Title: Default for Input samples
Course: Demonstrating STACK (v4.4.7)

Calculate \[ [value].[value]\]
Feedback: To multiply matrices \(A\) and \(B\) we need to remember that the \((i,j)\)th entry is the scalar product of the \(i\)th row of \(A\) with the \(j\)th column of \(B\). \[ [value].[value] = [value] = [value].\]

Differentiate [value] with respect to [value].
Feedback: Formally we need to use the Chain Rule to differentiate [value], but in this case it can be done directly to give [value].

Which method do you need to integrate [value] with respect to \(x\)?

Find the value of this integral:
Feedback: (a) Let \(u=\sin(t)\) so that \(u'=\cos(t)\). Hence \[ \int u \frac{\mathrm{d}u}{\mathrm{d}t} \mathrm{d}t = \int u \mathrm{d}u = \frac{1}{2}u^2+c = \frac{1}{2} \sin^2(t)+c.\] (b) Integrating by parts, let \(I=\int \sin(t)\cos(t)\mathrm{d}t\), then \[ I = \sin^2(t) - \int\cos(t)\sin(t)\mathrm{d}t + c'\] hence \( 2I = \sin^2(t) +c'\), or \[ \int \sin(t)\cos(t)\mathrm{d}t = \frac{1}{2}\sin^2(t)+\frac{c'}{2}.\] (c) Using \(\cos(t)\sin(t)=\frac{1}{2}\sin(2t)\) we have \[ \int \sin(t)\cos(t)\mathrm{d}t = \int \frac{1}{2}\sin(2t)\mathrm{d}t \] \[ = -\frac{1}{4}\cos(2t)+c'' = \frac{1}{2} \sin^2(t)-\frac{1}{2}+c'', \] using \(\cos(2t) = 1-2\sin^2(t)\). (d) Complex exponentials can be used directly in place of the trigonometrical formula. \[\int \cos(\theta)\sin(\theta) \mathrm{d}\theta = \int \frac{1}{2}\left( e^{i\theta}+e^{-i\theta} \right) \frac{1}{2i}\left( e^{i\theta}-e^{-i\theta} \right) \mathrm{d}\theta\\ = \frac{1}{4i} \int e^{2i\theta}-e^{-2i\theta} \mathrm{d}\theta = -\frac{1}{8}\left( e^{i2\theta}+e^{-i2\theta} \right) +c'''. \] It is standard practice to rewrite the answer in the form given in the statement of the question, we should transform this using \[ \cos(\theta) = \frac{1}{2}\left( e^{i\theta}+e^{-i\theta} \right)\] back into a final answer of \( -\frac{1}{4}\cos(2\theta)+c'''\).

Radio

This question illustrates the radio input type.  Note the options here have been randomly shuffled, with the "none" of these at the end.

Checkbox and Algebraic Input

In the question below, there are two inputs.  One is a checkbox and the other is a normal algebraic input.  The student can choose as many options as they like.

Note in STACK, the student's answer is always an algebraic expression.  Hence, even in the case where the options are text, the CAS still gets a list of values which must be valid CAS expressions.  Here they are just letters a-d.

How would you type in the following expression? \[ \frac{{1+a}}{b^2} \]
Feedback: To type in [value] in code we need to use brackets to group \(1+a\), and we can type this as [value].

This input demonstrates how STACK can use the *value* of an expression, not just its displayed form, to create input options in the radio input type.

Introduction

The whole point of STACK is that a student's answer should be typed as an algebraic expression.  The typical situation is that a student must input an algebraic expression.

This "quiz" provides examples of other input types supported by STACK.  E.g. multiple choice input and scientific units.

In this question you are expected to use correct units, and work to 2 significant figures.

Drop a small heavy object a short distance and measure the time taken to hit the ground.

How long did your object take to hit the ground?

Calculate the approximate distance your object fell.
Feedback: Let time taken in seconds \(\mathrm{s}\) be \(t\). Then distance \(s=\frac{1}{2}g t^2\), where \(g\) is the acceleration due to gravity.

Scientific Units

These input boxes look as if they are the same as algebraic inputs.  However, the student must type in units, otherwise their answer will be considered to be invalid, and STACK will not assess it.

Matrices

STACK can provide a grid into which students type their matrix elements.  This is a fixed size.

Where a student is required to know the size of the matrix, they will need to type in Maxima's matrix(...) command.

Write a sequence of statements that are equivalent.
Feedback: Really this input type is a test, and you can write any list of statements which are equivalent!

Working line by Line

In some input types you can work "line by line", showing your algebraic working.  The "validation" feedback shows if adjacent lines are equivalent.  In the input below you can type any chain of reasoning, and the validation to the right will decide if you are correct.  Normally students will be asked to solve a particular equation.

The input starts with a quadratic for you to solve, but you can try to solve other equations here.