2. The Fundamental Theorem of Calculus
Calculus was developed by the work of several mathematicians from the 17th to 18th century
Newton and Leibniz (Gottfried) are mainly credited with its definition.
So what is Calculus and what does it involve?

3. What’s the gradient of this line?

4. What is a Gradient? m = (y₂-y₁) (x₂-x₁)
The gradient of a line is the slope of the line. Gradient is defined change in y value/the change in x value
m = (y₂-y₁)
(x₂-x₁)

5. How did you calculate the gradient of the line?
Y2-y1
How did you calculate the gradient of the line?
Example:
Chose two points on the line: (2,1) & (3,3)
Apply:
m = (y₂-y₁)
(x₂-x₁)
= (3-1)
(3-2)
= 2 1
The line has a gradient of 2

6. What’s the gradient of this function?
What are the issues of finding gradients of quadratics?

7. Using Chords to approximate gradients at different points on a curve
Can we use the same method as we used for finding straight lines to obtain an approximation of a gradient?

8. Finding the gradient at a specific point
Can you find the best approximation for the gradient at point (2,4)?
How could you apply the technique using chords in order to find the gradient at (2,4)
Use resource handed out

9. Recap………….
Gradient of a curve at a specific point (A) is defined as being the same as the gradient of the tangent (t) to the curve at that point.
You cannot calculate the gradient of the tangent directly (need 2 points).
To find the gradient of the tangent at a point, you can find the gradient of the chords (c) joining the point (A) to other points on the curve (B).
The closer the chord gets to the point (A), the more accurate the approximation of the gradient of the tangent.
y = x²
M = 4 – 1
2 - 1
M = 3 B c t A

10. Using δ as an infinitesimal increase in x.P
The increase in x and y needed in order to calculate the exact gradient of the tangent at a point is so small it cannot be distinguished from 0.
Using the notation δ what are the (x,y) coordinates for point A…?
…And for point P?
How can we use this information to calculate the gradient at point A?
Y = x²
( x+δ,(x+δ)² ) c
(x, x²) A t

11. Finding the formula for the gradient of y = x²P
Gradient of the chord AP is:
y = x²
( x+δ,(x+δ)² ) c
(x, x²) A t

12. The gradient formula for y = f(x)
Gradient on AP =
f(x+δ) – f(x)
(x+δ) – x
This simplifies to?
So as δ becomes infinitesimally small and the gradient becomes close to the gradient of the tangent, the definition of f'(x) is given as:
Lim f(x+δ) – f(x)
δ δ P
( x+δ, f(x+δ) ) A
( x, f(x) )

13. Lim f(x+δ) – f(x) δ δ
Using this formula, differentiate the functions in front of you
Write derivative on the blue card
Do you notice anything about the relationship between the original function and the derivative?

14. An introduction to integration
Thursday 22nd September 2011
Newton Project

15. How to find the area under the curve
In this presentation we are going to look at how we can find the area under a curve. In this case the area we are looking to find is the area bounded by both the x – axis and the y-axis.
We will then consider how integration might help us do this.

16. Using rectangles to estimate the area
Maybe we could divide the area into rectangles?

17. Can we make the approximation better?
Insert YouTube Mr Barton’s Maths – Area under a curve
Use YouTube Mr Barton’s Maths – Area under a curve

18. Is there a better way? Hint .....Area of a trapezium
A trapezium is a quadrilateral that has only one pair of parallel sides.
Consider the area of the following trapezium. a
Area of a Trapezium = (a+b) x h 2 h b

19. Deriving the Formula
Area of a Trapezium: ½ h( a+b) T1 = ½ h(y0+y1) T2 = ½ h(y1+y2) T3 = ½ h(y2+y3) … T4 = ½ h(yn-1 +yn) Whole Area is the addition All of the Trapeziums: A= ½ h(y0+y1+y1+y2+y2+y3+ yn-1 +yn) A = ½ h(y0 + 2(y1+y2+y3+yn-1)+ yn)

20. Now some examples!
Use Lori’s handout of examples

21. Integration
The next part of this presentation explains the concept of integration, and how we can use integration to find the area under a curve instead of using the trapezium rule.
We saw earlier in the presentation that:
We will see later in the presentation that integration is the opposite of differentiation:
Where K is any constant, K is called the constant of integration
means the integral of .....with respect to x.
i.e. to integrate a power of x, increase the power by 1 and divide by the new power.

22. Integration
Consider a typical element bounded on the left by the ordinate through a general point P(x,y).
The width of the element represents a small increase in the value of x and so can be called
Also, if A represents the area up to the ordinate through P, then the area of the element represents a small increase in the value of A and so can be called
A typical strip is approximately a rectangle of height y and width Therefore, for any element
The required area can now be found by adding the areas of all the strips from x=a to x=b
P(x,y)
x=a
x=b

23. Integration The notation for the summation of Total Area is so
as gets smaller the accuracy of the results increases
Until in the limiting case
Total Area =

24. Appendix

25. Integration – Extra Material
can also be written as
As gets smaller
But so
Therefore,
The boundary values of x defining the total area are x=a and x=b so this is more correctly written as