Integrate the product of two functions by the method of integration by parts.

Integrate various functions by rewriting them as partial fractions.

Approximate the integral of a function by Riemann sums.

The derivative of $\displaystyle x ^ {m}(ax^2+b)^{n}$ is of the form $\displaystyle x^{m-1}(ax^2+b)^{n-1}g(x)$. Find $g(x)$.

Approximating integral of a quadratic by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.

Approximating integral of a linear function by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.

Differentiate $ (a+bx) ^ {m} \sin(nx)$

Differentiate $f(x)=x^{m}\sin(ax+b) e^{nx}$.

The answer is of the form:
$\displaystyle \frac{df}{dx}= x^{m-1}e^{nx}g(x)$ for a function $g(x)$.

Find $g(x)$.

Differentiate the function $(a + b x)^m  e ^ {n x}$ using the product rule.

Differentiate the function $f(x)=(a + b x)^m  e ^ {n x}$ using the product rule. Find $g(x)$ such that $f\;'(x)= (a + b x)^{m-1}  e ^ {n x}g(x)$.

Finding the coordinates and determining the nature of the stationary points on a polynomial function

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Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$. Solution involves the inverse trigonometric function $\arcsin$.

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$. Solution involves the inverse trigonometric function $\arcsin$.

An object moves in a straight line, acceleration given by:

$\displaystyle f(t)=\frac{a}{(1+bt)^n}$. The object starts from rest. Find its maximum speed. 

Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots. 

Find $\displaystyle \frac{d}{dx}\left(\frac{m\sin(ax)+n\cos(ax)}{b\sin(ax)+c\cos(ax)}\right)$. Three part question.

Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.

Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.

Inputting the values given into the partial derivatives to see if 0 is obtained is tedious! Could ask for the factorisation of equation 1 as the solution uses this. However there is a problem in asking for the input of the stationary points - order of input and also giving that there is two stationary points.

Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.

Inputting the values given into the partial derivatives to see if 0 is obtained is tedious! Could ask for the factorisation of equation 1 as the solution uses this. However there is a problem in asking for the input of the stationary points - order of input and also giving that there is two stationary points.

Dealing with functions in Numbas.

Multiple response question (2 correct out of 4) covering properties of continuity and limits of functions. Selection of questions from a pool.

A quadratic function $ax^2+bs+c$ is given. Six parabolas are sketched. Question is to select the correct parabola.  Need to consider the y-intercept, the coefficient of x^2, and the x-coordinate of the minimum/maximum point.

Differentiation of  polynomials, cos, sin, exp, log functions. Product, quotient and chain rules.

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A few simple functions are provided of the form ax, x+b and cx+d. Values of the functions, inverses and compositions are asked for. Most are numerical but the last few questions are algebraic.

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Students are given six graphs, corresponding to curves $\gamma(t)$. They must match each with its signed curvature function, $\kappa(t)$.

The graphs are generated by calculating $\theta(t)=\int \kappa(t) \mathrm{d}t$ (by hand: these are given to the question as functions of a variable '#', in string form), and solving $x^{\prime}=\cos(\theta(t)-\theta(0))$ and $y^{\prime}(t)=\sin(\theta(t)-\theta(0))$ numerically (using the RKF method) with a JavaScript extension.

Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$.

A (quadratic) function is skethed sketched. Three equations are given that can be solved using the graph. There is a chance there will only be one solution.

A graph of an (invertible) cubic is given. The question is to determine values of $f$ from graph.