2The Fundamental Theorem of Calculus Calculus was developed by the work of several mathematicians from the 17th to 18th centuryNewton and Leibniz (Gottfried) are mainly credited with its definition.So what is Calculus and what does it involve?
4What 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 valuem = (y₂-y₁)(x₂-x₁)
5How did you calculate the gradient of the line? Y2-y1How 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)= 21The line has a gradient of 2
6What’s the gradient of this function? What are the issues of finding gradients of quadratics?
7Using 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?
8Finding 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
9Recap………….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 = x²M = 4 – 12 - 1M = 3BctA
10Using δ as an infinitesimal increase in x. PThe 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)At
11Finding the formula for the gradient of y = x² PGradient of the chord AP is:y = x²( x+δ,(x+δ)² )c(x, x²)At
12The gradient formula for y = f(x) Gradient on AP =f(x+δ) – f(x)(x+δ) – xThis 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 δP( x+δ, f(x+δ) )A( x, f(x) )
13Lim f(x+δ) – f(x) δδUsing this formula, differentiate the functions in front of youWrite derivative on the blue cardDo you notice anything about the relationship between the original function and the derivative?
14An introduction to integration Thursday 22nd September 2011Newton Project
15How 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.
16Using rectangles to estimate the area Maybe we could divide the area into rectangles?
17Can we make the approximation better? Insert YouTube Mr Barton’s Maths – Area under a curveUse YouTube Mr Barton’s Maths – Area under a curve
18Is 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.aArea of a Trapezium = (a+b) x h2hb
19Deriving 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)
20Now some examples!Use Lori’s handout of examples
21IntegrationThe 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 integrationmeans 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.
22IntegrationConsider 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 calledAlso, 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 calledA typical strip is approximately a rectangle of height y and width Therefore, for any elementThe required area can now be found by adding the areas of all the strips from x=a to x=bP(x,y)x=ax=b
23Integration The notation for the summation of Total Area is so as gets smaller the accuracy of the results increasesUntil in the limiting caseTotal Area =