5.1 Limits and the derivativeIB Maths: Analysis and Approaches SL: Revision notes

A limit is the value that an expression gets closer and closer to as its input approaches some value. We write $\lim_{h\to0}(4+h)=4$ and read it as 'the limit of $4+h$ as $h$ tends to 0 is 4'.

A limit can exist even when the expression itself cannot be evaluated at that point. For example $\frac{4h+h^{2}}{h}$ is undefined at $h=0$, but for every other $h$ it equals $4+h$, so its limit as $h\to0$ is 4.

At SL you only need an informal idea of a limit: formal analytic methods of calculating limits are not required.

You may be given values of an expression as the input gets closer to a point, often from both sides. Look for the value the outputs settle towards.

For the cooling coffee, average rates of change over intervals starting at $t=10$ were $-2.0707,\,-2.1176,\,-2.1223$, and over intervals ending at $t=10$ were $-2.1768,\,-2.1282,\,-2.1234$. Both sequences close in on about $-2.12$, so this is the estimate of the limit.

The values closest to the point are the most reliable; the widest interval gives the worst estimate.

The gradient of a straight line is constant, but the gradient of a curve changes from point to point. To find it at a point $P$, take a nearby point $Q$ on the curve and find the gradient of the chord $PQ$: $\frac{f(x+h)-f(x)}{h}.$ As $Q$ slides towards $P$ ($h\to0$), the chord approaches the tangent at $P$, and its gradient approaches the gradient of the curve at $P$.

Example: for $f(x)=x^{2}-2x$ at $x=3$, the chord gradient simplifies to $4+h$, which tends to 4. So the gradient of the curve at $x=3$ is 4.

The derivative of $f$ gives the gradient of the curve at every point, so it is also called the gradient function. It is the instantaneous rate of change of one quantity with respect to another.

Notation: if $y=f(x)$, the derivative is written $\frac{\mathrm{d}y}{\mathrm{d}x}$ or $f'(x)$. Other letters are used in context: $\frac{\mathrm{d}V}{\mathrm{d}r}$ is the rate of change of volume with respect to radius, and $\frac{\mathrm{d}s}{\mathrm{d}t}$ is the rate of change of displacement with respect to time (velocity).

The units of a derivative are (units of the top quantity) per (unit of the bottom quantity), for example m s$^{-1}$ or °C per minute. A negative derivative means the quantity is decreasing.

The average rate of change over an interval is the gradient of a chord: $\frac{\text{change in }y}{\text{change in }x}.$ The instantaneous rate of change at a point is the gradient of the tangent: the limit of average rates over shorter and shorter intervals.

For a stone falling with $s=4.9t^{2}$, the average speed over the first 2 s is 9.8 m s$^{-1}$, but the average speed from $t=2$ to $t=2+h$ is $19.6+4.9h$, which tends to 19.6 m s$^{-1}$. The speed at the instant $t=2$ is twice the average over the first 2 seconds.