I. Introduction

A functional relation is defined between two quantities if and only if, for each applicable value of quantity one (called the independent variable) there is one and only one value of quantity 2 (called the dependent variable).

Examples: Given x (t) = 2t (where x is the dependent variable and t is the independent variable)

for t = 2 , x (2) = 2 (2) = 4, x = 4

for t = 3 , x ( 3) = 2 (3 )= 6, x = 6

for x (t) =  , t = 0 is not applicable

Example: For every hour of the day, at a given point, there is one temperature. For every age, an individual has a specific height.

*In science (physics), we express the relationships between quantities with mathematical formulas. Many of these formulas define functions.

Example: The area of a square = S = a² ( where a is equal to one side of the square). S is the dependent variable and a is the independent variable.

Example: Given , r is the independent variable and g is the dependent variable.

II. VARIATIONS (changes)

Let f (t) be a function of t. If t = t_{1 , then f(t) = f(t1) and if t = t2 , then f(t) = f (t2) . From t1 to t2 , the change in t is  The change in f(t) is .}

_{Note:  of anything is the second value minus the first value.}

_{Example: If t1 = 6 and t2 = 1, then} 

_{From t1 to t2 , the average rate of change of f(t) is :}

_{average rate of change =} 

_{Note: The average rate of change depends on} 

_{* The rate of change of f (t) , for a value t0 of t , is the value of  when  goes to zero.}

_{Refer to the second page of handout!}

_{Example: Given x (t) = mt}² (t1 = t and t2= t + )

when  goes to 0, you get  This is called the rate of change.

Given any function of the form x = mtⁿ , then .

Example: x (t) = 3t⁴, 

Example: x (t) = 4t^-2.5, 

The rate of change of x (t) at t = t0 is therefore 

Recall:

r1 and r2 

g1 = g(r1 ) and g2 = g(r2 ) 

Average Rate Of Change 

Rate Of Change: For functions of the form x (t)= mtⁿ , , where n is a real number.

Note: For other forms of functions, see your future math courses.

PART 2