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

2. Achilles and the tortoise: Zeno’s paradox
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3. 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?

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

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

6. 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?

7. The gradient of a quadratic functionP2 P1

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

9. 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

10. 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?

11. 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?

12. 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:

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

15. What is special about these points?

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

17. 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?

18. 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

19. Sketching more difficult graphs+ - + + -

20. 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

21. 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: