Average Rate of Change and Function Operations

Simplification of expressions with a unit denominator:
- When an expression is divided by $1$ , the denominator does not alter the value, making explicit division by $1$ unnecessary.
Evaluation of total function change between $x_1 = 1$ and $x_2 = 2$ :
- Expression evaluated: $2 \times 2^3 - 2 \times 2^2 - 3$ along with subtracting $2$ three times ( $-2 - 2 - 2$ ).
- The calculation involves evaluating $2 \times 2^3$ (noted as $60$ in computation), subtracting $2 \times 2^2 = 8$ , and subtracting $3$ .
- The resulting net increase in the function value from $x_1 = 1$ to $x_2 = 2$ is $8$ units.

Application Scenario Setup:
- A rocket is launched vertically straight up into the air.
- The elapsed flight time after launch is represented by variable $t$ , measured in seconds ( $\text{s}$ ).
- The rocket returns and hits the ground at $t = 11.5\text{ s}$ , at which point its height is $0\text{ m}$ .
- The height of the rocket changes over time, represented as a function of time $h(t)$ .
Definition and Formula for Average Rate of Change:
- Average rate of change measures how much a dependent variable (height $h(t)$ ) changes relative to an independent variable (time $t$ ) over a specified interval $[t_1, t_2]$ .
- Formula: $\text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}$

Interval from $t_1 = 0\text{ s}$ to $t_2 = 2.3\text{ s}$ :
- Height at initial time $t_1 = 0\text{ s}$ is $h(0) = 0\text{ m}$ .
- Height at time $t_2 = 2.3\text{ s}$ is $h(2.3) = 92\text{ m}$ .
- Substitution into average rate of change formula: $\text{Average Rate of Change} = \frac{h(2.3) - h(0)}{2.3 - 0}$ $\text{Average Rate of Change} = \frac{92 - 0}{2.3 - 0} = \frac{92}{2.3}$
- Evaluation of numerical result: $\frac{92}{2.3} = 40$
- Units and Dimensional Analysis:
  - The numerator represents distance in meters ( $\text{m}$ ).
  - The denominator represents time in seconds ( $\text{s}$ ).
  - The unit for the average rate of change is meters per second ( $\text{m/s}$ ).
  - Final evaluated rate of change for interval $[0\text{ s}, 2.3\text{ s}]$ : $40\text{ m/s}$ .
Interval from $t_1 = 4.6\text{ s}$ to $t_2 = 11.5\text{ s}$ :
- Initial time $t_1 = 4.6\text{ s}$ corresponds to a height $h(4.6) = 138\text{ m}$ .
- Final time $t_2 = 11.5\text{ s}$ corresponds to a height $h(11.5) = 0\text{ m}$ (rocket returning to ground level).
- Substitution into average rate of change formula: $\text{Average Rate of Change} = \frac{h(11.5) - h(4.6)}{11.5 - 4.6}$ $\text{Average Rate of Change} = \frac{0 - 138}{11.5 - 4.6}$

General Principles of Constructing New Functions:
- New functions can be constructed by combining existing functions using fundamental arithmetic operations.
- Permissible operations include addition, subtraction, multiplication, and division.
- Division constraint: A function $f(x)$ divided by another function $g(x)$ to form $\frac{f(x)}{g(x)}$ is defined provided that the denominator is non-zero ( $g(x) \neq 0$ ).
Algebraic Operations Examples:
- Given base functions:
  - $g(x) = 2x^2$
  - $h(x) = 3x$
- Difference of Functions $(h - g)(x)$ :
  - Calculated as $h(x) - g(x)$ .
  - Expression: $(h - g)(x) = 3x - 2x^2$
  - Factored form (optional): $x(3 - 2x)$ .
- Product Evaluation at a Specific Input $(h \times g)(-1)$ :
  - The notation $-1$ indicates a negative input value $x = -1$ substituted into the product function.
  - To evaluate $(h \times g)(-1)$ , substitute $x = -1$ into the product expression $(h \cdot g)(x)$ .

Question regarding formula provision on examinations:
- Prompt: "Are you gonna give us that equation on the test or we have to memorize it?"
- Context: Inquiries whether the rate of change equation will be provided on the test or if students are expected to memorize it.