Find the eigenfunctions of an irregular Sturm–Liouville problem and hence solve an inhomogeneous boundary value problem by writing the solution as an eigenfunction expansion.

The student has to write the general solution of a 2nd order PDE. They can choose the names of their arbitrary functions of $x$ and $y$.

The marking algorithm finds the names of the functions of $x$ and $y$ in the student's answer, and replaces them with $\sin(x)$ and $\cos(y)$ (these could be changed) so that the expression can be evaluated.

Calculating the derivative of a function of the form $\frac{(x-a)^2}{bx} + c$, finding its stationary points and determining their nature.

Using basic derivatives to calculate the gradient function of a hill $y=-e^x+b\ln\left(x\right)+c$, and then substituting values to find the gradient at specific distance from the sea. 

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Given a graph of the form either a.cos(bx) or a.sin(bx), identify the amplitude and period.

Shows how to use the lpad function to format a time in HH:MM format so that leading zeros are used for numbers less than 10.

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Homework set.  Use integration to find the centroid of shapes bounded by functions.

Homework set.  Use integration to find the centroid of shapes bounded by functions.

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.

Find the stationary point $(p,q)$ of the function: $f(x,y)=ax^2+bxy+cy^2+dx+gy$. Calculate $f(p,q)$.

Find the stationary point $(p,q)$ of the function: $f(x,y)=ax^2+bxy+cy^2+dx+gy$. Calculate $f(p,q)$.

An application of quadratic functions based on the Golden Gate Bridge in San Francisco, USA. Student is given an equation representing the suspension cable of the bridge and asked to find the width between the towers and the minimum height of the cable above the roadway. Requires and understanding of the quadratic function and where and how to apply correct formulae.

Multiple choice questions. Given randomised trig functions select the possible ways of writing the domain of the function.

Asks students to find the partil fraction decomposition for a rational function Denominator is a quadratic with distinct factors.

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

This question uses a Geogebra applet to solve a linear program with two variables using the graphical method. It contains three steps:

1. Construct the feasible area (polygon) by adding the constraints one by one. The students can see what happens when the constraints are added.
2. Add the objective function, and the level set of the objective value is shown, as well as its (normalised) gradient.
3. Compute the optimal solution by moving the level set of the objective around.

Calculate the marginal and average cost for a given cost function. Find the corresponding startup/shutdown price.
Maximize the profit function at a given price.

Given a randomised square root function select the possible ways of writing the domain of the function.

This is a copy of a question from the Numbas demos project, with references to the editor removed.

The student is shown a plot of a mystery function. They can enter values of $x$ check, within the bounds of the plot.

They're asked to give the formula for the function, and then asked for its value at a very large value of $x$.

A plot of the student's function updates automatically as they type. Adaptive marking is used for the final part to award credit if the student gives the right value for their incorrect function.

Unit 1: Sequences and Functions

Dealing with functions in Numbas. (Translation to German.)

A random dataset given by a linear function with noise (gradient and y-intercept of the linear function are randomised as is distribution of x values).

More work on differentiation with trigonometric functions

Compute a table of values for a quadratic function. A JSXgraph (the graph paper) plot shows the curve going through the entered values. The student input is now disconnected from the graph so that they slide the points usually after they input the values and the answer fields are not updated.

Compute a table of values for a quadratic function. A JSXgraph (the graph paper) plot shows the curve going through the entered values.

More work on differentiation with trigonometric functions