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RULES

In the usual game of Countdown you have to hit certain numerical targets using a set of starting numbers and the operations $+, -, \div, \times$ of addition, subtraction, division and multiplication, with each initial number or result of an operation being used at most one time.

In the game of Calculus Countdown, we need to hit certain target functions using a set of starting functions and the operations of differentiation, implicit integration (no constant of integration), product of two functions and reciprocal of a function. We can call these operations
$$D(\cdot), I(\cdot), P(\cdot, \cdot), R(\cdot),$$
where the dots $\cdot$ indicate that a function needs to be slotted in.You can use ONLY these operations: for example, 'addition of functions' or 'function of a function' are not allowed.

For example, $D(x^3) = 3x^2$, $I(x^{\frac{1}{2}}) = \frac{2}{3}x^{\frac{3}{2}}$, $P(\sqrt{x}, \frac{1}{\sqrt{x}}) = 1$ and $R(3x^{-7})=\frac{1}{3}x^7$.

Each initial function can be used at most one time in any attempt to hit the target, but you are allowed to use the results of any operations as an input to a new operation as in usual countdown.


Now play the countdown game!

$$\mbox{Starting functions: }x, x^2, 4, \ln(x), \exp(x), \exp(x) $$

Target a) $8$

Target b): $x^4$

Target c): $0.5$

Target d): $x^6/36$

Target e) $-\frac{32}{x^5}$

Target f) $x(2-x)$


Can you hit any of these targets in more than one way?

What is the most amazing target that you could hit using these initial functions?

What numbers would be possible targets?

Make up your own set of starting functions and targets. Be as devious as possible!

p.s. there are supposed to be two copies of $\exp(x)$!

See also the problem Operating Machines

Polynomials. Simultaneous equations. Manipulating algebraic expressions/formulae. Integration. Differentiation. Calculus generally. Generalising. Mathematical reasoning & proof. Creating expressions/formulae. Factorisation (algebraic).