3 Describing Change: Rates

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3 Describing Change Rates
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3.1 - Rates of Change
Rates of change is one of the foundations of
Calculus. If, over the interval of input values
(x1, x2) the output changes from y1 to y2, then
Change y2 y1 (in output units)
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3.1 - Rates of Change
If one of the previous rates of change is
negative, that represents a decrease in the
quantity. If one of the previous rates is
positive, that represents an increase in the
quantity.
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3.1 - Rates of Change

• When describing change over an interval, be sure
  to answer the following questions
• When? Specify the interval.
• What? Specify the quantity that is changing.
• How? Indicate whether the change is an increase
  or a decrease.
• By how much? Give the numerical answer labeled
  with proper units.

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3.1 Rates of Change The Secant Line
f(x)
Secant Line A line passing thorough two points
on a graph of a function. The slope of the secant
line is called the average rate of change.
Q
P
f(x1)
f(x2)
x1
x2
Average Rate of Change In Terms of Functions
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Examples
Page 167 6, 14, 24
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3.2 Instantaneous Rates of Change
f(x)
If you zoom in very, very close to any point on
the graph of f (x), it will appear to be linear.
This is known as local linearity.
a
The line that the graph resembles when zoomed in
very, very close, is called the tangent line to
the graph of f at x a.
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3.2 - Instantaneous Rates of Change
The slope of a curve at a point is called the
instantaneous rate of change of the function at
that point.
The slope of the curve at a point is the slope of
the tangent line to the curve at that point. Put
in a more formal way
Given a function f and a point P on the graph of
f, the instantaneous rate of change at point P is
the slope of the graph at P and is the slope of
the line tangent to the graph at P (provided the
slope exists).
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3.2 - Facts About Tangent Lines
Lines tangent to a smooth, non-linear 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.
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Facts About Tangent Lines
If a function is not continuous at a point, then
the instantaneous rate of change does not exist
at that point.
If a function is continuous at a point, but comes
to a sharp point at that point (called a cusp
point), then the instantaneous rate of change
does not exist at that point.
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Facts About Tangent Lines
P
f(x1)
x1
As x2 approaches x1, Q approaches P and the
secant line approaches the tangent line.
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Finding Slopes of Tangent Lines
The slope of the tangent line at x a can be
estimated by choosing a point very, very close to
a and determining the slope of the secant line
through these points.
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3.2 - Finding Slopes of Tangent Lines
A symmetric difference quotient can be used as
well to estimate the slope of a tangent line at x
a by taking two point on either side of a that
are equal distance from a and computing the slope
of the secant line through these two points.
f(x1)
f(x2)
x1
a
x2
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Examples
Page 184 8, 10, 14, 16, 26, 32
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3.3 - Derivatives

• Average Rates of Change
• Measure how rapidly (on average) a quantity
  changes over an interval.
• Can be obtained by calculating the slope of the
  secant line between two points.
• Requires discrete data points, a continuous
  curve, or a piecewise continuous function.

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3.3 - Derivatives

• Instantaneous Rates of Change
• Measure how rapidly a quantity is changing at a
  point.
• Can be obtained by calculating the slope of the
  tangent line at a single point.
• Require continuous or piecewise continuous curves
  to calculate.

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3.3 - Derivatives

• Each of the following phrases have the same
  meaning.
• Instantaneous rate of change
• Rate of change
• Slope of the curve
• Slope of the tangent line
• Derivative

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3.3 - Derivatives
The following notations all have the same meaning
and stand for the derivative of f with respect to
x.
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3.3 - Derivatives
Percent Rate of Change
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Examples
Page 201 4, 8, 10, 18
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3.4 Numerically Finding Slopes
The slope of the tangent line at x a can be
estimated by choosing a point very, very close to
a and determining the slope of the secant line
through these points.
Secant Line Green Tangent Line - Red
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3.4 Numerically Finding Slopes
A symmetric difference quotient can be used as
well to estimate the slope of a tangent line at x
a by taking two point on either side of a that
are equal distance from a and computing the slope
of the secant line through these two points.
f(x1)
f(x2)
x1
a
x2
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Examples
Page 213 4, 8, 12, 16
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3.5 Algebraically Finding Slopes

• Do the following to determine f ?(x).
• Begin with a typical point (x, f (x)).
• Choose a close point (x h, f (x h)).
• Write a point of the slope of the secant line
  between the points and simplify.
• Evaluate the limit of the slope as h approaches
  0.
• The limiting value is the derivative formula at
  each input where the limit exists.

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3.5 Algebraically Finding Slopes

• If y f (x), then the derivative dy/dx is given
  by the formula
• provided the limit exists.