§2.2: Estimating Instantaneous Rate of Change

September 30, 2010

Review of AROC

2112

12 , xxxxx

xfxf

x

y

Review of AROC

graphically related to the secant line intersecting the graph of a function at two points

Difference Quotient

basically the same as delta x over delta y but combines the concept the delta concept with function notation

∆x is the size of our interval and we replace that expression with h

x

xfxxf

h

xfhxf

x

y

)()()()(

Instantaneous Rate of Change we estimate the IROC of a function f(x) at a

point x = a by examining the AROC with a very small interval around the value of x = a

represented graphically by a tangent line to the curve f(x) at the point x = a

Tangent Line

a line that intersectsthe curve at asingle point

Interval Method of Estimating IROC to estimate the IROC of a function at a point,

we need to first talk about the intervals we can use…

Intervals

preceding interval an interval having an upper bound as the value of

x in which we are interested

following interval an interval having a lower bound as the value of x

in which we are interested

Intervals (cont.)

centred interval an interval containing the value of x in which we

are interested

Method for Determining IROC easiest way is with a centred interval you must “look” on both sides of the point two successive approximations

one is insufficient, you are looking for convergence we want our ∆x or “h” to be as small as possible,

(∆x < 0.1 is usually safe) at least on the second approximation

want to see if the difference quotient gets closer to a certain value as the size of the interval becomes smaller, 3 successive approximations allows us to perform more careful trend analysis

Graphically…

Example

Determine the IROC of f(x) = x2 + 1 at x=2

Difference between AROC and IROC AROC → over an interval

IROC → at a point although technically IROC is an estimation in this

course so it is over a small interval as an approximation to a point

Advanced Algebraic Method

doesn’t use actual numerical values of h or ∆x for the interval but is based on the idea that the size of the interval becomes infinitely small in size

requires solid algebraic skills relies on the difference quotient definition allows you to calculate the exact IROC at a

point and avoid an estimation

Example

Determine the exact IROC of f(x) = x2 + 1 at x=2

What do I need to know?

you MUST be able to estimate the instantaneous rate of change of a function at a point via the method of successive approximations

Homework

§2.2 p.85 #1-4, 6, 7, 9, 10, 12, 15 Reading p.89-91