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2.1 Rates of Change Wed Sept 10 Do Now Given f(x) = x^2 + 3 Find the slope of the secant line through (0, f(0)) and (3, f(3))

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Rates of Change What is a rate of change? –If x and y are related quantities, then a rate of change should tell us how much y changes in response to a unit change in x Examples: –Velocity –Slope

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Average Rate of Change The average rate of change is one measured over a certain interval –Slope of a line between 2 points –Change in velocity, etc between two times

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Instantaneous Rate of Change An instantaneous rate of change is one measured at a particular point in time –Tangent lines to a curve give the instantaneous slope of the curve at the point of tangency –It can be difficult to find instantaneous rates of change because rates tend to change

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Average Velocity Average velocity = change in position / length of time interval EX: an automobile travels 200 km in 4 hours, then its average velocity during this period is 50 km/h

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Average vs Instantaneous Average rates of change can be easy to measure, but leave room for error Instantaneous rates of change are very precise, but can be difficult to measure This relationship will lead us to the concept of a limit

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Graphical Interpretation A graphical interpretation of a rate of change is the slope of a function The slope between 2 points is the average rate of change The slope of the tangent line of a curve at a point is the instantaneous rate of change

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Using Average Rates to Find Instantaneous Rates The closer the 2 points of an average rate are, the more accurately it resembles the instantaneous rate

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Ex 1 A stone released from a state of rest falls to earth. Estimate the instantaneous velocity at t = 0.8 sec if the stone’s distance traveled is modeled by the function D = 4.9t^2

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Closure Journal Entry: What is the difference average and instantaneous rates of change? How can we estimate instantaneous rates of change? HW: p.64 #1 5 7 11 13 15 19 25

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AIM : How do we find tangent lines to a graph? Do Now: Find the slope of the line shown below that goes through points (0,6) and (3,0). HW2.1: p66-67 #9,10,19,22,23.

AIM : How do we find tangent lines to a graph? Do Now: Find the slope of the line shown below that goes through points (0,6) and (3,0). HW2.1: p66-67 #9,10,19,22,23.

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