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

Calculating gradients - polynomials

Identify whether or not an expression is a polynomial. Part of HELM Book 1.2

Identify whether or not an expression is a polynomial. Part of HELM Book 1.2

Given an arbitrary polynomial, identify its degree. Part of HELM Book 1.2

Is this polynomial a quadratic, linear or constant? Part of HELM Book 1.2

Part of HELM Book 1.2

Differentiate $\displaystyle e^{ax^{m} +bx^2+c}$

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

Calculating gradients - polynomials

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Fairly simple questions using differentiation "power rule" and "sum or difference rules" to differentiate single term functions and polynomials.

Some co-efficients and indices can be negative and/or fractional.

3 Repeated integrals of the form $\int_a^b\;\int_c^{f(x)}g(x,y)\;dy \;dx$ where $g(x,y)$ is a polynomial in $x,\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.

Find the remainder when dividing two polynomials, by algebraic long division.

Using a given list of four complex numbers, find by inspection the one that is a root of a given cubic real polynomial and hence find the other roots.

Integrating a polynomial functions which describe the rate of change of a population over time to find and use an equation that describes the total population according to time.

Calculating $\frac{dy}{dx}$ from an implicit polynomial function.

Calculating $\frac{dy}{dx}$ from an implicit polynomial function.

Calculating $\frac{dy}{dx}$ from an implicit polynomial function.

Example of an explore mode question. Student is given a 2x2 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 eigenvector is recognised by the marking algorithm, also multiples of the "obvious" one(s) (given the reduced row echelon form that we use to calculate them).

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

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

Factorising polynomials using the highest common factor.

Adapted from 'Factorisation' by Steve Kilgallon.

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".

Three parts, where the student has to collect together polynomials in $\mathbb{Z}_3$, $\mathbb{Z}_5$ and $\mathbb{Z}_7$, respectively.

The answer to part a has no $X$ term, because they cancel out.

Simple application of "Power Rule" to differentiate polynomials.

Some co-efficients and powers are non-integer and some may be negative.

Quotient and remainder, polynomial division.

Exam Question

5 questions on definite integrals - integrate polynomials, trig functions and exponentials; find the area under a graph; find volumes of revolution.

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

Missing: Application with bacteria, turning points, difficult chain rule

In this assignment, try and find the roots of a randomly generated polynomial, using the quadratic equation.

A test assignment to see if it integrates properly into Canvas.

Multiple choice question. Given a randomised polynomial select the possible ways of writing the domain of the function.