2.2: Estimating Instantaneous Rate of Change September 30, 2010 Review of AROC y f x2 f x1 x x2 x1
x x1 , x2 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 y f ( x h) f ( x) f ( x x) f ( x)
x h x x is the size of our interval and we replace that expression with h 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 intersects the curve at a single 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 we want our x or h to be as small as possible, (x < 0.1 is usually safe)
one is insufficient, you are looking for convergence 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 doesnt 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
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