Calculus Countdown

6552 /www/nrich/html/content/id/6552/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> In the game of Calculus Countdown you are given the following four machines into which you insert cards with functions written on them; the four machines chew up the input card(s) and spit out new cards with functions written on them. You can put any output cards back into the machines if you like. The idea of the game is to hit certain target cards given a set of initial cards. <mdo:image alt="" height="291" src="Images.png" width="594"></mdo:image> Let's play a game. You are given the following initial seven cards (no constants of integration from the integral machine and no repeats of cards, other than the pair of $e^x$s) <mdo:image alt="" height="35" src="Starting%20Cards.png" width="542"></mdo:image> Which of the following targets could you hit starting with these cards? You can use a fresh set of seven cards for each new target. <mdo:image alt="" height="36" src="Targets.png" width="463"></mdo:image> Can you make a smaller set of cards which could hit each of these targets? Why not invent your own set of starting cards and targets? </mdoxml> <?xml version="1.0" encoding="ISO-8859-1"?> <mdoxml version="1.0"> Game a): $\textrm{Target} = 8$ $\mathrm{D} \left( x^2 \right) = 2x$ $\mathrm{D} \left( 2x \right) = 2$ $\mathrm{P} \left( 4,2 \right) = 8$ Game b): $\textrm{Target} = x^4$ $\mathrm{P} \left( x, x^2 \right) = x^3$ $\mathrm{I} \left( x^3 \right) = \frac{x^4}{4}$ $\mathrm{P} \left( \frac{x^4}{4}, 4 \right) = x^4$ Game c): $\textrm{Target} = \frac{1}{2}$ $\mathrm{D} \left( x^2 \right) = 2x$ $\mathrm{D} \left( 2x \right) = 2$ $\mathrm{R} \left( 2 \right) = \frac{1}{2}$ Game d): $\textrm{Target} = \frac{x^6}{36}$ Method 1: $\mathrm{I} \left( \mathrm{I}(x) \right) = \frac{x^3}{6}$ $\mathrm{P} \left( x^2,\frac{x^3}{6} \right) = \frac{x^5}{6}$ $\mathrm{I} \left( \frac{x^5}{6} \right) = \frac{x^6}{36}$ Method 2: $\mathrm{I} \left( x^2 \right) = \frac{x^3}{3}$ $\mathrm{D} \left( \ln(x) \right) = \frac{1}{x}$ $\mathrm{R} \left( \frac{1}{x} \right) = x $ $\mathrm{P} \left( x, x \right) = x^2 $ $\mathrm{I} \left( x^2 \right) = \frac{x^3}{3}$ $\mathrm{P} \left( \frac{x^3}{3}, \frac{x^3}{3} \right) = \frac{x^6}{9}$ $\mathrm{R} \left( 4 \right) = 0.25 $ $\mathrm{P} \left( 0.25,\frac{ x^6}{9} \right) =\frac{x^6}{36}$ Game e): $\textrm{Target} = \frac{-32}{x^5}$ $\mathrm{D} \left( \mathrm{D} \left( \mathrm{D} \left( \ln(x) \right) \right) \right) = 2x^{-3}$ $\mathrm{R} \left( x \right) = \frac{1}{x}$ $\mathrm{P} \left( x^{-1}, 2x^{-3} \right) = 2x^{-4}$ $\mathrm{P} \left( 4, 2x^{-4} \right) = 8x^{-4}$ $\mathrm{D} \left( 8x^{-4} \right) = -32x^{-5}$ Game f): $\textrm{Target} = x(2 - x)$ $\mathrm{P} \left( x^2, \mathrm{R}(\exp(x)) \right) =x^2 \exp(-x))$ $\mathrm{D} \left( x^2 \exp(-x) \right) = 2x \exp(-x) - x^2 \exp(-x)$ $\mathrm{P} \left( \exp(x),2x \exp(-x) - x^2 \exp(-x) \right) = 2x-x^2$ </mdoxml> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=6552&part="> This problem</a> gives a great engaging context in which to practise calculus. It is very useful for getting across important ideas concerning integration and differentiation as operations, which will be useful at university in both science and maths courses. <h3>Possible approach</h3> <div>Initially play as a straightforward game to try to hit the targets.</div> <div>Students could then try to create new targets for their friends to try to hit using the same initial functions.</div> <div>You could then give the students the task of creating their own game of Calculus Countdown.</div> <div>Once students have hit a target they will need to communicate their answers clearly. How might they write down an answer in a clear and unambiguous way that can easily be interpreted by someone else?</div> <h3>Key questions</h3> <div>What happens when you integrate or differentiate the starting functions?</div> <div>What would you have to integrate or differentiate in one step to hit the targets? Does this help?</div> <h3>Possible extension</h3> <div>Interesting side questions which might emerge are: What targets are possible? Can you prove that certain targets (such as $\sqrt{x}$ or $3$) are impossible to hit? Clearly, proving the impossibility of a target will require some clear thinking; be sure that students concentrate on clearly writing down their arguments.</div> <h3>Possible support</h3> <div>Focus on the first 3 targets.</div> </mdoxml> <mdoxml version="1.0">To begin with, try acting on the starting functions with various operators to see the sorts of possibilities that occur. To hit a target, think what functions would be needed to make them in one step. How might I arrive at these inputs? </mdoxml> <mdoxml version="1.0"> Game 1 P(x, x^2) = x^3 I(x^3) = x^4/4 P(x^4/4, 4) = x^4 Game 2 D(x^2) = 2x R(2x) = 1/(2x) P(1/(2x), x) = 0.5 Game3 I(x^2) = x^3/3 Also, D(ln(x)) = 1/x R(1/x) = x P(x, x) = x^2 I(x^2) = x^3/3 P(x^3/3, x^3/3) = x^6/9 R(4) = 1/4 P(1/4, x^6/9) =x^6/36 Game 4 D(D(D(ln(x))) = 2/x^3 R(x) = 1/x P(1/x, 2/x^3) = 2/x^4 P(4, 2/x^4) = 8/x^4 D(8/x^4) = -32/x^5 Game 5 P(x^2, R(exp(x))) = x^2/exp(x) D(x^2/exp(x)) = 2x/exp(x) - x^2/exp(x) P(exp(x), last answer) = 2x-x^2 </mdoxml> 2 3 0 0 0 0 1 Calculus Countdown Can you hit the target functions using a set of input functions and a little calculus and algebra? Pre-Calculus and Calculus Calculus generally Pre-Calculus and Calculus Integration Pre-Calculus and Calculus Differentiation Algebra Manipulating algebraic expressions/formulae Applications Maths in STEM Admin Discussion