Questions and General Questions