Ben McGovern

Find the unit vectors in the direction of four 3-dimensional vectors. Three of the vectors are given, and the fourth is expressed as a linear combination of two of the other vectors.

Calculating the derivative of a function of the form $ax^n \ln(bx)$ using the product rule.

Calculating the derivative of a function of the form $\frac{(x-a)^2}{bx}$, finding its stationary points and determining their nature.

Find the derivative of a function of the form $y=a \sin(bx+c)$ using a table of derivatives.

Calculating several linear combinations of three 3-dimensional vectors. 

Calculating several linear combinations of three 2-dimensional vectors. 

Calculate the magnitude of a 3-dimensional vector $\mathbf v$, where $\mathbf v$ is written in the form $v_1 \mathbf i+v_2 \mathbf j + v_3 \mathbf k$.

Calculate the magnitude of a 3-dimensional vector, where $\mathbf v$ is written in the form $\pmatrix{v_1\\v_2\\v_3}$.

Calculate the magnitude of a 2-dimensional vector $\mathbf v$, where $\mathbf v$ is written in the form $v_1 \mathbf i+v_2 \mathbf j$.

Calculate the magnitude of a 2-dimensional vector, where $\mathbf v$ is written in the form $\pmatrix{v1\\v2}$.

Given the position vectors of three 2-dimensional points $A$, $B$ and $C$, find the coordinates of a fourth point $D$ such that $ABCD$ forms a parallelogram. 

Find the unit vectors in the direction of four 2-dimensional vectors. Three of the vectors are given, and the fourth is expressed as a linear combination of two of the other vectors.

Rewriting expressions of the form $n\log(a)\pm m\log(b)$ as a single logarithm.

Calculating the derivative of a function of the form $(ax+b)^n$ using the chain rule.

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Rewriting $\log_{10}(x)$ in terms of $\log_{10}(a)$ and $\log_{10}(b)$, where $a$, $b$ and $x$ are given.

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Calculating the derivative a function of the form $ax^n \sin(bx)$ using the product rule.

Finding the arc length of a sector of a circle when given the radius of the circle and angle of the sector.

Calculating the amount of money in a savings account after a given amount of time, and calculating how long it will take for the amount of savings to exceed a given value. 

Rewrite the expression $\frac{cx+d}{(x+a)(x+b)}$ as partial fractions in the form $\frac{A}{x+a}+\frac{B}{x+b}$.

Simplifying expressions from $\frac{x^mx^n}{x^p}$ to $x^{m+n-p}$. 

Rewrite the expression $\frac{cx+d}{(x+a)^2}$ as partial fractions in the form $\frac{A}{x+a}+\frac{B}{(x+a)^2}$.

Given 3 vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$, find the constants $p$, $q$ and $k$ such that $ p\mathbf a + q \mathbf b = \mathbf c$, where $k$ is an unknown component of $\mathbf c$ .

Given 3 vectors $\mathbf v$, $\mathbf a$ and $\mathbf b$, find the constants $c_1$ and $c_2$ such that $\mathbf v = c_1 \mathbf a + c_2 \mathbf b$ .

Rewriting fractions of the form $\frac{\sqrt[m]{x^n}}{\sqrt[p]{x^q}}$ to $x^{\frac{n}{m}-\frac{q}{p}}$.

Rewriting expressions from $\sqrt[m]{x^n}$ to $x^\frac{n}{m}$.

Factorising a quadratic expression of the form $ax^2+bx+c$ by completing the square.

Factorising a quadratic expression of the form $x^2+bx+c$ by completing the square.

Factorising a quadratic expression with the $x^2$-term having a coefficient of 1.