Unit B - Differentiation
B3.3 - Rates of Change
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Lesson Objectives
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1. Calculate an average rate of change
2. Calculate an instantaneous rate of change using
difference quotients and limits
3. Calculate instantaneous rates of change
numerically, graphically, and algebraically
4. Calculate instantaneous rates of change in real
world problems
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(A) Exploration
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To investigate average and instantaneous
rates of change, you will complete the
following investigation with your group in your
own
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Ask questions of me only if the instructions
are not clear
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Rates of Change - An Investigation
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PURPOSE  predict the rate at which the
height is changing at 2 seconds
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Consider the following data of an object
falling from a height of 45 m.
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Table of Data
Time
Height (in meters)
0
45
0.5
43.8
1.0
40
1.5
33.8
2.0
25
2.5
16.3
3.0
0
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Table of Data
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Scatterplot and Prediction
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1. Prepare a scatter-plot of the data.
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2. We are working towards finding a good estimate for the rate of
change of the height at t = 2.0 s.
So from your work in science courses like physics, you know that
we can estimate the instantaneous rate of change by drawing a
tangent line to the function at our point of interest and finding the
slope of the tangent line.
So on a copy of your scatter-plot, draw the curve of best fit and
draw a tangent line and estimate the instantaneous rate of
change of height at t = 2.0 s.
How confident are you about your prediction.
Give reasons for your confidence (or lack of confidence).
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Algebraic Estimation – Secant Slopes
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To come up with a prediction for the instantaneous rate of change
that we can be more confident about, we will develop an algebraic
method of determining a tangent slope.
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So work through the following exercise questions:
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We will start by finding average rates of change, which we will use
as a basis for an estimate of the instantaneous rate of change.
Find the average rate of change of height between t = 0 and t = 2 s
Mark both points, draw the secant line and find the average rate of
change.
Now find the average rate of change of height between
 (i) t = 0.5 and t = 2.0,
 (ii) t = 1.0 and t = 2.0,
 (iii) t = 1.5 and t = 2.0,
Draw each secant line on your scatter plot
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Prediction Algebraic Basis
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Now using the work from the previous slide,
we can make a prediction or an estimate for
the instantaneous rate of change at t = 2.0 s.
(i.e at what rate is the height changing 1990)
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Explain the rationale behind your prediction.
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Can we make a more accurate prediction??
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Algebraic Prediction – Regression Equation
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Unfortunately, we have discrete data in our example,
which limits us from presenting a more accurate
estimate for the instantaneous rate of change.
If we could generate an equation for the data, we
may interpolate some data points, which we could
use to prepare a better series of average rates of
change so that we could estimate an instantaneous
rate of change.
So now find the best regression equation for the
data using technology.
Justify your choice of algebraic model for the height
of the object.
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Algebraic Prediction – Regression Equation
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Algebraic Prediction – Regression Equation
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Algebraic Prediction – Regression Equation
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Now using our equation, we can generate
interpolated values for time closer to t = 2.0 s (t =
1.6, 1.7, 1.8, 1.9 s). Now determine the average
rates of change between
(i) t = 1.6 s and t = 2.0 s,
(ii) t = 1.7 s and t = 2.0 s etc...
We now have a better list of average rates of
change so that we could estimate an instantaneous
rate of change.
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Algebraic Prediction – Regression Equation
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Algebraic Prediction – Regression Equation
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Best Estimate of Rate of Change
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Finally, what is the best estimate for the
instantaneous rate of change at t = 2.0 s?
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Has your rationale in answering this question
changed from previously?
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How could you use the same process to get an even
more accurate estimate of the instantaneous rate of
change?
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Another Option for Exploration
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One other option to explore:
Using our equation, generate other interpolated values for
time close to but greater than t = 2.0 s (i.e. t = 2.5, 2.4, 2.3,
2.2, 2.1 s).
Then calculate average rates of change between
(i) t = 2.5 and t = 2.0,
(ii) t = 2.4 and t = 2.0, etc.... which will provide another list of
average rates of change.
Provide another estimate for an instantaneous rate of change
at t = 2.0 s.
Explain how this process is different than the option we just
finished previously?
How is the process the same?
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A Third Option for Exploration
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Another option to explore is as follows:
(i) What was the average rate of change between t = 1.0 and t = 2.0?
(ii) What was the average rate of change between t = 3.0 and t = 2.0?
(iii) Average these two rates. Compare this answer to your previous
estimates.
(iv) What was the average rate of change between t = 1.0 and t = 3.0?
Compare this value to our previous estimate.
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(v) Now repeat the process from Question (iv) for the following:
 (a) t = 1.5 and t = 2.5
 (b) t = 1.75 and t = 2.25
 (c) t = 1.9 and t = 2.1
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(vi) Explain the rational (reason, logic) behind the process in this third option
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Summary
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From your work in this investigation:
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(i) compare and contrast the processes of manually estimating a
tangent slope by drawing a tangent line and using an algebraic
approach.
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(ii) Explain the meaning of the following mathematical statement:
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slope of tangent = m  lim
t2
Or more generalized, explain
h ( t )  h (2 )
t2
m  lim
xa
f ( x )  f ( a)
xa
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Summary
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Summary
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(B) Instantaneous Rates of Change Numeric Calculation
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Ex 1. The point (2,2) lies on
the curve f(x) = 0.25x3. If Q
is the point (x,f(x)), find the
average rate of change (or
the secant slope of the
segment PQ) of the function
f(x) = 0.25x3 if the x coordinate of P is:
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Then, predict the
instantaneous rate of
change at x = 2
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(i) x = 3
(ii) x = 2.5
(iii) x = 2.1
(iv) x = 2.01
(v) x = 2.001
(vi) x = 1
(vii) x = 1.5
(viii) x = 1.9
(ix) x = 1.99
(x) x = 1.999
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(B) Instantaneous Rates of Change Numeric Calculation
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Ex 2.
For the function f(x),
f (x) 
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sin( x )
x
Pick some appropriate
x values to allow for a
valid numerical
investigation of
lim
x0
determine instantaneous
rate of change at x = 0
sin( x )
x
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(C) Instantaneous Rates of Change Algebraic Calculation
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Find the instantaneous
rate of change of the
function f(x) = -x2 + 3x 5 at x = -4
m  lim
f ( 4  h )  f ( 4 )
h0
m  lim
h
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  4  h   3  4  h   5   33 
2
h0
m  lim
h
 h
2
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 8 h  16    12  3 h   5   33 
h0
m  lim
h
 h
2
 11 h  33    33 
h0
m  lim
h0
h
h h  11 
h
m  lim  h  11 
h0
m  11
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(C) Instantaneous Rates of Change Algebraic Calculation
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Make use of the limit
definition of an
instantaneous rate of
change to determine
the instantaneous rate
of change of the
following functions at
the given x values: 
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f ( x )  2 x  5 x  40
2
g( x )  1 
h(x) 
2x
1
2 x
at
at
at
x  4
x 1
x 2
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(D) Internet Links - Applets
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The process outlined in the previous slides is
animated for us in the following internet links:
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Secants and tangent
A Secant to Tangent Applet from David Eck
JCM Applet: SecantTangent
Visual Calculus - Tangent Lines from Visual Calculus
– Follow the link for the Discussion
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(E) Homework
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S3.3, p173-178
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(1) average rate of change: Q1,3,6,8
(2) instantaneous rate of change:
Q9,12,14,15 (do algebraically)
(3) numerically: Q17,20
(4) applications: Q27-29,32,34,39,41
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NOTE: I AM MARKING EVEN #s
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