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Differentiate the following functions: $\displaystyle x ^ n \sinh(ax + b),\;\tanh(cx+d),\;\ln(\cosh(px+q))$

Write complex numbers in real-imaginary form.

Poles, residues, and contour integral of a complex-valued function.  Pair of pure imaginary poles.

Poles, residues, and contour integral of a complex-valued function.  Pair of real poles.

Poles, residues, and contour integral of a complex-valued function.  Single, simple pole.

Contour integral of $z^2$ along any path.

Contour integral of $\mathrm{e}^{-z}$ along any path.

Contour integral of a complex-valued function $f(z)$ with the poles of $f(z)$ either inside or outside the path $C$.

Given a payoff table with two states and five actions, identify which actions are admissible, then the maximax, maximin, and minimax regret actions.

Given a payoff table with two states and five actions, identify which actions are admissible, then the maximax, maximin, and minimax regret actions.

Given a payoff table with two states and five actions, identify which actions dominate others, and identify admissible actions.

Express $f(z)$ in real-imaginary form, given that $z=x+iy$.

Given $f(x)=(x+b)^n$. Find the gradient and equation of the chord between $(a,f(a))$ and $(a+h,f(a+h))$ for randomised values of $a$, $b$ and $h$.

Express $f(z)$ in real-imaginary form, given that $z=x+iy$, where $f(z)$ involves hyperbolic functions.

Modulus and argument of a single complex number, where $\mathrm{Re}(z)=\mathrm{Im}(z)$.

Expressing $\log(f(i))$ in the form $u+iv$.  Principal values of log.

Find the roots of $\sin(z)=a$.

Questions on Pearson and Spearman correlation coefficients.

For given optimisation problems, determine maximin, maximax, and minimax regret actions, expected value criteria, expected value of perfect information.

Questions on differentiation from first principles, and continuity and differentiability.

Use the chain rule to differentiate various functions.

Use the product rule to differentiate various functions.

Elementary examples of multiplication and addition of complex numbers. Four parts.

Composite multiplication and division of complex numbers. Two parts.

Direct calculation of low positive and negative powers of complex numbers. Calculations involving a complex conjugate. Powers of $i$. Four parts.

Find modulus and argument of two complex numbers. Then use De Moivre's Theorem to find negative powers of the complex numbers.

Determine the long-term behaviour of 1D dynamical systems.

Fixed points of 2D dynamical systems.

Eight questions on finding least upper bounds and greatest lower bounds of various sets.

Fixed points of a 1D dynamical system.