This is the question for week 3 of the MA100 course at the LSE. It looks at material from chapters 5 and 6. The following describes how two polynomials were defined in the question. This may be helpful for anyone who needs to edit this question.

In part a we have a polynomial. We wanted it to have two stationary points. To create the polynomial we first created the two stationary points as variables, called StationaryPoint1 and StationaryPoint2 which we will simply write as s1 ans s2 here. s2 was defined to be larger than s1. This means that the derivative of our polynomial must be of the form a(x-s1)(x-s2) for some constant a. The constant "a" is a variable called PolynomialScalarMult, and it is defined to be a multiple of 6 so that when we integrate the derivative a(x-s1)(x-s2) we only have integer coefficients. Its possible values include positive and negative values, so that the first stationary point is not always a max (and the second always a min). Finally, we have a variable called ConstantTerm which is the constant term that we take when we integrate the derivative derivative a(x-s1)(x-s2). Hence, we can now create a randomised polynomial with integers coefficients, for which the stationary points are s1 and s2; namely (the integral of a(x-s1)(x-s2)) plus ConstantTerm.

In part e we created a more complicated polynomial. It is defined as -2x^3 + 3(s1 + s2)x^2 -(6*s1*s2) x + YIntercept on the domain [0,35]. One can easily calculate that the stationary points of this polynomials are s1 and s2. Furthermore, they are chosen so that both are in the domain and so that s1 is smaller than s2. This means that s1 is a min and s2 is a max. Hence, the maximum point of the function will occur either at 0 or s2 (The function is descreasing after s2). Furthermore, one can see that when we evaluate the function at s2 we get (s2)^2 (s2 -3*s1) + YIntercept. In particular, this is larger than YIntercept if s2 > 3 *s1, and smaller otherwise. Possible values of s2 include values which are larger than 3*s1 and values which are smaller than 3*s1. Hence, the max of the function maybe be at 0 or at s2, dependent on s2. This gives the question a good amount of randomisation.

Slope of a curve at a point

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.

Implicit differentiation.

Given $x^2+y^2+dxy +ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

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

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

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

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

This question asks you to identify a point on a graph.

This question asks you to place a point on a number line.

This question allows you to practice plotting points on graphs.

This question allows you to practice placing points on a number line.

This question allows you to practice identifying points on graphs.

This question allows you to practice identifying points on graphs.

Slope of a curve at a point

Questions asking you to find the equation of a line between two points, in Cartesian coordinates.

Slope of a curve at a point

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

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

Uses JSXGraph to generate a plot for a cubic, with given critical points, along with three other incorrect graphs with modified properties. JSXGraph code is commented.

Implicit differentiation.

Given $x^2+y^2+dxy +ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

Given a description in words of the costs of some items in terms of an unknown cost, write down an expression for the total cost of a selection of items. Then simplify the expression, and finally evaluate it at a given point.

The word problem is about the costs of sweets in a sweet shop.

Given a description in words of the costs of some items in terms of an unknown cost, write down an expression for the total cost of a selection of items. Then simplify the expression, and finally evaluate it at a given point.

The word problem is about the costs of sweets in a sweet shop.

Application of differentitaion in geology. A projectile problem.

Implicit differentiation.

Given $x^2+y^2+dxy +ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

Slope of a curve at a point

Student is asked to drag points onto the unit circle, to represent sin(x) and cos(x), where x is a multiple of 45 degrees.

Critical points, absolute minimum, local maximum and minimum points, increasing and decreasing, concavity

rebel

rebelmaths

Implicit differentiation.

Given $x^2+y^2+dxy +ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

A graph (of a cubic) is given. The question is to determine the number of roots and number of stationary points the graph has. Non-calculator. Advice is given.

A cubic with a maximum and minimum point is given. Question is to determine coordinates of the minimum and maximum point.

Student is asked to drag points onto the unit circle, to represent sin(x) and cos(x), where x is a multiple of 45 degrees.

A function of the form f(x)= sin(ax+b) is given and plotted. A few points are plotted on the curve. $x$-coordinates are provided for two of them and $y$-coordinate provided for third. Student is required to determine other coordinates.