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.

Calculate the functional value at a point.

Calculate the functional value at a point.

Characterise the cosh function as continuous, many-to-one, even, and find the limit as x approaches 1. Part of HELM book 2.4.3.

Given a linear or a quadratic function and asked whether it is continuous. Part of HELM Book 2.4.

Determine expressions for internal shear and bending moment as a function of $x$ for a beam with two vertical loads.

This shows how to take an expression given by the student and plot the implicit curve of points where it equals zero.

Find the area, first moment of area, and coordinates of a general spandrel.  The area may be above or below the function.

When saving the data necessary to resume an attempt, Numbas only saves the values of variables which use the random number generator.

If you naively generate a large random sample in one variable, it'll be marked as a source of randomness so Numbas will save every value in the sample. Loading this value when you resume the attempt can be very slow.

Instead, you can generate a seed value in one variable and then use it with the seedrandom function in the variable that stores your random list. Only the value of the seed needs to be saved.

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.

Calculating the derivative of a function of the form $ax^n \ln(bx)$ using the product rule.

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

Find the derivative of a function of the form $y=a \sin(bx+c)$ using a table of derivatives.

Solve equations involving logs and exponential functions, by using inverse operations.

Calculating the derivative of a function of the form $(ax+b)^n$ using the chain rule.

Calculating the derivative a function of the form $ax^n \sin(bx)$ using the product rule.

Finding the zero of a non-polynomial function

This question asks the student to give a function with a particular root. It then asks them to divide by (x-{root}), and uses adaptive marking to mark against the previous answer.

This uses the "expression" data type, which is currently undocumented and experimental.

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

rebelmaths

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).

Using the unit circle definition of sin, cos and tan, to calculate the exact value of trig functions evaluated at angles that depend on 0, 30, 45, 60 or 90 degrees.

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.

Harder inequalities that involve terms like $\frac{1}{x}$ where multiplication by the denominator requires you to split into cases for positive or negative, or you multiply by the square of the denominator.

Inequality involving a single absolute value, question solution uses the piecewise nature of the absolute value function.

Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.

The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.

This particular example has a positive gradient.

 Students must derive the shear and bending moment functions for beam loaded with a parabolic distributed load described with a loading function.  Diagrams are given, but the students must derive the equations by integration.

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.

Shear and bending moment diagram for a beam loaded with an arbitrary parabolic distributed load described with a loading function.  Integrate to find reactions and equations for shear and bending moment.