CIE Centre A-level Pure Maths P1 Chapter 6 CIE Centre A-level Pure Maths © Adam Gibson

Achilles and the tortoise: Zeno’s paradox 1m/s 5m/s 100m 40m START

Paradoxes are resolved using limits We can use this mathematical notation: As you can easily prove, because the series is convergent, the sum is not infinity! Using the same idea, we can easily give the value of expressions like: What is the value of this limit?

The meaning of gradient Suppose you were climbing a hill where the gradient is steepest Where is the climb most difficult?

The gradient is changing y B We can estimate the gradient by drawing small triangles A ∆y Where is the gradient zero? ∆x x

Gradients and tangents The tangent to a curve at a point P is the straight line which touches the curve at P, but does not cross it P The gradient can be estimated using a secant line, but the exact value of the gradient is the gradient of the tangent. How can we use the idea of a limit to find the gradient of the tangent?

The gradient of a quadratic function P2 P1

Differentiating a curve Remembering that The gradient of the secant between P1 and P2 is: Now … what happens as P2 slowly approaches P1 ?

Answer: we get a limit! We have a special way of writing the limit of the secant’s gradient as h gets smaller: This is the “derivative” of f(x). You cannot cancel “d” Careful! dy means: “an infinitesimally small change in y” dx means: “an infinitesimally small change in x” These expressions are not a product involving a number d

Therefore: is the magic equation that can give us the gradient of the tangent at x=x1 Can you now calculate dy/dx for f(x)=x2, at x=x1?

Let’s examine the graph again: Can you see how the gradient changes as we move P1 along the curve? P2 P1 P3 What is the gradient when x=-2?

Summary and vocabulary The gradient of a function at a point is the gradient of the tangent to the curve at that point To find the gradient, we must differentiate the function The differential or derivative is written like this: Say “dy by dx or just dydx” It can be inferred that the derivative of f(x) is:

TASKS Use the same procedure to find the derivative of the following functions: What patterns or rules do you notice?

What is special about these points?

? So far we have seen: ? … but what if Example: Use the limit formula to find the derivative ?

It turns out we can use any rational number n – except? .. can use the “difference of two squares” We can cancel “h” again! Amazingly, still : (with n= 0.5) It turns out we can use any rational number n – except?

TASKS Sketch the graph of the function Prove that Use this to find the equation of the tangent at x = 2 and sketch it on your graph Extra task Use the limit formula to find the derivative of

Sketching more difficult graphs + - + + -

Increasing and decreasing functions When the function is increasing , the gradient dy/dx is positive When the function is decreasing , the gradient dy/dx is negative Formally, f(x) is increasing over the interval f(x) is decreasing over the interval

A tractable example Consider Over which intervals is it decreasing? Find the derivative: Set dy/dx = 0. What are the solutions? So the function is decreasing in the interval: