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Unformatted text preview: 2.3 Instantaneous Rates of Change.notebook September 18, 2009 2.2/2.3 Examining Instantaneous Rates of Change
Instantaneous rate of change:
the exact rate of change of a function y = f(x) at a specific value of the independent variable x = a How can we use the average rate of change to give us an estimate of the rate of change at exactly one point (i.e. instantaneous rate of change)? Complete the Investigation in partners (to be handed in). Investigation Question: How can you use the slopes of secant lines to estimate the instantaneous rate of change?
A golf ball lying on the grass is hit so that its initial vertical velocity is 25 m/s. The height, h, in metres, of the ball after t seconds can be modeled by the function h(t) = ‐4.9t2 + 25t. Interval Λ h Λ t AROC = Λ h
Λ t 1 ≤ t ≤ 2 1 ≤ t ≤ 1.5 1 ≤ t ≤ 1.1
1 ≤ t ≤ 1.01 1 ≤ t ≤ 1.001 1 2.3 Instantaneous Rates of Change.notebook September 18, 2009 The instantaneous rate of change is represented by a tangent line to a curve at a point.
We cannot find the slope of the tangent because we only know one point on the tangent line, P. We can estimate the slope of the tangent at P (ie. the instantaneous rate of change at P) by finding the slopes of secants for smaller and smaller intervals around P.
Demonstration What happens as the point Q approaches the point P? As the point Q approaches the point P, the slope of the secant line PQ approaches the slope of the tangent to the point P. The best estimate occurs when the interval used to calculate the rate of change is as small as possible. If we want to know the instantaneous rate of change of f(x) at x = a, consider the intervals between x = a and x = a ± h, where h is a really small number. Instantaneous Rate of Change This is called the difference quotient. Mathematically: (1) select points closer and closer to the point P
(2) calculate the slope of each secant (3) the slopes of the secants will approach a value, which is the slope of the tangent
2 2.3 Instantaneous Rates of Change.notebook September 18, 2009 Example #1:
Estimate the instantaneous rate of change of f(x) = x2 at the point x = 2. METHOD ONE: Intervals on one side of 2 METHOD TWO: Difference quotient GRAPH Example #2:
A soccer ball is kicked into the air such that its height, h, in metres, after time t seconds can be modelled by the function h(t) = ‐4.9t2 + 15t + 1.
(a) Determine the average rate of change over the interval 1 ≤ t ≤ 3.
(b) Estimate the instantaneous rate of change when t = 1. (c) For this situation, what does the instantaneous rate of change represent when t = 1? 3 2.3 Instantaneous Rates of Change.notebook September 18, 2009 Assigned Work:
page 86 #3, 4a, 6, 9,15
page 91 #3 ‐ set C 4 ...
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- Spring '11