We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byKaylin Crull
Modified about 1 year ago
© Boardworks Ltd of 54 Contents © Boardworks Ltd of 54 Using graphs to solve equations The change-of-sign rule Solving equations by iteration Review of the trapezium rule Simpson's rule Examination-style questions Simpson's rule
© Boardworks Ltd of 54 Simpson's rule When we used the trapezium rule we split the area that we were trying to find equal strips and then fitted straight lines to the curve. This led to approximations that often had a large percentage error. Simpson's rule also works by dividing the area to be found into equal strips, but instead of fitting straight lines to the curve we fit parabolas. The general form of a parabolic curve is y = ax 2 + bx + c. If we are given the coordinates of any three non-collinear points we can draw a parabola through them. We can find the equation of this parabola using the three points to give us three equations in the three unknowns, a, b and c.
© Boardworks Ltd of 54 Defining a parabola using three points
© Boardworks Ltd of 54 Simpson’s rule Consider the following parabola passing through the three points P, Q and R with coordinates (– h, y 0 ),(0, y 1 ) and ( h, y 2 ). If we take the equation of the parabola to be y = ax 2 + bx + c then we can write the area, A, of the two strips from – h to h as: x 0 y y0y0 y1y1 y2y2 P Q R We can use these points to define the area, A, divided by the ordinates y 0, y 1 and y 2, into two strips of equal width, h. –h–h h A
© Boardworks Ltd of 54 Simpson’s rule We can find a and c using the points (– h, y 0 ), (0, y 1 ) and ( h, y 2 ) to write three equations: When y = y 0 and x = – h : y 0 = a (– h ) 2 + b (– h ) + c = ah 2 – bh + c y 1 = c y 2 = ah 2 + bh + c When y = y 1 and x = 0: When y = y 2 and x = h : Adding the first and last equations together gives: y 0 + y 2 = 2 ah c y 0 + y 2 = 2 ah y 1 Since c = y 1
© Boardworks Ltd of 54 Simpson’s rule So, We can now use this to write the area, A, in terms of the ordinates y 0, y 1 and y 2 : In general, the area under any quadratic function, q ( x ), divided into two equal strips from x = a to x = b is given by: 2 ah 2 = y 0 + y 2 – 2 y 1 It is best not to isolate a because we actually want an expression for. where h =.
© Boardworks Ltd of 54 Simpson’s rule This forms the basis of Simpson’s rule where we divide the area under a curve into an even number of strips and fit a parabola to the curve across every pair of strips. The area of each pair of strips is taken to be approximately: If there are four strips (with 5 ordinates) the area will be: Adding another pair of strips would give the area as:
© Boardworks Ltd of 54 Simpson’s rule In general, for n strips the approximate area under the curve y = f ( x ) between the x -axis and x = a and x = b is given by: Where n is an even number and. This can be more easily remembered as: ( y first + y last + 4(sum of y odds ) + 2(sum of y evens )) This is Simpson’s rule. Don’t forget that Simpson’s rule can only be used with an even number of strips (or an odd number of ordinates).
© Boardworks Ltd of 54 x y 0 Simpson’s rule Let’s apply Simpson’s rule to approximate the area under the curve y = e –2 x between x = 0, x = 2 and the x -axis. Let’s use four strips as we did when we approximated this area before. 2 y = e –2 x y1y1 1 y2y2 y3y3 y4y4 y0y0 These five points define two parabolas over each pair of strips. The coordinates of the five points can be found using a table: x
© Boardworks Ltd of 54 Simpson’s rule Using Simpson’s rule with h = gives: = (to 3 s.f.) When we calculated this area using the same number of strips with the trapezium rule we obtained an area of (to 3 s.f.). This result had a percentage error of 8.15%. Comparing the area given by Simpson’s rule to the actual area gives us a percentage error of: Therefore, Simpson’s rule is much more accurate than the trapezium rule using the same number of strips.
© Boardworks Ltd of 54 Contents © Boardworks Ltd of 54 Using graphs to solve equations The change-of-sign rule Solving equations by iteration Review of the trapezium rule Simpson's rule Examination-style questions
© Boardworks Ltd of 54 Examination-style question 1 a)Use Simpson's rule with five ordinates to find an approximate value for to 3 decimal places. b) Use integration by parts to find the exact value of the definite integral given in part a). c) Give the percentage error of the approximation found in part a) to 2 decimal places. a) If there are five ordinates there are four strips. The width of each strip is therefore
© Boardworks Ltd of 54 Examination-style question 1 Using Simpson's rule: 321 x 0 b)Let and So and
© Boardworks Ltd of 54 Examination-style question 1 Integrating by parts: c) The percentage error is 3.28%
Integrals 5. Approximate Integration Approximate Integration There are two situations in which it is impossible to find the exact value of a definite.
© Boardworks Ltd of 49 Approximating the area under a curve Sometimes the area under a curve cannot be found by integration. This may be because.
Graphing Quadratic Functions y = ax 2 + bx + c. Quadratic Functions The graph of a quadratic function is a parabola. A parabola can open up or down. If.
© Boardworks Ltd of 36 © Boardworks Ltd of 36 AS-Level Maths: Core 2 for Edexcel C2.8 Integration This icon indicates the slide contains.
Topic: U2 L1 Parts of a Quadratic Function & Graphing Quadratics y = ax 2 + bx + c EQ: Can I identify the vertex, axis of symmetry, x- and y-intercepts,
M.M. 10/1/08 What happens if we change the value of a and c ? y=3x 2 y=-3x 2 y=4x 2 +3 y=-4x 2 -2.
Graphing Quadratic Functions Definitions Rules & Examples Practice Problems.
Find the x -intercept and y -intercept 1.3x – 5y = 15 2.y = 2x + 7 ANSWER (5, 0); (0, –3) ANSWER (, 0) ; (0, 7) 7 2 –
Using the Quadratic Formula to Solve a Quadratic Equation.
By definition 25 is the number you would multiply times itself to get 25 for an answer. Because we are familiar with multiplication, we know that 25.
Simpson’s Rule Mini Whiteboards To check your understanding of the video.
You can use a quadratic polynomial to define a quadratic function A quadratic function is a type of nonlinear function that models certain situations.
SAT Problem of the Day. 5.5 The Quadratic Formula 5.5 The Quadratic Formula Objectives: Use the quadratic formula to find real roots of quadratic equations.
D MANCHE Finding the area under curves: There are many mathematical applications which require finding the area under a curve. The area “under”
Lesson 10-1 Graphing Quadratic Equations. Definitions Quadratic Function - can be written in the form of y = ax 2 + bx + c, where a 0. Parabola - The.
Table of Contents Quadratic Equation: Solving Using the Quadratic Formula Example: Solve 2x 2 + 4x = 1. The quadratic formula is Here, a, b, and c refer.
Graphing Quadratics. Finding the Vertex We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x – coordinate.
Essential Question: How do you graph a quadratic function in standard form? Students will write a summary on graphing quadratic functions in standard form.
3.1 Notes: Solving Quadratic Equations By graphing and square roots.
8 TECHNIQUES OF INTEGRATION. There are two situations in which it is impossible to find the exact value of a definite integral. TECHNIQUES OF INTEGRATION.
~ Chapter 10 ~ Quadratic Equations and Functions Algebra I Lesson 10-1 Exploring Quadratic Graphs Lesson 10-2 Quadratic Functions Lesson 10-3 Finding &
Quadratic Functions By: Cristina V. & Jasmine V..
Finding Areas Numerically. y4y4 h y5y5 The basic idea is to divide the x-axis into equally spaced divisions as shown and to complete the top of these.
Finding Approximate Areas Under Curves. The Trapezium Rule y 0 y 1 y 2 y 3 y 4 y 5 This curve has a complicated equation so instead of integrating split.
Graphing Quadratic Functions y = ax 2 + bx + c. All the slides in this presentation are timed. You do not need to click the mouse or press any keys on.
© Boardworks Ltd of 33 CO-ORDINATE GEOMETRY OF THE LINE.
Graphs of Quadratic Equations. Standard Form: y = ax 2 +bx+ c Shape: Parabola Vertex: high or low point.
Exam Questions relay. Simpson’s Rule Objectives: To recognise and apply Simpson’s rule to approximate areas bounded by curves.
QUADTRATIC RELATIONS. A relation which must contain a term with x2 It may or may not have a term with x and a constant term (a term without x) It can.
Name: Date: Topic: Solving & Graphing Quadratic Functions/Equations Essential Question: How can you solve quadratic equations? Warm-Up : Factor 1. 49p.
Contents 8.2 Problems Leading to Quadratic Equations 8.3 Solving Simultaneous Equations by Algebraic Method 8.4 Graphical Solutions of Simultaneous Equations.
QUADTRATIC RELATIONS Standard Form. GRAPHING QUADRATICS IN STANDARD FORM Step 1: determine the line of symmetry and the vertex This is a little more difficult.
Algebra 9.3 Graphing Quadratic Functions. CLASSIFYING EQUATIONS LINEAR QUADRATIC y = 2x + 4 y = 2x²+ 7x + 3 y = 5x²y = 5x y = x² - 4 y = x - 4 What is.
Integration. Problem: An experiment has measured the number of particles entering a counter per unit time dN(t)/dt, as a function of time. The problem.
Graphing Quadratic Functions Quadratic functions have the form: y = ax 2 + bx + c When we graph them, they make a parabola!
21: Simpson’s Rule © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.
The General Quadratic Function Students will be able to graph functions defined by the general quadratic equation.
© Boardworks Ltd of 43 A5 Simultaneous equations KS4 Mathematics.
Vertex Form of A Quadratic Function. y = a(x – h) 2 + k The vertex form of a quadratic function is given by f (x) = a(x - h) 2 + k where (h, k) is the.
Homework questions thus far??? Section 4.10? 5.1? 5.2?
Copyright © Cengage Learning. All rights reserved. 4 Integrals.
© Boardworks Ltd of 45 © Boardworks Ltd of 45 AS-Level Maths: Core 1 for Edexcel C1.7 Differentiation This icon indicates the slide contains.
Write an equation for a function EXAMPLE 3 Tell whether the table of values represents a linear function, an exponential function, or a quadratic function.
© Boardworks Ltd of 50 © Boardworks Ltd of 50 AS-Level Maths: Core 1 for Edexcel C1.2 Algebra and functions 2 This icon indicates the slide.
Lesson 10-1 Graphing Quadratic Functions. Objectives Graph quadratic functions Find the equation of the axis of symmetry and the coordinates of the vertex.
© Boardworks Ltd of 48 Solving by factorization 1.Solving by factorization 2.Solving by completing the square 3.Solving by using the quadratic equation.
5.1 – Introduction to Quadratic Functions Objectives: Define, identify, and graph quadratic functions. Multiply linear binomials to produce a quadratic.
Solving Equations Containing To solve an equation with a radical expression, you need to isolate the variable on one side of the equation. Factored out.
© 2017 SlidePlayer.com Inc. All rights reserved.