the Further Mathematics network www.fmnetwork.org.uk the Further Mathematics network www.fmnetwork.org.uk Teaching Topics: Calculus and Maclaurin Series in FP2 Let Maths take you Further FP2 in General For many students, FP2 is the end point for their school career in pure mathematics. It s excellent for improving students fluency in lots and lots of topics and send them off to their university or career destination as a confident mathematician. Don t worry about setting routine exercises, in my experience students seem to like them. Calculus in FP2 Students starting FP2 are hopefully about to become very fluent, confident users of calculus. My interest is in applications of calculus in computing. Good tip - start FP2 lessons with quickfire calculus. For a very good mark in FP2 a very high level of fluency with calculus and algebra is needed. Calculus in FP2 Calculus in FP2 In FP2 students find out how to differentiate arcsinx, arccos x, arctan x, sinh x, cosh x, tanh x, arsinh x, arcosh x, artanx x. This enables them to extend the class of functions that they can integrate enormously. 1
Revisiting and developing understanding of the chain rule In C3 and C4 calculus it s possible for students to learn types of derivative and almost begin to forget about the chain rule. e.g the derivative of sin(f(x)) is f`(x)cos(f(x)). In FP2 test students out by asking them to differentiate e.g. f(x) = arctan (4x) Graphs and Derivatives It s good to discuss with students the graphs of the inverse trig functions and their derivatives. Deriving the derivatives of arcsinx and arccosx using the inverse function theorem is recommended, students will enjoy this! Integration Proving these results A nice way to have students prove these results is by differentiation. e.g. Care with constants Examiners Report 2
Past Paper Question Examiners Report Using the standard formulae Move any constant off the x 2 term (Complete the square and substitute if there is an x term) Use the standard formula More difficult examples, completion of square Ask students to give you any quadratic, f(x), with no real roots, then integrate 1/f(x) Some examples require completion of square for a negative x 2 term. Even though this isn t examined very often, doing this with students is thoroughly recommended. Integrating rational functions Consider integration of 1/(ax 2 + bx + c). If b 2 4ac >= 0 then this can be done using partial fractions or by inspection. If b 2 4ac < 0, then, as seen, this can be done by completing the square, then using the derivative of arctan. This deals with ANY rational functions If f(x) = p(x)/q(x), then by using the fundamental theorem of algebra, long division and partial fractions it will be possible to integrate f(x). 3
Example Substitution requiring some thought Use of integration by parts Solution Examiner s Report Past Paper Question 4
Examiners Report Past Paper Question Examiner s Report Past Paper Questions Examiners Report Past Paper Question 5
Examiners Report Maclaurin Series A wonderful topic for discussion Motivates entirely the years that students have spent studying polynomials, polynomials are everything. Graphical approach is great. Idea that the whole function is captured in the information about it at 0. Graphical Demonstration f(x) = sin x Get students to do this by hand up to the x 2 and x 3 term. Ask them where the curves are cutting the x- axis. Get them to substitute values in to see how good the approximation is and how it is improving. Delving a bit deeper Define f 1 (x) = x sin x, f 2 (x) = cos x 1 + x 2 /2 f 3 (x) = sin x x + x 3 /6 f 4 (x) = 1 x 2 /2 + x 4 /24 cos x Differentiating these shows that they are strictly increasing for x > 0. It follows that for x > 0 x > sin x > x x 3 /6 Maclaurin Series - Advice Past Paper Question Remember the range of examples that students are expected to deal with Make sure that students know the short cuts for getting series from other series 6
Examiners Report Past Paper Questions Examiners Report The Story of Sine From opposite over hypotenuse..to the Maclaurin Series. First Definition Extending this to all the reals sin x is defined to be the ratio of the opposite side to angle x and the hypotenuse in a right angled triangle. sin x is now defined as in the diagram below (a, b) sin x = a cos x = b x 7
To Maclaurin Series Continuity Differentiability Rolle s Theorem The Mean Value Theorem Taylor s Theorem 8