Specialist Mathematics flashcards

74 TCE Specialist Mathematics cards across 2 modules (10 definitions, 34 key facts, 6 fill-the-gap, 24 formulas). Each session gives you 20 cards, due cards first. Swipe right for easy, left for hard, down for again. Keyboard: Space to flip, 1 to 4 to rate.

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Definition
Absolute value functions - TCE Mathematics Specialised (Tasmania)
Define Absolute value
Equivalently $|x| = \sqrt{x^2}$ , and $|x|$ is the distance from $x$ to $0$ on the number line.
Definition
Differential equations - TCE Mathematics Specialised (Tasmania)
Define Exponential growth and decay
"The rate of change is proportional to the amount" becomes $\dfrac{dy}{dt} = ky$ , solving to $y = y_0 e^{kt}$ . A positive $k$ is growth (populations, compound processes); a negative $k$ is decay (radioactivity, discharge).
Definition
Complex arithmetic and conjugates - TCE Mathematics Specialised (Tasmania)
Define Complex conjugate
The conjugate of $z = x + iy$ is $\bar z = x - iy$ . Geometrically it is the reflection of $z$ in the real axis.
Fill the gap
Statistical inference - TCE Mathematics Specialised (Tasmania)
The bridge is the ___ of the sample mean.
The bridge is the sampling distribution of the sample mean.
Definition
Composite and inverse functions - TCE Mathematics Specialised (Tasmania)
Define Domain and range swap
The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y = x$ .
Key fact
Integration techniques and applications - TCE Mathematics Specialised (Tasmania)
Tell me about Integration by parts
Choose $u$ to be the factor that becomes simpler when differentiated. A useful priority for picking $u$ is logarithms, then inverse trigonometric functions, then algebraic, then trigonometric, then exponential.
Definition
Factorising polynomials over the complex numbers - TCE Mathematics Specialised (Tasmania)
Define Fundamental theorem of algebra
Every polynomial of degree $n \ge 1$ with complex coefficients has exactly $n$ roots in the complex numbers, when each root is counted as many times as its multiplicity. Equivalently, it factorises completely into $n$ linear factors over the complex field.
Definition
Proof by mathematical induction - TCE Mathematics Specialised (Tasmania)
Define Principle of mathematical induction
If both steps hold, then $P(n)$ is true for all $n \ge n_0$ .
Definition
Rational functions and asymptotes - TCE Mathematics Specialised (Tasmania)
Define Domain and vertical asymptotes
Set $Q(x) = 0$ . Each zero of the denominator that is not cancelled by the numerator gives a vertical asymptote.
Definition
Roots of unity - TCE Mathematics Specialised (Tasmania)
Define Primitive root and the standard notation
The number $\omega$ is called a primitive $n$ th root of unity because its powers generate them all.