Glossary

Average Rate of Change (AROC)

Criticality: 2

The slope of the secant line between two points on a function, representing the average speed or change over a given interval.

Example:

If a car travels 100 miles in 2 hours, its Average Rate of Change (average speed) is 50 mph.

Common Denominator (for limits)

Criticality: 2

An algebraic technique used to combine fractions within a limit expression by finding a common denominator, often helpful when dealing with complex fractions that result in indeterminate forms.

Example:

When evaluating $\lim_{x \to 0} \frac{\frac{1}{x+1}-1}{x}$ , finding a common denominator for the numerator helps simplify the expression.

Conjugate Multiplication (for limits)

Criticality: 2

An algebraic technique involving multiplying the numerator and denominator by the conjugate of an expression containing square roots, used to rationalize the expression and resolve indeterminate forms.

Example:

To evaluate $\lim_{x \to 0} \frac{\sqrt{x+4}-2}{x}$ , you would use conjugate multiplication by $\sqrt{x+4}+2$ to simplify the expression.

Continuity Over an Interval

Criticality: 2

A function is continuous over an interval if it is continuous at every single point within that interval.

Example:

The function $f(x) = \sin(x)$ demonstrates continuity over an interval like $(-\infty, \infty)$ because it has no breaks, holes, or jumps anywhere.

Continuity at a Point

Criticality: 3

A function is continuous at a point x=a if three conditions are met: f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a).

Example:

For $f(x) = x^2$ , it exhibits continuity at a point like x=3 because f(3)=9, $\lim_{x \to 3} x^2 = 9$ , and they are equal.

Exponential Functions (as continuous functions)

Criticality: 1

Functions of the form $f(x) = a^x$ (where a > 0 and a \ne 1), which are continuous everywhere across their entire domain.

Example:

The function $f(x) = e^x$ is an exponential function and is continuous for all real numbers.

Factoring (for limits)

Criticality: 3

An algebraic technique used to simplify rational functions by factoring the numerator and/or denominator to cancel common factors, often resolving indeterminate forms like 0/0.

Example:

To evaluate $\lim_{x \to 3} \frac{x^2-9}{x-3}$ , you can use factoring to simplify the expression to $\lim_{x \to 3} (x+3)$ , which then evaluates to 6.

Horizontal Asymptotes

Criticality: 2

Horizontal lines that a function's graph approaches as x tends towards positive or negative infinity, indicating the end behavior of the function.

Example:

The function $f(x) = \frac{2x^2+1}{x^2-5}$ has a horizontal asymptote at y=2.

Infinite Discontinuity

Criticality: 2

A type of discontinuity where the function approaches positive or negative infinity as x approaches a certain value, typically occurring at a vertical asymptote.

Example:

The function $f(x) = \frac{1}{x-2}$ has an infinite discontinuity at x=2, as the function values shoot off to infinity or negative infinity.

Instantaneous Rate of Change (IROC)

Criticality: 3

The slope of the tangent line at a single point on a function, representing the rate of change at a specific moment. It is found using limits.

Example:

The speedometer in your car shows your Instantaneous Rate of Change (speed) at a precise moment.

Intermediate Value Theorem (IVT)

Criticality: 3

A theorem stating that if a function is continuous on a closed interval [a, b], then it must take on every y-value between f(a) and f(b) at least once within that interval.

Example:

If a continuous function $f(x)$ has $f(1)=2$ and $f(5)=10$ , the Intermediate Value Theorem guarantees there's some x-value between 1 and 5 where $f(x)=7$ .

Jump Discontinuity

Criticality: 2

A type of discontinuity where the function 'jumps' from one y-value to another at a specific x-value, meaning the one-sided limits exist but are not equal.

Example:

A piecewise function defined as $f(x) = x$ for $x<0$ and $f(x) = x+2$ for $x \ge 0$ has a jump discontinuity at x=0.

Limit

Criticality: 3

The value a function approaches as its input (x-value) gets arbitrarily close to a certain point, regardless of whether the function is defined at that point.

Example:

As x approaches 0, the function sin(x)/x approaches 1, so the limit is 1.

Limit Laws (Algebraic Properties of Limits)

Criticality: 2

Rules that allow for the evaluation of limits of sums, differences, products, quotients, and powers of functions by breaking them into simpler limits.

Example:

Using Limit Laws, you can find $\lim_{x \to 2} (x^2 + 3x)$ by evaluating $\lim_{x \to 2} x^2$ and $\lim_{x \to 2} 3x$ separately and adding the results.

Limit Notation

Criticality: 3

The mathematical way to express a limit, written as $\lim_{x \to a} f(x) = L$, meaning 'the limit of f(x) as x approaches 'a' is L.'

Example:

Writing $\lim_{x \to 5} (2x+1) = 11$ uses limit notation to show the function approaches 11 as x approaches 5.

One-Sided Limit

Criticality: 3

The value a function approaches as the input approaches a point from either the left side (e.g., $\lim_{x \to a^-} f(x)$) or the right side (e.g., $\lim_{x \to a^+} f(x)$).

Example:

For a piecewise function, you might check the one-sided limit from the left at x=2 to see what value the function approaches from that direction.

Polynomials (as continuous functions)

Criticality: 1

Functions of the form $a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, which are continuous everywhere across their entire domain.

Example:

The function $f(x) = 3x^4 - 2x^2 + 7$ is a polynomial and is therefore continuous for all real numbers.

Rational Functions (as continuous functions)

Criticality: 2

Functions expressed as a ratio of two polynomials, continuous everywhere except at points where the denominator is zero.

Example:

The function $f(x) = \frac{x+1}{x-4}$ is a rational function and is continuous everywhere except at x=4.

Removable Discontinuity

Criticality: 3

A 'hole' in the graph of a function at a specific point, where the limit exists but either the function is undefined at that point or its value does not match the limit. It can be 'filled' by redefining the function.

Example:

The function $f(x) = \frac{x^2-1}{x-1}$ has a removable discontinuity at x=1 because there's a hole in the graph, but the limit as x approaches 1 exists.

Secant Line

Criticality: 2

A line that connects two distinct points on a curve. Its slope represents the average rate of change between those two points.

Example:

Drawing a line connecting (1, f(1)) and (3, f(3)) on the graph of y = x^2 creates a secant line.

Slant Asymptote

Criticality: 1

A diagonal line that a rational function approaches as x tends towards positive or negative infinity, occurring when the degree of the numerator is exactly one greater than the degree of the denominator.

Example:

The function $f(x) = \frac{x^2+1}{x}$ has a slant asymptote at y=x.

Squeeze Theorem (Sandwich Theorem)

Criticality: 2

A theorem stating that if a function is 'sandwiched' between two other functions that both approach the same limit at a point, then the function in the middle must also approach that same limit.

Example:

If you know that $-x^2 \le f(x) \le x^2$ for all x, then by the Squeeze Theorem, $\lim_{x \to 0} f(x) = 0$ .

Substitution (for limits)

Criticality: 2

A direct method for evaluating limits by plugging the approaching value into the function, applicable when the function is continuous at that point and doesn't result in an indeterminate form.

Example:

To find $\lim_{x \to 1} (x+5)$ , you can use substitution to get $1+5=6$ .

Trigonometric Functions (as continuous functions)

Criticality: 1

Functions like sine, cosine, tangent, etc., which are continuous over their respective domains.

Example:

The function $f(x) = \cos(x)$ is a trigonometric function and is continuous for all real numbers.

Two-Sided Limit

Criticality: 3

A limit that exists only if the function approaches the same value from both the left and the right sides of a given point.

Example:

If $\lim_{x \to 3^-} f(x) = 7$ and $\lim_{x \to 3^+} f(x) = 7$ , then the two-sided limit $\lim_{x \to 3} f(x) = 7$ .

Vertical Asymptotes

Criticality: 2

Vertical lines that a function's graph approaches but never touches, typically occurring where the denominator of a rational function is zero and the numerator is non-zero.

Example:

The function $f(x) = \frac{1}{x}$ has a vertical asymptote at x=0.