Rates of Change, Tangent Lines, and Limits - Vocabulary

Vocabulary flashcards covering key terms from Sections 2.1 and 2.2 on rates of change, tangent lines, limits, limit laws, and continuity.

Average speed

The rate of change of distance with respect to time over a given interval; computed as (distance at t2 − distance at t1) / (t2 − t1).

Instantaneous speed

The rate of change of position at a single moment; the limit of average speeds as the time interval approaches zero.

Average rate of change

The change in the function value divided by the change in the input over an interval; the slope of the secant line through the interval.

Secant line

A line through two points on a curve; its slope equals the average rate of change over the interval.

Slope of a curve (tangent slope)

The derivative at a point; the limit of secant slopes as the second point approaches the first.

Tangent line

The line through a point on a curve with slope equal to the curve’s derivative at that point.

Limit

The value that f(x) approaches as x approaches c; the idea of approaching, regardless of the function’s value at c.

Continuity

A function is continuous at c if lim f(x) as x→c equals f(c).

Hole in the graph

A point where the function is undefined but the limit exists; represented as a hole in the graph.

Polynomial function

A function defined by a polynomial; the limit of a polynomial as x→c equals P(c).

Rational function

A ratio of two polynomials; if Q(c) ≠ 0, then lim P(x)/Q(x) as x→c = P(c)/Q(c).

Limit laws

Rules that allow taking limits through common algebraic operations (e.g., sums, products, etc.).

Sum Rule

lim [f(x) + g(x)] = lim f(x) + lim g(x), provided the limits exist.

Difference Rule

lim [f(x) − g(x)] = lim f(x) − lim g(x).

Constant Multiplication Rule

lim [k f(x)] = k lim f(x).

Product Rule

lim [f(x) g(x)] = (lim f(x))(lim g(x)).

Quotient Rule

lim [f(x)/g(x)] = lim f(x) / lim g(x), provided lim g(x) ≠ 0.

Power Rule

lim [f(x)]^n = [lim f(x)]^n for a positive integer n.

Root Rule

lim [ (f(x))^(1/n) ] = [lim f(x)]^(1/n), with domain considerations.

Limits of Polynomials

If P is a polynomial and c is real, lim P(x) as x→c = P(c).

Limits of Rational Functions

If P and Q are polynomials and Q(c) ≠ 0, lim P(x)/Q(x) as x→c = P(c)/Q(c).

Squeeze Theorem (Sandwich Theorem)

If g(x) ≤ f(x) ≤ h(x) near c and lim g(x) = lim h(x) = L, then lim f(x) = L.

Eliminating common factors from zero denominators

Canceling a common factor to obtain a simpler expression with the same limit for x ≠ c.

Numerical estimation of limits

Using calculator or computer values of f(x) for x near c to estimate lim f(x) as x→c.