# Section 2.2 Average and Instantaneous Rate of Change The Derivative of a Function at a Point 3.1.

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Section 2.2 Average and Instantaneous Rate of Change The Derivative of a Function at a Point 3.1

You are based in Indonesia, and you monitor the value of the US Dollar on the foreign exchange market very closely during a rather active five-day period. Suppose you find that the value of one US dollar can be well approximated by the function (The rupiah is the Indonesian currency), where t is time in days. (t = 0 represents the value of the Dollar at noon on Monday) a) What was the value of the Dollar at noon on Tuesday?

b)According to the graph, when was the value of the Dollar rising most rapidly? Monday at Noon

c) Compute the average rate of change of R(t) over the interval [1, 1+h] for h = 1, h = 0.01, h = 0.001, and h = 0.0001 HUH?

[1, 1 + 1] = [1, 2] [1, 1 + 0.01] = [1, 1.01] [1, 1 + 0.001] = [1, 1.001] [1, 1 + 0.0001] = [1, 1.0001] So what are the conclusions from this?

AS h -> 0……. h = 1, h = 0.01, h = 0.001, and h = 0.0001 The slope of the secant line approaches 300…. 200, 299, 299.9, 299.99 Based upon this conclusion, we can say: Back to the problem…….. On Tuesday, the Rupiah was increasing in value at A rate of 300 rupiahs per day. General Conclusion…..

A look at the definition of the first derivative at a point in action Note: Scroll down for the second

Let f(x) = ln x a) Find the average rate of change of f between x = 0.99 and x = 1. b) Find the average rate of change of f between x = 1 and x = 1.01. c) Explain why the answer in (a) is to large of an estimate for f (x) and the answer in (b) is too small of an estimate for f (x).

Let f(x) = ln x a)Find the average rate of change of f between x = 0.99 and x = 1 b)Find the average rate of change of f between x = 1 and x = 1.01

Let f(x) = ln x a) The average rate of change between x = 0.99 and x = 1 is 1.005 b) The average rate of change between x = 1 and x = 1.01 is 0.995 c) Explain why the answer in (a) is to large of an estimate for f (x) and the answer in (b) is too small of an estimate for f (x).

Given the graph below, where does the derivative NOT exist? X X X X X X X X X

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