MAT 212

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• MAT 212
• Brief Calculus

Section 3.2Instantaneous Rates of Change
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At the instant the horse crossed the finish
line, it was traveling at 42 miles per hour.
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There is a paradox in trying to study motionat a
particular instant in time.
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There is a paradox in trying to study motionat a
particular instant in time.
By focusing on a single instant, we stop
motion!!!
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How can we know the speed of a horse at the
instant it crosses the finish line?
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How can we know the speed of a horse at the
instant it crosses the finish line?
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Instantaneous Rate of Change The instantaneous
rate of change at a point on a curve is the slope
of the curve at that point.
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Instantaneous Rate of Change The instantaneous
rate of change at a point on a curve is the slope
of the curve at that point. Slope is the measure
of tilt of a LINE!
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Local Linearity If we look close enough near any
point on a smooth curve, the curve will look like
a straight line! Example Grapefruit
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Local Linearity If we look close enough near any
point on a smooth curve, the curve will look like
a straight line! This is called the tangent
line at that point The slope of a graph at a
point is the slope of the tangent line at that
point
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Example On this graph, indicate where the
slope is positive
greatest negative zero Find two points
on the curve where the slopes are about the
same.
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Average Rate of Change Graphically, the average
rate of change over the interval a x b is the
slope of the secant line connecting f(a) with
f(b). Instantaneous Rate of Change Instantaneous
rate of change is the slope of the line TANGENT
to the curve at a single point.
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The Tangent Line The tangent line at a point Q on
a smooth, continuous graph is the limiting
position of the secant lines between point Q and
point P as P approaches Q along the graph (if
the limiting position exists)
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The Tangent Line General Rule Lines tangent to a
smooth, nonlinear curve do not cut through the
graph of the curve at the point of tangency and
lie completely on one side of the graph near the
point of tangency Except at an inflection
point. Examples Concave upConcave down
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• Where instantaneous rate of change DOES NOT
  exist
• Point of discontinuity
• Sharp point
• Vertical tangent (no run)
• Examples

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Symmetric Difference Quotient To estimate the
rate of change from a set of data without taking
the time to construct a model. Example Book
problem 32 Pg 190