Net Change, Average Rate of Change, and Difference Quotient Flashcards

Vocabulary flashcards covering core definitions, formulas, step-by-step procedures, and specific numerical examples for Net Change, Average Rate of Change, Difference Quotient, and Graph Reading concepts from the lecture notes.

Net Change

The difference between what something ended at and what it started at, defined as $f(b) - f(a)$ given a starting point $(a, f(a))$ and an ending point $(b, f(b))$.

Average Rate of Change

The ratio of the net change of a function to the difference between its endpoints, calculated using the formula $\frac{f(b) - f(a)}{b - a}$.

Difference Quotient

An expression for the average rate of change formulated by writing the second point in terms of the first point $(a+h, f(a+h))$, defined as $\frac{f(a+h) - f(a)}{h}$.

Graph Height $f(a)$

The height above or below the x-axis at the input value $a$ on the graph of a function.

Graph Endpoint Conventions

Graphical notation where visible dots mean the graph ends at those points in the picture, and no dots mean the graph continues forever.

Domain (Page 6 Graph)

The set of x-values represented on the example graph, given as the interval $[1, 5]$.

Range (Page 6 Graph)

The set of y-values represented on the example graph, given as the interval $[-2, 3]$.

Net Change Calculation Example ($f(t) = 3t^2 + 2t + 1$)

The evaluation of net change between $t = 1$ and $t = 4$, given by $f(4) - f(1) = 57 - 6 = 51$.

Average Rate of Change Calculation Example ($f(t) = 3t^2 + 2t + 1$)

The evaluation of average rate of change from $t = 1$ to $t = 4$, given by $\frac{f(4) - f(1)}{4 - 1} = \frac{51}{3} = 17$.

Average Rate of Change Example ($f(x) = x^3 - 2x$)

The evaluation of average rate of change from $x = -2$ to $x = 2$, given by $\frac{f(2) - f(-2)}{2 - (-2)} = \frac{4 - (-4)}{4} = \frac{8}{4} = 2$.

Step 1 of Difference Quotient for $f(x) = 3 - 2x^2$

Evaluating and expanding $f(a+h) = 3 - 2(a+h)^2 = 3 - 2(a^2 + 2ah + h^2) = 3 - 2a^2 - 4ah - 2h^2$.

Step 3 of Difference Quotient for $f(x) = 3 - 2x^2$

Subtracting $f(a)$ from $f(a+h)$, which yields $f(a+h) - f(a) = (3 - 2a^2 - 4ah - 2h^2) - (3 - 2a^2) = -4ah - 2h^2$.

Step 4 of Difference Quotient for $f(x) = 3 - 2x^2$

Dividing the simplified expression $f(a+h) - f(a)$ by $h$, which yields $\frac{-2h^2 - 4ah}{h} = -2h - 4a$.