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MathematicsNATIONALMATH + SCIENCEINITIATIVE
x
y
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LEVELAlgebra 2 or Math 3 in a unit on quadratic functions
MODULE/CONNECTION TO AP*Rate of Change: Average and
Instantaneous
*Advanced Placement and AP are registered trademarks of the
College Entrance Examination Board. The College Board was not
involved in the production of this product.
MODALITYNMSI emphasizes using multiple representations to
connect various approaches to a situation in order to increase
student understanding. The lesson provides multiple strategies and
models for using those representations indicated by the darkened
points of the star to introduce, explore, and reinforce
mathematical concepts and to enhance conceptual understanding.
P
G
N A
V
P – Physical V – VerbalA – AnalyticalN – NumericalG –
Graphical
Investigating Average Rate of ChangeABOUT THIS LESSON
This lesson examines the average and instantaneous rates of
change of linear and quadratic functions by calculating the slopes
of secant lines and estimating the slopes of tangent lines. First,
students consider linear functions and conclude that the slopes of
secant lines for any interval of a linear function are equal and
that the average and instantaneous rates of change are the same.
The second section of the lesson focuses on quadratic functions so
that students can observe how secant line slopes change, depending
on the interval selected. Students are led to discover a unique
property of quadratic functions: the slope of the secant line for
any particular interval is equal to the slope of the tangent line
at the midpoint of that interval. Students then apply this property
to solve a real-world situation. Throughout the lesson, students
have opportunities to reinforce their skills in determining
function values and calculating slopes.
OBJECTIVESStudents will
● determine the slope of a secant line.● estimate the
instantaneous rate of change of
a function.● write the equation for a tangent line to
a function.● discover and apply in a real-world situation
a unique property of quadratic functions: the slope of the
secant line for any interval is equal to the slope of the tangent
line at the midpoint of that interval.
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Mathematics—Investigating Average Rate of Change
COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson
addresses the following Common Core State Standards for
Mathematical Content. The lesson requires that students recall and
apply each of these standards rather than providing the initial
introductiontothespecificskill.Thestarsymbol(★)
attheendofaspecificstandardindicatesthatthehigh school standard is
connected to modeling.
Targeted StandardsF-IF.6: Calculate and interpret the
average
rateofchangeofafunction(presentedsymbolically or as a table)
over a specifiedinterval.Estimatetherateofchange from a graph.★ See
questions 1-3, 4b-i, 4k-m, 5b-f, 5i, 6a-b, 6d-e, 7a-b, 7e-g
Reinforced/Applied StandardsF-IF.7a: Graph functions expressed
symbolically
and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.
(a)Graphlinearandquadraticfunctions and show intercepts, maxima,
and minima.★ See questions 4a-b, 4e, 5a-b, 5e, 7c
A-CED.2: Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate
axes with labels and scales.★ See questions 4m, 5j, 7c
F-IF.2: Use function notation, evaluate functions for inputs in
their domains, and interpret statements that use function notation
in terms of a context. See questions 1a, 2a, 4b, 4e, 4g, 4i, 4m,
5b, 5d, 5f, 5j, 7d
S-ID.6a: Represent data on two quantitative variables on a
scatter plot, and describe how the variables are related.
(a)Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsin
the context of the data. Use given functions or choose a function
suggested bythecontext.Emphasizelinear,quadratic, and exponential
models.★ See questions 7c-g
N-Q.1: Use units as a way to understand problems and to guide
the solution of multi-step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and the
origin in graphs and data displays.★ See questions 7e, 7g
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Mathematics—Investigating Average Rate of Change
COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICEThese
standards describe a variety of instructional
practicesbasedonprocessesandproficienciesthat are critical for
mathematics instruction. NMSI incorporates these important
processes andproficienciestohelpstudentsdevelopknowledge and
understanding and to assist them in making important connections
across grade levels. This lesson allows teachers to address the
following Common Core State Standards for Mathematical
Practice.
MP.2: Reason abstractly and quantitatively. Students progress
from a computational understanding to a verbal generalization then
to a real-world application. In question 7, students convert
real-world data into a scatterplot, create a regression equation,
and then interpret values in terms of the problem situation.
MP.4: Model with mathematics. Students test the car company’s
claim by creating a regression function to fit the data and using
the model to refute the claim.
MP.5: Use appropriate tools strategically. Students use a
graphing calculator to fit a function to data and use the function
to predict additional values.
MP.8: Look for and express regularity in repeated reasoning.
Students determine that, for quadratic functions, the average rate
of change over an interval equals the instantaneous rate of change
at the midpoint of that interval, based on repeated calculations,
and then use this rule in an applied situation.
FOUNDATIONAL SKILLSThe following skills lay the foundation for
concepts included in this lesson:
● Calculate the slope of a line● Write a linear equation● Sketch
graphs of simple quadratic functions
ASSESSMENTSThe following types of formative assessments are
embedded in this lesson:
● Students engage in independent practice.● Students apply
knowledge to a new situation.● Students summarize a process or
procedure.
The following assessments are located on our website:
● Rate of Change: Average and Instantaneous – Algebra 2 Free
Response Questions
● Rate of Change: Average and Instantaneous – Algebra 2 Multiple
Choice Questions
MATERIALS AND RESOURCES● Student Activity pages ● Straight
edges● Coloredpencils(optional)● Graphing calculators
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Mathematics—Investigating Average Rate of Change
TEACHING SUGGESTIONS
This lesson offers the advantage of requiring students to
practice and apply a variety of essential skills, such as working
with function notation, calculating function values, interpreting
interval notation, and computing slopes, while exploring new
situations, recognizing patterns, and drawing generalized
conclusions. There are also ample opportunities for students to
develop graphing calculator skills and expertise. Students should
be
encouraged to use the symbol yx
∆∆
to represent the
average rate of change over an interval. The lesson can be
easily divided into three separate activities to be completed on
three different occasions: questions 1 – 3 address linear
functions, questions 4 – 6 use quadratic functions, question 7
applies the conclusions from questions 4 – 6 to a real-world
situation.
To avoid rounding errors and emphasize the use of function
notation when evaluating the difference quotient, type the function
in . From the home
screen type the following command:
or .
For example to calculate the rate of change of
for the interval , enter
then from the home screen
type .
Question 7 is a calculator-active question that provides a
real-world application for the skills students have practiced in
the earlier questions. Students may need instruction in using the
calculator’s regression feature. Rounded values should not be used
in subsequent calculations. On the TI84 calculator, enter the
x-values in List 1, the y-values in List 2, then use the command
“QuadReg L1, L2, Y1” to calculate the regression equation and
to store the equation in the graphing menu. This will
avoidtheissueofroundingthecoefficientsintheequation. After storing
the regression equation in Y1, use the home screen program to
calculate the slope.
You may wish to support this activity with TI-Nspire™
technology. See Storing Values and Expressions and Finding
Regression Equations in the NMSI TI-Nspire Skill Builders.
Suggestedmodificationsforadditionalscaffoldinginclude the
following:4 Modify the graph to provide the sketch of the
quadraticin(a)andatleastoneofthesecantlinesin(b).
7 Provide a written summary of the calculator procedures that
are needed for this question.
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Mathematics—Investigating Average Rate of Change
NMSI CONTENT PROGRESSION CHARTIn the spirit of NMSI’s goal to
connect mathematics across grade levels, a Content Progression
Chart for
eachmoduledemonstrateshowspecificskillsbuildanddevelopfromsixthgradethroughpre-calculusinanaccelerated
program that enables students to take college-level courses in high
school, using a faster pace to compress content. In this sequence,
Grades 6, 7, 8, and Algebra 1 are compacted into three courses.
Grade 6 includes all of the Grade 6 content and some of the content
from Grade 7, Grade 7 contains the remainder of the Grade 7 content
and some of the content from Grade 8, and Algebra 1 includes the
remainder of the content from Grade 8 and all of the Algebra 1
content.
The complete Content Progression Chart for this module is
provided on our website and at the beginning of the training
manual. This portion of the chart illustrates how the skills
included in this particular lesson develop as students advance
through this accelerated course sequence.
6th Grade Skills/Objectives
7th Grade Skills/Objectives
Algebra 1 Skills/Objectives
Geometry Skills/Objectives
Algebra 2 Skills/Objectives
Pre-Calculus Skills/Objectives
From graphical or tabular data or from a stated situation
presented in paragraph form, calculate or compare the average rates
of change and interpret the meaning.
From graphical or tabular data or from a stated situation
presented in paragraph form, calculate or compare the average rates
of change and interpret the meaning.
From graphical or tabular data or from a stated situation
presented in paragraph form, calculate or compare the average rates
of change and interpret the meaning.
From graphical or tabular data or from a stated situation
presented in paragraph form, calculate or compare the average rates
of change and interpret the meaning.
From graphical or tabular data or from a stated situation
presented in paragraph form, calculate or compare the average rates
of change and interpret the meaning.
From graphical or tabular data or from a stated situation
presented in paragraph form, calculate or compare the average rates
of change and interpret the meaning.
Recognize intervals of functions with the same average rate of
change.
Recognize intervals of functions with the same average rate of
change.
Recognize intervals of functions with the same average rate of
change.
Compare average rates of change on different intervals in a
table or graph.
Compare average rates of change on different intervals in a
table or graph.
Compare average rates of change on different intervals in a
table or graph.
Estimateand/or compare instantaneous rates of change at a point
based on the slopes of the tangent lines.
Estimateand/or compare instantaneous rates of change at a point
based on the slopes of the tangent lines.
Estimateand/or compare instantaneous rates of change at a point
based on the slopes of the tangent lines.
Use and interpret average rate of change as
( ) ( )y f b f ax b a
∆ −=∆ −
Use and interpret average rate of change as
( ) ( )y f b f ax b a
∆ −=∆ −
Use and interpret slopes of secant and tangent lines.
Use and interpret slopes of secant and tangent lines.
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Mathematics—Investigating Average Rate of Change
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MathematicsNATIONALMATH + SCIENCEINITIATIVE
Investigating Average Rate of Change
Answers
1. a. i. yx
∆∆
= ( 5) ( 1) 14 2 35 1 4
f f− − − − += =− + −
ii. yx
∆∆
= (2) (8) 7 25 32 8 6
f f− −= =− −
iii. Set up varies; answer is 3
iv. yx
∆∆
= (0.9) (1) 3.7 4 30.9 1 0.1
f f− −= =− −
v. yx
∆∆
= (0.999) (1) 2.997 4 30.999 1 0.001
f f− −= =− −
b. 3
2. a. i. yx
∆∆
= ( 6) (3) 7 1 26 3 9 3
f f− − −= = −− − −
ii. yx
∆∆
= (3) (9) 1 3 23 9 6 3
f f− += = −− −
iii. Set up varies; answer is 23
−
iv. yx
∆∆
= (0.9) (1) 20.9 1 3
f f− = −−
v. yx
∆∆
= (0.999) (1) 20.999 1 3
f f− = −−
b. 23
−
3. a. For a linear function, the average rate of change is the
same between any two points on the line.
b. The average rate of change is the value of the slope in the
equation of the line.
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Mathematics—Investigating Average Rate of Change
4. a. See the drawing at the right
b. i. 3yx
∆ = −∆
ii. 1yx
∆ = −∆
iii. 2yx
∆ =∆
c. no; nod. no
e. i. y = 2; y=5; 1yx
∆ =∆
ii. y = 1; y = 2; 1yx
∆ =∆
f. yes; yes
g. i. (0.4) (0.6) 1.16 1.36 10.4 0.6 0.2
y f fx
∆ − −= = =∆ − −
ii. (0.49) (0.51) 1.2401 1.2601 10.49 0.51 0.02
y f fx
∆ − −= = =∆ − −
iii. (0.499) (0.501)0.499 0.501
y f fx
∆ −= =∆ −
1.249001 1.251001 10.002
− =−
h. yes; yes
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Mathematics—Investigating Average Rate of Change
i.
First Point Second Point Δy Δx ΔyΔx
x-coordinate of the midpoint of
the segment(–1,2) (2,5) 3 3 1 0.5(0,1) (1,2) 1 1 1 0.5(0.4,1.16)
(0.6,1.36) 0.2 0.2 1 0.5(0.49,1.2401) (0.51,1.2601) 0.02 0.02 1
0.5(0.499,1.249001) (0.501,1.251001) 0.002 0.002 1 0.5
j.
Thepointsaregettingclosertooneanother.Theyareapproachingthepoint(0.5,1.25).
k. 1
l. 1 5,2 4
;allintervalslistedinparts(e)and(g)
m. 1 52 4
y x = − +
5. a. See the graph of the parabola
b. i. (−3,5.75)and(5,−6.25)
(5) ( 3) 6.25 5.75 35 ( 3) 8 2
y f fx
∆ − − − −= = = −∆ − −
x-value of the midpoint is 1.ii. (–2,6)and(4,–3)
(4) ( 2) 3 6 34 ( 2) 6 2
y f fx
∆ − − − −= = = −∆ − − x-value of the midpoint is 1.
iii. (0,5)and(2,2) (2) (0) 2 5 3
2 0 2 2y f fx
∆ − −= = = −∆ − x-value of the midpoint is 1.
c. yes; yes
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Mathematics—Investigating Average Rate of Change
d. i. (0.9) (1.1) 3.8975 3.5975 30.9 1.1 0.2 2
y f fx
∆ − −= = = −∆ − −
ii. (0.99) (1.01)0.99 1.01
y f fx
∆ −= =∆ −
3.764975 3.734975 30.02 2− = −
−
iii. (0.999) (1.001)0.999 1.001
y f fx
∆ −= =∆ −
3.75149975 3.74849975 30.2 2− = −
−
e. yes; yes
f.
First Point Second Point Δy Δx ΔyΔx
x-coordinate of the midpoint of
the segment
(−3,5.75) (5,−6,25) −12 832
− 1
(−2, 6) (4,−3) −9 632
− 1
(0,5) (2,2) −3 232
− 1
(0.9,3.8975) (1.1,3.5975) −0.3 0.232
− 1
(0.99,3.764975) (1.01,3.734975) −0.03 0.0232
− 1
(0.999,3.7514998) (1.001,3.7484998) −0.003 0.00232
− 1
g. yes; 31, 34
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Mathematics—Investigating Average Rate of Change
h. 32
−
i. (1,3.75)
j. 3 3( 1) 32 4
y x= − − + See the graph.
6. a. slope b. constant; constant c. the same; zero d. constant
e. midpoint
7. a. The average rates of change are not constant; therefore,
the function is not linear. Forexample,on[0,10], 22.5d
t∆ =∆
whileon[10,20], 42.5dt
∆ =∆
b. The average rate of change represents the average speed of
the car in meters per second.
For[0,20], m32.5sec
dt
∆ =∆
.
c. 2( ) 0.983 12.883 0.263R t t t= + − (Thisisarounded version
of the answer.)
d. (20) 650.562R = meters. According to the regression function,
the distance that the car hastraveledin20secondsis0.562meters more
than the value given in the table. Regression functions model the
data, and the data points are not necessarily points on the
function.
e. On[0,20], m32.541sec
Rt
∆ =∆
; 32.541 m 1km 60sec 60min 0.6214mi mi73sec 1000m 1min 1hr 1km
hr
≈
Onthis20-secondinterval,thecar’saveragerateofchangewithrespecttotime
(speedorvelocity)is mi73hr
.
f. Atthemidpointoftheinterval[0,20],att=10seconds
g. (12) (0) meters24.67812 second
R R− = Thecarhasnotacceleratedto60mphat6sec.
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Mathematics—Investigating Average Rate of Change
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1Copyright © 2014 National Math + Science Initiative, Dallas,
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Mathematics NATIONALMATH + SCIENCEINITIATIVE
Investigating Average Rate of Change
For ( )y f x= on the interval [ , ]a b , the average rate of
change is ( ) ( )y f b f ax b a
∆ −=∆ −
. This quotient is the
slope of the secant line. In other words, this is the slope
calculated between two points on the function f(x). The
instantaneous rate of change, the slope of the tangent line at one
point, will be explored in this lesson.
1. ( ) 3 1f x x= +
a. Calculate the average rate of change, yx
∆∆
, of the function over each of the given intervals.
i. [–5,–1]
ii. [2,8]
iii. Choose any different interval.
iv. [0.9,1]
v. [0.999,1]
b. What is the instantaneous rate of change at x = 1?
2. 2( ) 33
f x x= − +
a. Calculate the average rate of change, yx
∆∆
, of the function over each of the given intervals.
i. [–6,3]
ii. [3,9]
iii. Choose any different interval.
iv. [0.9,1]
v. [0.999,1]
b. What is the instantaneous rate of change at x = 1?
3. a.
Explainwhytheanswersinquestion1arethesameandwhytheanswersinquestion2arethesame.
b. Describe an easy method for determining the average rate of
change of a linear function.
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Mathematics—Investigating Average Rate of Change
4. 2( ) 1f x x= +a. Sketch the function by carefully plotting
the points at integer values of x.
b. Draw a secant line for each of the following intervals and
graphically determine the average rate of
changeofthefunction(slopeofthesecantline)overeachinterval.
i. [–2,–1]
ii. [–1,0]
iii. [0,2]
c.
Arethesecantlinesinpart(b)parallel?Dothesecantlinesinpart(b)havethesameslope?
d.
Basedontheanswersforpart(b),aretheaverageratesofchangeforaquadraticfunctionconstant?
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Mathematics—Investigating Average Rate of Change
e. Using a colored pencil, draw a secant line for each interval
given. What are the coordinates of the points on the graph of the
function where the secant line intersects the curve? Calculate the
averagerateofchangeofthefunction(slopeofthesecantline)overeachintervalanddeterminethe
x-coordinate of the midpoint of each segment. Record your
information in the table provided in part(i).
i. [–1,2]
ii. [0,1]
f.
Arethesecantlinesinpart(e)parallel?Dothesecantlinesinpart(e)havethesameslope?
g. Using a colored pencil, draw a secant line for each of the
following intervals. What are the coordinates of the points on the
graph of the function where the secant line intersects the curve?
Calculatetheaveragerateofchangeofthefunction(slopeofthesecantline)overeachintervalanddetermine
the x-coordinate of the midpoint of each segment. Record your
information in the table in part(i).
i. [0.4,0.6]
ii. [0.49,0.51]
iii. [0.499,0.501]
h.
Arethesecantlinesinpart(g)parallel?Dothesecantlinesinpart(g)havethesameslope?
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Mathematics—Investigating Average Rate of Change
i.
Completethetableincludingyourinformationfromparts(e)and(g).
First Point Second Point Δy ΔxΔyΔx
x-coordinate of the midpoint of the segment
(–1,____) (2,_____)
(0,_____) (1,_____)
(0.4,______) (0.6,_______)
(0.49,________) (0.51,________)
(0.499,_________) (0.501,_________)
j. Do the coordinates in the table seem to approach a certain
point? What is that point?
k.
Estimatetheinstantaneousrateofchange(slopeofthetangentline)atx=0.5.
l. Atwhatspecificpointof ( )f x
on[–1,2]istheinstantaneousrateofchangeofthefunctionequaltotheaveragerateofchangeofthefunctionontheinterval[–1,2]?Forwhatotherintervalsgiveninthis
question is this same relationship also true?
m. Using your estimate for the instantaneous rate of change at
x=0.5foundinpart(k),writetheequation of the tangent line through
the point . Using a colored pencil, draw this line on your
graph.
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Mathematics—Investigating Average Rate of Change
5. 21( ) ( 2) 64
f x x= − + +
a. Sketch the function by carefully plotting the points at
integer values of x.
b. Using a colored pencil, draw a secant line for each given
interval. What are the coordinates of the points on the graph of
the function where the secant line intersects the curve? Calculate
the
averagerateofchangeofthefunction(slopeofthesecantline)overeachintervalandcalculatethex-coordinateofthemidpointofeachsegment.Recordyourinformationinthetableinpart(f).
i. [–3,5]
ii. [–2,4]
iii. [0,2]
c.
Arethesecantlinesinpart(b)parallel?Dothesecantlinesinpart(b)havethesameslope?
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Mathematics—Investigating Average Rate of Change
d. Using a colored pencil, draw a secant line for each given
interval, calculate the average rate of
changeofthefunction(slopeofthesecantline)overeachinterval,andrecordyouranswersinthetableinpart(f).
i. [0.9,1.1]
ii. [0.99,1.01]
iii. [0.999,1.001]
e.
Arethesecantlinesinpart(d)parallel?Dothesecantlinesinpart(d)havethesameslope?
f.
Completethetabletoincludeyourinformationfromparts(b)and(d).
First Point Second Point Δy Δx ΔyΔx
x-coordinate of the midpoint of the segment
(–3,_______) (5,________)
(–2,________) (4,________)
(0,________) (2,_________)
(0.9,__________) (1.1,_________)
(0.99,___________) (1.01,__________)
(0.999,__________) (1.001,_________)
g. Do the coordinates in the table seem to approach a certain
point? What is that point?
h.
Estimatetheinstantaneousrateofchange(slopeofthetangentline)atx =
1.
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Mathematics—Investigating Average Rate of Change
i.
Atwhatspecificpointon[–3,5]istheinstantaneousrateofchangeofthefunctionequaltotheaverage
rate of change of the function?
j. Using your estimate for the instantaneous rate of change at
x=1foundinpart(h),writetheequationof the tangent line through the
point . Using a colored pencil, draw this line on your graph.
6. Fill in the blanks for each statement using the choices
provided. Note: Some choices may be used more than once and some
may not be used at all.
constant different endpoint lengthmidpoint slope the same
zero
a.
Theaveragerateofchangebetweentwopointsofafunctionisthe____________ofthesecantline.
b.
Sincetheslopeofalinearfunctionis___________,theaveragerateofchangeis_____________.
c. For a constant function, the
y-coordinateis____________foreverypairofpointsselected,sotheaveragerateofchangealwayshasavalueequalto____________.
d.
Theaveragerateofchangeforaquadraticfunctionisnot_______________foreverypairofpointsselected.
e. For a quadratic function, the x-value of the point where the
average rate of change over a given
intervalequalstheinstantaneousrateofchangeofthatintervalisthe_____________oftheinterval.
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Mathematics—Investigating Average Rate of Change
7. A car company is testing the speed and acceleration of one of
its new sports cars. The table shows the distance the car travels
when it accelerates from a standstill. Use a graphing calculator to
answer the following questions.
Elapsed time in seconds (t) Distance in meters (d)0 010 22514
37518 55020 650
a.
Explainwhythisdataisnotlinearandjustifyyouranswermathematicallyusingtheslopesofapairof
secant lines.
b. In the context of the problem, what does dt
∆∆
represent? What is the average rate of change, dt
∆∆
, on
the interval [0, 20]? Indicate appropriate units of measure.
c. Determine the quadratic regression function, ( )R t , for the
data and superimpose its graph on a scatterplot of the data. Copy
the graph and the data from your calculator onto the grid
provided.
QuadraticRegressionEquation _______________________________
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Texas. All rights reserved. Visit us online at www.nms.org.
Mathematics—Investigating Average Rate of Change
d. What is (20)R
?Explainthemeaningofthisvalueintermsoftheproblemsituation,andexplainwhythis
value is different from the value in the table.
e. According to
R(t),whatistheaveragerateofchangeoverthe20-secondtimeintervalfrom
0secondsto20seconds?Converttheanswertothenearestwholenumberinmilesperhourandexplainitsmeaningintermsoftheproblemsituation.(1km=0.6214miles)
f. Since the regression function is quadratic, where should the
average rate of change be equal to the
instantaneousrateofchangefortheinterval[0,20]?
g.
Thecarcompanyclaimsthecarcanacceleratefrom0to60mphin6seconds.Thismeansthattheinstantaneousrateofchangeat6secondsmustbe60mph.Proveordisprovethisclaimbyexaminingthe
average rate of change over an interval for which 6 seconds is the
midpoint.
Note: miles meters60 26.821hour second
≈ .
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Copyright © 2014 National Math + Science Initiative, Dallas,
Texas. All rights reserved. Visit us online at www.nms.org.10
Mathematics—Investigating Average Rate of Change