TCE Math Methods exam 2026Exam: Tue 17 Nov · TASC timetable

$v(t) = \dfrac{dx}{dt}$ (velocity is the rate of change of position)

$a(t) = \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}$ (acceleration is the rate of change of velocity)

$E(X) = \mu = \displaystyle\sum x\,P(X=x)$

$\operatorname{Var}(X) = \sigma^2 = \displaystyle\sum (x-\mu)^2 P(X=x) = E(X^2) - \mu^2$

Standard deviation: $\sigma = \sqrt{\operatorname{Var}(X)}$

$e \approx 2.71828$ is the base for which the curve $y = e^x$ has gradient exactly equal to its own height at every point. This is what makes $e$ "natural" for calculus.

Chain rule

If $y = f(g(x))$ then $\dfrac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

Product rule

If $y = u\,v$ then $\dfrac{dy}{dx} = u'v + uv'$.

Quotient rule

If $y = \dfrac{u}{v}$ then $\dfrac{dy}{dx} = \dfrac{u'v - uv'}{v^{2}}$.

The standard deviation is also called the standard error. It shrinks as $n$ grows, in proportion to $\dfrac{1}{\sqrt{n}}$.

$\displaystyle\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C$

$\displaystyle\int \sin(kx)\,dx = -\frac{1}{k}\cos(kx) + C$

$\displaystyle\int \cos(kx)\,dx = \frac{1}{k}\sin(kx) + C$

where $h = \dfrac{b-a}{n}$ and $y_i = f(a + ih)$ are the $n+1$ ordinates.

where $f$ is the upper curve, $g$ is the lower curve, and $a$ and $b$ are the $x$-coordinates of the points of intersection.