Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

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.

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Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Some basic tasks involving vectors, including converting to/from component form, scalar product, resultant vectors.

Some basic tasks involving vectors, including converting to/from component form, scalar product, resultant vectors.

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

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Directional derivative of a scalar field.

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

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

Perform basic operations with 2x2 matrices - addition, subtraction, multiplication by scalar, matrix multiplication, determinant, transpose

Quiz to assess matrix addition, subtraction, multiplication and multiplication by scalar, determinants and inverses, solving a system of simultaneous equations.

Vector addition, multiplication by a scalar, magnitude, dot product, Perpendicular vectors, angle between vectors, cross product.

rebel

rebelmaths

Addition, subtraction and multiplication of 2 x 2 matrices and multiplication by a scalar.

(Last three parts of original question removed.)

5 questions on vectors. Scalar product, angle between vectors, cross product, when are vectors perpendicular, combinations of vectors defined or not.

5 questions on vectors. Scalar product, angle between vectors, cross product, when are vectors perpendicular, combinations of vectors defined or not.