1. §2.2: Estimating Instantaneous Rate of Change
September 30, 2010

2. Review of AROC

3. Review of AROC
graphically related to the secant line intersecting the graph of a function at two points

5. 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

6. 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

7. Tangent Line
a line that intersects the curve at a single point

8. 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…

9. Intervals preceding interval following 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

10. Intervals (cont.) centred interval
an interval containing the value of x in which we are interested

11. 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

12. Graphically…

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

14. 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

15. 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

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

17. 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

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