Nuffield Mathematics teaching resources are for use in secondary and further education

# FSMQ Level 3 (pilot) Calculus scheme of work

Before starting this A2 level FSMQ students are expected to:

• Be able to use algebraic methods to rearrange and solve linear and quadratic equations including use of at least one of: graphic calculator, the formula $\frac{-b&space;\pm&space;\sqrt{b^2&space;-&space;4ac}}{2a}$ (must be memorised) and completing the square.  (Factorisation may be used where appropriate.)
• Be familiar with the graphs of basic functions (powers of x, quadratic, trigonometric, exponential and logarithmic functions) and how geometric transformations can be applied to them using transformations by the vectors $\begin{bmatrix}&space;a&space;\\&space;0&space;\end{bmatrix}$ and $\begin{bmatrix}&space;0&space;\\&space;a&space;\end{bmatrix}$ and stretches of scale factor a with the invariant lines x = 0 and y = 0.
• Be able to do graph plotting by hand and using either computer software or a graphical calculator (including zoom and trace facilities)

Students who have completed the AS level Use of Mathematics Algebra unit will have covered these topics.

A suggested work scheme showing topics and methods to be covered is given below, but the order and time allocations can be varied to suit different groups of students.

Note that AQA assessment of this FSMQ is by examination only.

The following techniques should be introduced as soon as possible and used throughout the course:

• Using a calculator effectively and efficiently, including the use of memory and function facilities (recording the working as well as the result)
• Doing calculations without a calculator using written methods and mental techniques
• Checking calculations using estimation, inverse operations, and different methods.
 Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Introduction to calculus (4 hours) What is calculus? Brief revision of gradients of straight lines and curves (including real contexts).  Positive, negative and zero gradients and their interpretation. Sketch graphs of gradient functions. Find and interpret area under graphs (eg speed/time, acceleration/time) using areas of triangles, rectangles and trapezia. Speed and distance  Students explore the idea that the area under speed-time graphs can be used to find the distance travelled.

 Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Gradient functions (7 hours) Calculate the gradient at a point a on a function y = f(x) using the numerical approximation: gradient of tangent $\approx&space;\frac{f(a&space;+h)&space;-&space;f(a)}{h}$ where the interval  h is small. Improve the result by using smaller intervals. Interpret gradients in terms of their physical significance and use the correct units to measure gradients/rates of change. Sketch graphs of gradient functions (including curves not given as functions as well as curves defined as functions).  Identify the key features of gradient functions in terms of gradients of the original functions (including zeros of gradient functions linking to local turning points). Use $\frac{f(x&space;+&space;h)&space;-&space;f(x)}{h}$   to generate gradient functions leading to gradient of $y&space;=&space;x^n$ being  $\frac{dy}{dx}&space;=&space;nx^n^-^1$ Differentiate polynomials and sums and differences of other powers of x using notation $\frac{dy}{dx}$  and ${f}'(x)$ . Gradients   Presentation introducing differentiation, and a worksheet and Excel spreadsheet which students can use to calculate gradients from increments. Derivative matching Twelve sets of cards, each consisting of a polynomial function, its graph, its gradient function and the graph of the gradient function – for students to match. Areas under curves (4 hours) Estimate the areas under graphs of functions using numerical methods (including the trapezium rule).  Consider over and under-estimates and improving accuracy by using a smaller interval. Coastal erosion A    This activity uses the context of coastal erosion to introduce the trapezium rule for estimating the area under a curve.  (Could be combined with Coastal Erosion B.) Integration (7 hours) Find areas under curves, between x = a and x = b using  $\int&space;\begin{matrix}&space;b&space;\\&space;a&space;\end{matrix}&space;f(x)dx$,  $\left&space;(&space;f&space;(x)&space;\geq&space;0&space;\right&space;)$ Integration as the reverse of differentiation for xn (excluding n = – 1) and constants. Simple integration rules including sums, differences and multiplication of powers of x by a constant.  Include correct notation and constant of integration. Definite integration Coastal erosion B   Students use integration to estimate loss of land due to coastal erosion.  (Could be combined with 'Coastal erosion A'.) Area under a graph     Introduces integration using the area under velocity-time graphs. This extends the content of the 'Speed and distance' activity. Slide presentation, notes and exercises. Mean values   Students use area formulae and integration to find mean values of speed, share prices and water depth.  Supported by slide presentation.

 Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Second derivatives (6 hours) Find second derivatives using notation:  $\frac&space;{d^2y}{dx^2}$ and ${f}''(x)$ Identify key features of a second derivative – linking positive values to increasing gradient, negative values to decreasing gradient and zeros to points of inflexion. Apply second derivatives to gradients, maxima, minima and stationary points, increasing and decreasing functions.  Include examples where zero values of second derivatives occur at maximum and minimum points as well as points of inflexion. Stationary points Presentation, examples and practice questions on sketching graphs. Maxima and minima Presentation and practice questions using differentiation to solve maxima and minima problems. Containers    Activity in which students use differentiation to minimise surface area. Maximising and minimising  Presentation including worked examples for use in discussion about maximum and minimum problems that students could investigate. More rules of differentiation (8 hours) Differentiate: · trigonometric functions $A~\textup{sin}(mx&space;+&space;c)$  $A~\textup{cos}(mx&space;+&space;c)$ using radians) · exponential functions $ke^m^x$ (m positive or negative) · sums and differences of functions · products of functions. Exponential rates of change   Introduction to differentiation of exponential functions.  Students draw tangents to curves, then investigate gradient functions using a spreadsheet. More integration (8 hours) Integrate: · trigonometric functions $A~\textup{sin}(mx&space;+&space;c)$, $A~\textup{cos}(mx&space;+&space;c)$ · exponential functions $ke^m^x$ (m positive or negative) · x–1 · sums and differences of functions Include integration by inspection  such as $\int&space;e^-^5^xdx$ , $\int\textup{sin&space;}6x~dx$ Integrate by one use of integration by parts, such as $\int&space;xe^-^5^xdx$ , $\int~x~\textup{cos&space;}4x~dx$ , $\int&space;x&space;~\textup{ln&space;}x~dx$ Include the constant of integration and calculate it in known situation. Find definite integrals.  Apply integration in real contexts. That’s a lot of rock!  Students fit a curve to the cross section of a tunnel, then use integration and numerical methods to estimate the volume of rock removed and the time taken. This could be used as a classroom activity or for homework. Differential equations (8 hours) Use integration to find families of solutions to  first order differential equations with separable variables.  Find particular solutions when boundary conditions are given. Drug clearance   Data sheet shows how drug clearance after taking a painkiller can be modelled by exponential decay. Students investigate further (including the clearance of caffeine after a variety of drinks). What's it worth? Students investigate a variety of suggested models for the depreciation of a car. Involves solving differential equations, drawing sketch graphs and comparison with real data. Revision (8 hours) Revise topics.  Work through revision questions and practice papers. Discuss the data Sheet - make up and work through questions based on it.

Page last updated on 25 January 2012