Section 1.3 Flashcards - Difference Quotients and Piecewise Functions

Practice flashcards covering difference quotients, average rates of change, piecewise functions, and absolute value conversions from Section 1.3 Auto Notes.

What is the difference quotient formula, and what does it geometrically represent?

The formula is $\frac{f(x+h) - f(x)}{h}$. It geometrically represents the slope of the secant line ($m_{sec}$), which is the average rate of change (AROC) between $(x, f(x))$ and $(x+h, f(x+h))$.

What is the fully simplified difference quotient for the function $f(x) = 4x$?

$$4$$

What is the fully simplified difference quotient for the function $f(x) = 1 - 2x - 3x^2$?

$$-2 - 6x - 3h$$

What is the fully simplified difference quotient for the function $g(x) = \frac{1}{x+1}$?

$$\frac{-1}{(x+h+1)(x+1)}$$

What is the fully simplified difference quotient for the function $f(x) = \sqrt{-2x}$?

$$\frac{-2}{\sqrt{-2(x+h)} + \sqrt{-2x}}$$

For $f(x) = \sqrt{-2x}$, what is the average rate of change (AROC) on the interval $[-8, -2]$?

$$-\frac{1}{3}$$

How is the absolute value function $f(x) = |2x+6|$ expressed as a piecewise function without absolute value signs?

$$f(x) = \begin{cases} 2x+6, & x \ge -3 \\ -2x-6, & x < -3 \end{cases}$$

How is the function $f(x) = |2x - 4| + 1$ written as a piecewise function without utilizing absolute value signs?

$$f(x) = \begin{cases} 2x - 3, & x \ge 2 \\ -2x + 5, & x < 2 \end{cases}$$

Given the piecewise function $h(x) = \begin{cases} \frac{x^2-9}{x-3}, & \text{if } x \neq 3 \\ 6, & \text{if } x = 3 \end{cases}$, what are the evaluated values for $h(5)$, $h(0)$, and $h(3)$?

$h(5) = 8$, $h(0) = 3$, and $h(3) = 6$

What is the simplified difference quotient for the parent function $f(x) = x^2$?

$$2x + h$$

What is the simplified difference quotient for the parent function $f(x) = \frac{1}{x}$?

$$\frac{-1}{x(x+h)}$$

According to Section 1.3 Auto Notes part 2, what key terms and phrases are listed to know regarding function behavior and analysis?

Increasing/Decreasing/Constant, Relative and Absolute Extrema, Relative Maximum/Minimum, Even/Odd/Neither, Piecewise Function, Difference Quotient, and Point of Discontinuity.

How is the piecewise function $f(x) = \begin{cases} 1 - x, & x \le 1 \\ x^2 - 1, & x > 1 \end{cases}$ defined based on the domains?

It equals $1 - x$ for $x \le 1$ and $x^2 - 1$ for $x > 1$