Bill Foster

The derivative of $\displaystyle \frac{ax^2+b}{cx^2+d}$ is $\displaystyle \frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.

Contains a video solving a similar quotient rule example. Although does not explicitly find $g(x)$ as asked in the question, but this is obvious.

Find the gradient of  $ \displaystyle ax^b+\frac{c}{x^{d}}+f$ at $x=a$

Find the stationary points of a cubic which has 2 turning points.

Examples on differentiation using the quotient rule and chain rule.

Differentiate $\displaystyle ax^b, ax^b+cx^{1/d}, \frac{a}{x^{1/c}}+\frac{b}{x^{-1/d}}+c$ with respect to $x$.

Differentiate $f(x)=x^{m}\sin(ax+b) e^{nx}$.

The answer is of the form:
$\displaystyle \frac{df}{dx}= x^{m-1}e^{nx}g(x)$ for a function $g(x)$.

Find $g(x)$.

Application of the Poisson distribution given expected number of events per interval.

Finding probabilities using the Poisson distribution.

Application of the binomial distribution given probabilities of success of an event.

Finding probabilities using the binomial distribution.

Given a random variable $X$  normally distributed as $\operatorname{N}(m,\sigma^2)$ find probabilities $P(X \gt a),\; a \gt m;\;\;P(X \lt b),\;b \lt m$.

Exercise using a given uniform distribution $X$, calculating the expectation and variance. Also finding $P(X \le a)$ for a given value $a$.

Question on the exponential distribution involving a time intervals and arrivals application, finding expectation and variance. Also finding the probability that a time interval between arrivals is less than a given period. All parameters and times randomised. 

Divide $ f(x)=x ^ 4 + ax ^ 3 + bx^2 + cx+d$ by $g(x)=x^2+p $ so that:
$\displaystyle \frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)}$

Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.

Split $\displaystyle \frac{ax+b}{(cx + d)(px+q)}$ into partial fractions.

Questions testing addition, subtraction, multiplication of numerical fractions and reduction to lowest terms. They also test BIDMAS in the context of fractions.

Reducing fractions to their lowest form by cancelling common factors in the numerator and denominator. There are 4 questions. 

Add/subtract fractions and reduce to lowest form.

Questions testing understanding of numerators and denominators of numerical fractions, and reduction to lowest terms.

Find $\displaystyle \frac{a} {b + \frac{c}{d}}$ as a single fraction in the form $\displaystyle \frac{p}{q}$ for integers $p$ and $q$.

Questions testing understanding of numerators and denominators of numerical fractions.

Questions testing understanding of numerators and denominators of numerical fractions, and reduction to lowest terms.

Given sample data find mean, standard deviation, median, interquartile range.

Note that there are different versions of the upper and lower quartiles, so you may want to include your own versions - see the user defined functions in the question.

Given a table of the number of days in which sales were between £x1000 and £(x+1)1000 find the relative percentage frequencies of these volume of sales.

Deciding whether or not  three sampling methods are simple random sampling, stratified sampling, systematic or judgemental sampling. Also whether or not the method of selection is random, quasi-random or non-random.

Choosing whether given random variables are qualitiative or quantitative.

Solve for $x$ each of the following equations: $n^{ax+b}=m^{cx}$ and $p^{rx^2}=q^{sx}$.

Express $\log_a(x^{c}y^{d})$ in terms of $\log_a(x)$ and $\log_a(y)$. Find $q(x)$ such that $\frac{f}{g}\log_a(x)+\log_a(rx+s)-\log_a(x^{1/t})=\log_a(q(x))$.

There is a video included explaining the rules of logarithms by going through simplification of logs of numbers rather than algebraic expressions.

Solve for $x$: $\log(ax+b)-\log(cx+d)=s$

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Solve for $x$:  $\log_{a}(x+b)- \log_{a}(x+c)=d$