Ugur Efem

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$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

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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".

Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.

Double integral in polar coordinates - concentric circles

alternating series 

ratio test question

Calculation of the length and alternative form of the parameteric representation of a curve, involving trigonometric functions.

Curl and divergence of a vector field.  Determine whether the vector field is irrotational or solenoidal.

moment of inertia - cylinder

moment of inertia - cylinder

set of exercises on double and triple integrals

mass of tetrahedron via integration

Taylor Approximation ofr $\cos(2x)$

power series expansions for sine

Various question on power series 

Multiple response question (4 correct out of 8) covering properties of convergent and divergent series and including questions on power series. Selection of questions from a pool.

Find the first 3 terms in the MacLaurin series for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.

Find the first 3 terms in the Taylor series at $x=c$ for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.

Series Homework 2

question involving caomparison test and absolute convergence

Questions on Vector Calculus

Parametric form of a curve, cartesian points, tangent vector, and speed.

Parametric form of a curve, cartesian points, tangent vector, and speed.

Intersection points, tangent vectors, angles between pairs of curves, given in parametric form.

Divergence of vector fields.

Directional derivative of a scalar field.

Gradient of $f(x,y,z)$.

Should warn that multiplied terms need * to denote multiplication.