Glossary
Boundary (in piecewise functions)
A specific x-value where the definition of a piecewise-defined function changes from one expression to another.
Example:
In f(x) = {x+1 if x<0, x² if x≥0}, x=0 is the boundary point where the function's rule changes.
Closed Interval
An interval that includes its endpoints, denoted by square brackets, e.g., [a, b].
Example:
When measuring temperature from 0 to 10 degrees Celsius, we are considering a closed interval of temperatures, including both 0 and 10.
Continuity theorem (for combinations of functions)
A theorem stating that if individual functions are continuous at a point, their sum, difference, product, and quotient (provided the denominator is non-zero) are also continuous at that point.
Example:
Since f(x) = x and g(x) = sin(x) are continuous everywhere, their sum h(x) = x + sin(x) is also continuous everywhere according to the continuity theorem.
Continuous Function
A function is continuous if its graph can be drawn without lifting the pen, meaning it has no breaks, jumps, or holes within its domain.
Example:
The function is a continuous function across all real numbers, as its graph is a smooth, unbroken parabola.
Continuous Function
A function is continuous if its graph can be drawn without lifting the pencil, meaning it has no breaks, jumps, or holes in its domain.
Example:
The path of a smoothly flying drone is a continuous function of its position over time, as it doesn't teleport or disappear.
Continuous Function
A function that can be drawn without lifting the pencil, meaning it has no breaks, jumps, or holes over its domain or a specified interval.
Example:
A polynomial function like f(x) = x³ - 2x + 1 is a continuous function over all real numbers.
Continuous Function
A *continuous function* is one whose graph can be drawn without lifting the pen, meaning it has no breaks, jumps, or holes over its domain.
Example:
Polynomials like are examples of continuous functions over all real numbers, as their graphs are smooth and unbroken.
Continuous at the point
A function f is continuous at a specific point x=c if f(c) exists, the limit of f(x) as x approaches c exists, and these two values are equal.
Example:
For f(x) = x + 2, at x = 3, f(3) = 5, and lim(x→3) f(x) = 5, so f is continuous at the point x=3.
Continuous at the point
A function is *continuous at the point* $x=c$ if its value at $c$ exists, its limit as $x$ approaches $c$ exists, and these two values are equal.
Example:
The function is continuous at the point because , , and .
Continuous over an interval
A function is continuous over an interval if it is continuous at every single point within that interval.
Example:
The function f(x) = x² is continuous over an interval like (-∞, ∞) because it has no breaks or jumps anywhere.
Cosine Function
A periodic trigonometric function that describes a smooth oscillation, similar to sine but shifted, represented as $k\cos(px)$.
Example:
The voltage in an AC circuit often follows a cosine function, oscillating smoothly between positive and negative peaks.
Denominator is Zero
A condition in a rational function where the expression in the denominator evaluates to zero, leading to an undefined value for the function and a point of discontinuity.
Example:
For the function , the function is undefined when the denominator is zero, specifically at , indicating a vertical asymptote.
Differentiable
A function is differentiable if its derivative exists at every point in its domain, implying the function is smooth and has no sharp corners or vertical tangents.
Example:
The velocity of a car accelerating smoothly is a differentiable function of time, as its rate of change (acceleration) is well-defined at every instant.
Discontinuity
A point or interval where a function is not continuous, meaning its graph has a break, jump, or hole.
Example:
The function has a discontinuity at , as the graph breaks at that point due to a vertical asymptote.
Domain
The set of all possible input values (x-values) for which a function is defined.
Example:
For the function , the domain is all real numbers , because the square root of a negative number is not a real number.
Domain
The complete set of all possible input values (x-values) for which a function is defined and produces a real output.
Example:
For the function f(x) = sqrt(x), the domain is all non-negative real numbers, as you cannot take the square root of a negative number in real analysis.
Essential Discontinuity
A discontinuity where at least one of the one-sided limits (left or right) does not exist or approaches infinity, often indicating a vertical asymptote.
Example:
The function f(x) = 1/(x-2) has an essential discontinuity at x=2 because as x approaches 2, the function's value shoots off to positive or negative infinity.
Exponential Constant (e)
An irrational mathematical constant approximately equal to 2.71828, often used as the base for natural logarithms and in continuous growth/decay models.
Example:
When calculating continuously compounded interest, the formula directly uses the exponential constant (e) to represent the continuous growth factor.
Exponential Function
A function where the independent variable appears as an exponent, typically in the form $ke^{px}$ or $ka^{px}$.
Example:
The growth of a bacterial colony can be modeled by an exponential function like , showing rapid increase over time.
Horizontal Asymptote
A horizontal line that the graph of a function approaches as the independent variable tends towards positive or negative infinity.
Example:
The function has a horizontal asymptote at , indicating that as increases, the function's value gets closer and closer to zero.
Intermediate Value Theorem
This theorem states that if a function is continuous on a closed interval, it must take on every value between its function values at the endpoints of that interval.
Example:
If a student's height was 150cm at age 10 and 170cm at age 15, then by the Intermediate Value Theorem, they must have been exactly 160cm tall at some point between those ages.
Jump Discontinuity
A type of discontinuity where a function's value abruptly changes, resulting in a 'leap' between two distinct values at a specific point.
Example:
Consider a function that outputs your grade based on score: if score < 50, grade is F; if score >= 50, grade is D. At score = 50, there's a sudden jump discontinuity from F to D.
Limit
The value that a function or sequence approaches as the input or index approaches some specific value.
Example:
As x gets closer and closer to 3, the function f(x) = x + 5 approaches the limit of 8.
Limit
The value that a function approaches as its input approaches a certain value, regardless of the function's actual value at that point.
Example:
The limit of f(x) = (x² - 1)/(x - 1) as x approaches 1 is 2, even though the function is undefined at x=1.
Limit
The *limit* of a function describes the value that the function's output approaches as its input gets arbitrarily close to a specific point.
Example:
For the function , the limit as approaches 1 is 3, meaning as gets closer and closer to 1, gets closer and closer to 3.
Limit from the left
The value a function approaches as the input variable approaches a specific point from values smaller than that point.
Example:
For a piecewise function, if you approach x=0 from negative values (e.g., -0.1, -0.01), you are finding the limit from the left.
Limit from the right
The value a function approaches as the input variable approaches a specific point from values larger than that point.
Example:
For a piecewise function, if you approach x=0 from positive values (e.g., 0.1, 0.01), you are finding the limit from the right.
Logarithmic Function
A function that is the inverse of an exponential function, typically in the form $k\ln(px)$ or $k\log_a(px)$.
Example:
The Richter scale, which measures earthquake intensity, uses a logarithmic function to compress a wide range of energy releases into a more manageable scale.
Odd Multiple of pi/2
Values such as $\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2}$, etc., where the cosine function is zero, causing the tangent function to be undefined.
Example:
The tangent function has vertical asymptotes at every odd multiple of pi/2, which means is undefined at or .
One-sided limits
The value a function approaches as the input approaches a point from either the left (values less than the point) or the right (values greater than the point).
Example:
For a function with a jump at x=0, the one-sided limits from the left and right of x=0 would be different, indicating a discontinuity.
Open Interval
An interval that does not include its endpoints, denoted by parentheses, e.g., (a, b).
Example:
If a race car's speed is strictly between 100 km/h and 200 km/h (not exactly 100 or 200), its speed is in an open interval (100, 200).
Piecewise-Defined Function
A function defined by different expressions or formulas for different parts of its domain.
Example:
A common example is a tax bracket system, where the tax rate changes based on income, forming a piecewise-defined function.
Polynomial Function
A function composed of sums or differences of positive integer powers of a variable, along with possible constant terms.
Example:
When analyzing the motion of a projectile, the height function is a polynomial function, ensuring a smooth trajectory without sudden jumps.
Positive Integer Powers
Exponents that are whole numbers greater than zero, such as $x^1, x^2, x^3$, etc.
Example:
In the expression , both and represent terms with positive integer powers, which is a defining characteristic of polynomial functions.
Rational Function
A function defined as a fraction where both the numerator and the denominator are polynomial functions.
Example:
To model the concentration of a drug in the bloodstream over time, you might use a rational function like , which helps predict how the concentration changes.
Redefine
To *redefine* a function at a point of removable discontinuity means to assign a specific value to the function at that point, typically the limit, to make the function continuous.
Example:
To fix the hole in at , we redefine to be 4, making the function continuous.
Removable Discontinuity
A type of discontinuity where a function has a 'hole' at a specific point, which can be 'removed' by redefining the function's value at that single point.
Example:
The function f(x) = (x^2 - 4)/(x - 2) has a removable discontinuity at x=2, as it can be simplified to f(x) = x + 2 for x ≠ 2, creating a hole at x=2.
Removable Discontinuity
A point in a function's graph where the function is undefined or has a value not equal to its limit, but the limit itself exists, creating a 'hole' that can be filled.
Example:
If you have the function , there's a removable discontinuity at because the limit exists as approaches 3, even though is undefined.
Sine Function
A periodic trigonometric function that describes a smooth oscillation, represented as $k\sin(px)$.
Example:
The height of a pendulum bob swinging back and forth can be accurately described by a sine function over time.
Tangent Function
A periodic trigonometric function defined as the ratio of sine to cosine, represented as $k\tan(px)$.
Example:
When calculating the angle of elevation to the top of a building from a certain distance, you would use the tangent function to relate the height and distance.
Trigonometric Functions
Functions that relate angles of a right-angled triangle to the ratios of its side lengths, including sine, cosine, and tangent.
Example:
When designing a Ferris wheel, engineers use trigonometric functions to model the height of a rider at any given time as the wheel rotates.
Twice-Differentiable
A function is twice-differentiable if its second derivative exists at every point in its domain, which means both the function and its first derivative are continuous.
Example:
The position of a pendulum swinging back and forth is often modeled as a twice-differentiable function, allowing us to analyze its velocity and acceleration.
Vertical Asymptote
A vertical line on a graph that the function approaches infinitely closely but never actually touches, typically occurring where the denominator of a rational function is zero.
Example:
The function y = 1/x has a vertical asymptote at x=0, meaning the graph gets infinitely close to the y-axis but never crosses it.
Vertical Asymptote
A vertical line that the graph of a function approaches but never touches, typically occurring at points where the function is undefined due to a zero in the denominator.
Example:
The graph of has a vertical asymptote at , illustrating how the function's values tend towards infinity as gets closer to zero.