Calc 11.3

Introduction to Average Rate of Change

• The concept is often denoted as delta y over delta x.

• Alternate representations include:

  • $\frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

  • $\frac{y<em>1 y</em>2}{x<em>1 x</em>2}$

Average Rate of Change Definition

• The average rate of change of a function $f(x)$ with respect to $x$ as $x$ changes from $a$ to $b$ is given by:

  • $AROC = \frac{f(b) - f(a)}{b - a}$

• This is analogous to the slope formula.

Rephrasing Average Rate of Change

• The notation can be reinterpreted as:

  • $f(b) - f(a)$ as the change in the function,

  • $b - a$ as the change in x values.

• Expressed this way, it can resemble:

  • $\frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

Difference Quotient

• This is denoted simply as a slope for analytical purposes.

• Understanding this formula provides a connection to slack and secant lines between two points on a curve.

Slope of the Secant Line

• Definition:

  • The secant line connects two points on a curve.

• Notation refers to calculating average rates of change using the secant line's slope formula.

Connecting Concepts

• Elasticity in economics can be related back to average rate of change.

• Different types of elasticity alter interpretations but fundamentally relate back to the same principles.

Example Problem 1: Functions and Rates of Change

Given Function: f(x) = 1 - 3x^2

• Interval: from $x = -2$ to $x = 0$

• Steps:

  1. Calculate $f(-2)$:

  • $f(-2) = 1 - 3(-2)^2 = 1 - 12 = -11$

  1. Calculate $f(0)$:

  • $f(0) = 1 - 3(0)^2 = 1 - 0 = 1$

  1. Average Rate of Change Calculation:

  • $AROC = \frac{1 - (-11)}{0 - (-2)} = \frac{12}{2} = 6$

Example Problem 2: Another Function and Rate of Change

Given Function: g(x) = e^{\sqrt{x}} + 1

• Interval: from $x = -1$ to $x = 3$

• Steps:

  1. Calculate $g(-1)$:

  • $g(-1) = e^{\sqrt{-1}} + 1 = e^{0} + 1 = 1 + 1 = 2$

  1. Calculate $g(3)$:

  • $g(3) = e^{\sqrt{3}} + 1 = e^{2} + 1$ (value will be calculated separately)

  1. Average Rate of Change Calculation:

  • $AROC = \frac{(e^2 + 1) - 2}{3 - (-1)} = \frac{e^2 - 1}{4}$

Example Problem 3: h(x) = ln(x)

• Interval: from $x = e$ to $x = e^2$

• Steps:

  1. Calculate $h(e)$:

  • $h(e) = ln(e) = 1$

  1. Calculate $h(e^2)$:

  • $h(e^2) = ln(e^2) = 2$

  1. Average Rate of Change Calculation:

  • $AROC = \frac{2 - 1}{e^2 - e} = \frac{1}{e^2 - e}$

Practical Example: Methamphetamine Seizure

• Data:

  • 2000: 1390 kg

  • 2004: 2300 kg

• Average Rate of Change Calculation:

  • $AROC = \frac{2300 - 1390}{2004 - 2000} = \frac{910}{4} = 227.5$ kg/year

Example with Travel Speed

• Travel distance: 90 miles in 1.5 hours

• Calculation of average speed:

  • $AROC = \frac{90 - 0}{1.5 - 0} = \frac{90}{1.5} = 60$ miles per hour.

Instantaneous Rate of Change

• Definition: For function $f$ at $x = a$,

  • $IROC = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

• This indicates the slope of the tangent to the curve at point $x = a$.

Deriving Instantaneous Rate of Change

Example Function: s(t) = 2t^2 - 5t + 40

• Average velocity from $t = 2$ to $t = 4$:

  • $s(2) = 2(2^2) - 5(2) + 40 = 8 - 10 + 40 = 38$

  • $s(4) = 2(4^2) - 5(4) + 40 = 32 - 20 + 40 = 52$

  • Average velocity calculation:

    • $AROC = \frac{52 - 38}{4 - 2} = \frac{14}{2} = 7$ feet/second.

Finding Instantaneous Velocity

• Calculate using limit as h approaches 0:

  • $s(2 + h) - s(2)$

  • Simplifying yields the derivative, leading to final values for velocity calculations.

Conclusion and Review

• Understanding average and instantaneous rate of changes provides critical insights into rates impacting differing functions within various fields such as physics and economics.

• Further problem-solving will solidify knowledge of these concepts leading into advanced derivatives and calculus concepts.