Algebraic properties of exponential functions
The algebraic properties of exponential functions are the index laws: `a^x a^y = a^(x + y)`, `a^(-x) = 1/(a^x)`, `(a^x)^y = a^(xy)`, `a^0 = 1`, where `x`, `y` and `a` are real.
Algebraic properties of logarithms
The algebraic properties of logarithms are the rules: `log_a (xy) = log_a x + log_a y`, `log_a (1/x) = -log_a x`, and `log_a 1 = 0`, for any positive real numbers `x`, `y` and `a`.
A straight line is an asymptote of the function `y = f(x)` if graph of `y = f(x)` gets arbitrarily close to the straight line. An asymptote can be horizontal, vertical or oblique. For example, the line with equation `x = pi/2` is a vertical asymptote to the graph of `y = tan x`, and the line with equation `y = 0` is a horizontal asymptote to the graph of `y = 1/x`.
The expansion `(x + y)^n = x^n + ((n),(1))x^(n - 1)y + ... + ((n),(r))x^(n - r)y^r + ... + y^n` is known as the binomial theorem. The numbers `((n),(r)) = (n!)/(r!(n-r)!) = (n xx (n - 1) xx ... xx (n - r + 1))/(r xx (r - 1) xx ... xx 2 xx 1)` are called binomial coefficients.
The Cartesian plane is a plane consisting of a set of two lines intersecting each other at right angles. The horizontal line is the `x`-axis and the vertical one is the `y`-axis, and the point of their intersection is called the origin with the coordinates `(0, 0)`.
A rotation, typically measured in radians or degrees.
Completing the square
The quadratic expression `ax^2 + bx + c` can be rewritten as `a(x + b/(2a))^2 + (c - (b^2)/(4a))`. Rewriting it in this way is called completing the square.
The probability of an event `A` occurring when it is known that some event `B` has already occurred, is given by `P(A|B) = (P(A nn B))/(P(B))`.
The discriminant (`Delta`) of the quadratic expression `ax^2 + bx + c` is the quantity `b^2 - 4ac`.
A function `f` is a rule such that for each `x`-value there is only one corresponding `y`-value. This means that if `(a, b)` and `(a, c)` are ordered pairs, then `b = c`.
The gradient of the straight line passing through points `(x_1, y_1)` and `(x_2, y_2)` is the ratio `(y_2 - y_1)/(x_2 - x_1)`. Slope is a synonym for gradient.
Graph of a function
The graph of a function `f` is the set of all points `(x, y)` in the Cartesian plane where `x` is in the domain of `f` and `y = f(x)`.
Two events are independent if knowing that one occurs tells us nothing about the other. The concept can be defined formally using probabilities in various ways: events `A` and `B` are independent if `P(A nn B) = P(A)P(B)`, if `P(A|B) = P(A)` or if `P(B) = P(B|A)`. For events `A` and `B` with non-zero probabilities, any one of these equations implies any other.
The index laws are the rules: `a^xa^y = a^(x + y)`, `a^(-x) = 1/(a^x)`, `(a^x)^y = a^(xy)`, `a^0 = 1`, and `(ab)^x = a^xb^x`, where `a`, `b`, `x` and `y` are real numbers.
Length of an arc
The length of an arc of a circle is given by `l = r theta`, where `l` is the arc length, `r` is the radius and `theta` is the angle subtended at the centre, measured in radians. This is simply a rearrangement of the formula defining the radian measure of an angle.
Linearity property of the derivative
The linearity property of the derivative is summarized by the equations:
`d/dx(ky) = kdy/dx` for any constant `k`, and `d/dx(y_1 + y_2) = (dy_1)/dx + (dy_2)/dx`.
Local and global maximum and minimum
A stationary point on the graph `y = f(x)` of a differentiable function is a point where `f'(x) = 0`.
We say that `f(x_0)` is a local maximum of the function `f(x)` if `f(x) <= f(x_0)` for all values of `x` near `x_0`.
We say that `f(x_0)` is a global maximum of the function `f(x)` if `f(x) <= f(x_0)` for all values of `x` in the domain of `f`.
We say that `f(x_0)` is a local minimum of the function `f(x)` if `f(x) >= f(x_0)` for all values of `x` near `x_0`.
We say that `f(x_0)` is a global minimum of the function `f(x)` if `f(x) >= f(x_0)` for all values of `x` in the domain of `f`.
Two events are mutually exclusive if there is no outcome in which both events occur.
Problems solved using procedures not regularly encountered in learning activities.
Pascal’s triangle is a triangular arrangement of binomial coefficients. The `n^"th"` row consists of the binomial coefficients `((n),(r))`, for `0 <= r <= n`; each interior entry is the sum of the two entries above it, and the sum of the entries in the `n^"th"` row is `2^n`.
Period of a function
The period of a function `f(x)` is the smallest positive number `p` with the property that `f(x + p) = f(x)` for all `x`. The functions `sin x` and `cos x` both have period `2pi`, and `tan x` has period `pi`.
Point of inflection
A point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.
If `ax^2 + bx + c = 0` with `a != 0`, then `x = (-b +- sqrt(b^2 - 4ac))/(2a)`. This formula for the roots is called the quadratic formula.
The radian measure `theta` of an angle in a sector of a circle is defined by `theta = l/r`, where `r` is the radius and `l` is the arc length. Thus an angle whose degree measure is `180^"o"` has radian measure of `pi`.
A random variable is a numerical quantity, the value of which depends on the outcome of a chance experiment. For example, the proportion of heads observed in 100 tosses of a coin.
A discrete random variable is one which can only take a countable number of value, usually whole numbers.
A continuous random variable is one whose set of possible values are all of the real numbers in some interval.
If an event `E` occurs `r` times in `n` trials of a chance experiment, the relative frequency of `E` is `r/n`.
Problems solved using procedures regularly encountered in learning activities.
A secant of the graph of a function is a straight line passing through two points on the graph. The line segment between the two points is called a chord.
Sine and cosine functions
In the unit circle definition of `"cosine"` and `"sine"`, `cos theta` and `sin theta` are the `x`- and `y`-coordinates of the point on the unit circle corresponding to the angle `theta` measured as a rotation from the ray `OX`. If `theta` is measured in the counter-clockwise direction, then it is said to be positive; otherwise it is said to be negative.
Sine rule and cosine rule
The lengths of the sides of a triangle are related to the sines of its angles by the equations
`a/(sinA) = b/(sinB) = c/(sinC)`
This is known as the sine rule.
The lengths of the sides of a triangle are related to the cosine of one of its angles by the equation
`c^2 = a^2 + b^2 - 2ab cos C`
This is known as the cosine rule.
The tangent line (or simply the tangent) to a curve at a given point `P` can be described intuitively as the straight line that has the same gradient at a curve where they meet. In this sense it is the best straight-line approximation to the curve at the point `P`.
Vertical line test
A relation between two real variables `x` and `y` is a function, and `y = f(x)` for some function `f`, if and only if each vertical line, i.e. each line parallel to the `y`-axis, intersects the graph of the relation in at most one point. This test to determine whether a relation is, in fact, a function is known as the vertical line test.