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§2.2: Estimating Instantaneous Rate of Change

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## §2.2: Estimating Instantaneous Rate of Change

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