MATH 1551 Day 2 – Rates of Change, Secant and Tangent Lines (Section 2.1)

Flashcards cover definitions and concepts from Day 2 notes on rates of change, average rate of change, difference quotients, secant/tangent lines, and related problems (including the Nyad data and Think-Pair-Share problem).

What is the formula for the average rate of change of y = f(x) over the interval [x1, x2] in terms of f?

AROC = (f(x2) − f(x1)) / (x2 − x1).

How can the average rate of change be written using h = x2 − x1?, and what is that called?

AROC = [f(x1 + h) − f(x1)] / h, with h ≠ 0. Difference quotioent

What is a secant line in relation to a function?

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

What is the tangent line to f(x) at x = x1?

The line that touches the graph at (x1, f(x1)); its slope is the instantaneous rate of change at x1 and is the limit of secant slopes as x2 → x1 because in a very small scale, it is parallel to that segment of the function

What does the instantaneous rate of change at x1 represent graphically?

The slope of the tangent line to f at x1

What is the relationship between the slope of a secant line and the average rate of change?

The slope of the secant line equals the average rate of change over the interval.

What is the difference quotient?

The expression [f(x2) − f(x1)] / (x2 − x1) (or [f(x1 + h) − f(x1)] / h) representing the average rate of change.

Why is the tangent line important in calculus?

Its slope represents the instantaneous rate of change; it is approached by secant slopes as the interval shrinks.

For f(x) = 6 − (x − 2)^2, order the following from smallest to largest: A) slope of secant between x = 1 and x = 2; B) slope of the tangent at x = 0; C) instantaneous rate of change at x = 2; D) [f(4) − f(2)]/2.

D < C < A < B.

Problem 4 (Extra): If you take 2 hours to cross a 1-mile bridge, how fast must you return to achieve an average round-trip speed of 1 mph?

You would need to return in 0 hours (instantaneously); impossible in practice—thus the target average speed cannot be achieved.

When can you use the AROC to aproximate the IROC?

When the interval x is extremely small, since the secant’s slope becomes very close to that of the tangent