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?

http://www.youtube.com/watch?v=iUFmtbTvTPU.

What’s the gradient of this line?

What is a Gradient?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)

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 = 2The line has a gradient of 2

Y2-y1

What’s the gradient of this function?

What are the issues of finding gradients of quadratics?

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?

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

Recap………….

A

M = 4 – 1

2 - 1

M = 3

•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.

t

c

B

y = x²

Using δ as an infinitesimal increase in x.

• 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?

c

t

A

PY = x²

(x, x²)

( x+δ,(x+δ)² )

Finding the formula for the gradient of y = x²

• Gradient of the chord AP is:

c

t

A

Py = x²

(x, x²)

( x+δ,(x+δ)² )

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 0 δ

A

P

( x, f(x) )

( x+δ, f(x+δ) )

Lim f(x+δ) – f(x) 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?

δ 0

1)( nn nxxdx

d

An introduction to integration

Thursday 22nd September 2011 Newton Project

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.

Using rectangles to estimate the area

Maybe we could divide the area into rectangles?

Can we make the approximation better?

• Insert YouTube Mr Barton’s Maths – Area under a curve

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.

Area of a Trapezium = (a+b) x h 2

h

a

b

Deriving the FormulaArea 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)

Now some examples!

• 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.

1)( nn nxxdx

d

Kxn

dxx nn

1

1

1

dx......

• 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

.x

.A

xyA

Y

Xx=a x=b.x

P(x,y)

x

.A

• 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 =

bx

ax

bx

ax

xyA

bx

ax

A

x

bx

axx

xy0

lim

Appendix

• 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

xyA yx

A

x yx

Ax

0lim

dx

dA

x

Ax

0lim y

dx

dA

xyA

xyAreaTotalb

a