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Module 4

Pre-Calculus

4th Nine Weeks Table of Contents Precalculus Module 4

Unit 9 Rational Functions Rational Functions with Removable Discontinuities (1‐5) End Behavior of Rational Functions (6) Rational Functions and Their Asymptote (7‐8) (RA)2‐TEY‐ A Graphic Organizer (9‐16) Unit 10 Limits and Derivatives Average Rate of Change versus Instantaneous Rate of Change (17‐20) Investigating Average Rate of Change (21‐29) Slope of the Secant Line and Limits (30‐33) Slopes of Curves (34‐37)

Student Activity

Rational Functions with Removable Discontinuities

1. a) Simplify the rational expression 2xx

and state any values of x where the expression is

undefined.

b) Using the simplified expression in part (a), predict the shape for the graph of the function 2( ) xf xx

= . Sketch your prediction.

c) Graph the function 2( ) xf xx

= using the calculator window 10 10x− ≤ ≤ and

Enter the function in your calculator as it is written in un-simplified form. Does the calculator graph appear the same as your graph in part (b)?

10 10y− ≤ ≤

d) Turn off the axes on your graphing calculator and graph the function again. What do you observe? Why does this happen?

e) Draw an open circle on your graph in part (b) where the graph appears to have a hole.

2. a) Simplify the rational expression 23x

xand state any values of x where the expression is

undefined.

b) Sketch a graph for the function 23( ) xf x

x= .

Do you think that the graph of the function will have a hole in it?

Draw an open circle on the graph where you think the hole will occur.

Turn off the axes on your graphing calculator and graph the function using a standard window. Compare the graph on the calculator to your graph in part (b).

c) Use the trace or the table feature on your calculator to help you determine the value of x where the function is discontinuous. Why is the function discontinuous at this value of x?

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Student Activity

d) Construct a table of values that shows values of f(x) when x is close to zero. Use a graphing calculator to help you construct the table of values.

x < 0 f(x) x > 0 f(x)

−0.5 0.5

−0.1 0.1

−0.01 0.01

−0.001 0.001

−0.0005 0.0005

−0.0001 0.0001

e) As x approaches 0, what value does f(x) appear to be approaching?

3. a) Simplify the rational expression 3x

xand state any values of x where the expression is

undefined.

b) Sketch a graph for the function 3

( ) xf xx

= .

Do you think that the graph of the function will have a hole in it? Draw an open circle on the graph where you think the hole will occur.

Turn off the axes on your graphing calculator and graph the function using a standard window. Compare the graph on the calculator to your graph in part (b).

c) Construct a table of values that shows values of f(x) when x is close to zero.

Use a graphing calculator to help you construct the table of values.

x < 0 f(x) x > 0 f(x)

−0.5 0.5

−0.1 0.1

−0.01 0.01

−0.001 0.001


Student Activity

d) As x approaches 0, what value does f(x) appear to be approaching?

e) What is the domain and range of f(x)?

f) For what intervals of x where f(x) is continuous?

4. a) Simplify the rational expression 2 4 3

3x x

x+ ++

and state any values of x where the

expression is undefined.

b) Sketch a graph for the function 2 4 3( )

3x xf x

x+ +

=+

. Show any points where the graph is

discontinuous with an open circle.

c) Verify your graph in part (b) using a graphing calculator.

d) Construct a table of values that shows values of f(x) when x is close to -3. Use a graphing calculator to help you construct the table of values, or use the simplified expression for f(x) you found in part (a).

x < -3 f(x) x > -3 f(x)

-3.1 -2.9 -3.01 -2.99 -3.001 -2.999 -3.0001 -2.9999 -3.00001 -2.99999

e) As x gets closer to -3, what value does f(x) appear to be approaching?

f) What is the domain and the range of f(x)?

g) For what intervals of x is f(x) continuous?


Student Activity

Note: This type of discontinuity is called removable because we could remove the discontinuity by redefining the function at just one number.

For example:

The function 2 4( )

3x xf x

x+ +

=+

3 is discontinuous at –3.

X

Y

-4 -3 -2 -1 1 2

-4

-3

-2

-1

1

2

0

WINDOW

xmin = –4

xmax = 2

xscl = 1

ymin = –4

ymax = 2

yscl = 1

By redefining f(x) at the number x = –3, g(x) is a continuous function.

The function

2 4 3if 33

( )2 if

x xxx

g xx

⎧ + +⎪

3

≠ −+⎪⎪= ⎨⎪ − =⎪⎪⎩

− is continuous. Note: g(x) = x + 1 for all x.

WINDOW

X

Y

-4 -3 -2 -1 1 2

-4

-3

-2

-1

1

2

0

xmin = –4

xmax = 2

xscl = 1

ymin = –4

ymax = 2

yscl = 1


Student Activity

5. Match each graph to the correct function.

y

x

y

x

y

x

( ) 2f x x= − 2 4( )

2xg xx−

=+

3

2

4( )2

x xh xx x−

=+

a) b) c) b) c)

For questions 6 – 8, follow the instructions listed below. For questions 6 – 8, follow the instructions listed below. a) Factor the numerator and denominator of the function and state the domain. a) Factor the numerator and denominator of the function and state the domain. b) Reduce any common factors and state any value(s) of x where the function is undefined. b) Reduce any common factors and state any value(s) of x where the function is undefined. c) State the range of the function. c) State the range of the function. d) Graph the function using paper and pencil. Draw an open circle and label any removable

discontinuities on the graph. d) Graph the function using paper and pencil. Draw an open circle and label any removable

discontinuities on the graph. e) Use your graphing calculator to check your answers. e) Use your graphing calculator to check your answers.

6. 6. 4 210 9( )

3x xf x

x− +

=−

7. 3 2 4 4( )

1x x xf x

x+ − −

=+

8. 3 2

2

6 19 10( )3 2

x x xf xx x− +

=−


Student Activity

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End Behavior of Rational Functions

1. Determine the horizontal asymptote of y = 2 43

xx−+

by evaluating the limit as x approaches

± infinity. Does the function intersect its horizontal asymptote? If so, at what point(s)?

2. Determine the horizontal asymptote of 2

2 43

xyx−

=+

by evaluating the limit as x approaches


3. Determine the horizontal asymptote of 2

2

23 10

x xyx x

+ −=

+ − by evaluating the limit as x approaches


4. The rational function,3

2

3 23 10

x xyx x

− +=

+ −, has an oblique asymptote. Use long division to divide the

numerator by the denominator. The line, y = quotient, is the oblique asymptote. Does the rational function intersect its oblique asymptote? If so, at what point(s)?

5. Determine the oblique asymptote of 3 2

2

2 5 620

x x xyx x+ − −

=− −

. Does the function intersect its

oblique asymptote? If so, at what point(s)? 6. What is the rule that determines when a rational function has a horizontal asymptote and when it

has an oblique asymptote? 7. When the degree of the numerator is two or more higher than the degree of the denominator, the

end behavior of a rational function is modeled by a polynomial. Use long division to divide the numerator by the denominator. The polynomial is y = quotient. What is the polynomial that

models the end behavior of the curve 3 22 5 6

1x x xy

x+ − −

=−

as x approaches infinity? Will the

graph of the rational function intersect this polynomial? If so, at what point(s)?

6

Mathematics

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Rational Functions and Their Asymptotes

Part 1

1. Given the function 2

1

1y

x

a. What are the horizontal asymptote(s)?

b. What are the vertical asymptote(s)?

c. What are the x- and y-intercepts?

d. Graph the function.

e. What is the maximum value of the function? Explain how you found this value.

2. Do all rational functions have vertical asymptotes? Explain your answer.

3. Given the function f(x) = 2

2

2 3x x

x

a. What are the horizontal asymptote(s)?

b. What are the vertical asymptote(s)?

c. What are the x- and y-intercepts?

d. Determine ( 1.5)f . What is interesting about this value?

e. Complete the table of values.

f. Use your graphing calculator to confirm your findings.

4. Can the graph of a rational function cross its horizontal asymptote? Explain your answer.

5. How can you determine if and where a function crosses its horizontal asymptote?

x 0 –0.5 –1 –1.5 –2 –2.5 –3 –4 –5 –6 –100

f (x)

7

Student Activity – Rational Functions and Their Asymptotes

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Part 2

For questions 6 – 11, complete parts a – f.

a. What is the equation of each vertical asymptote?

b. What is the equation of each horizontal asymptote?

c. Determine all x-intercepts.

d. Determine the y-intercept.

e. Does the function cross its horizontal asymptote? If so, where? Show the work that leads

to your answer(s).

f. Graph each function.

6. 2

8

2y

x

7. 2

1

1y

x

8. 2

13

1y

x

9. 2

2

2 7

3

xy

x

10. 2

2

6 9

2 4

x xy

x x

11. 3 2

3

2 3

1

x x xy

x

8

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Mathematics NATIONALMATH + SCIENCEINITIATIVE

RATEY – A Graphing Organizer

RATEY is an organizing tool for graphing rational functions. To use this tool effectively, the numerator and the denominator should be factored completely over the real numbers before beginning the graph. RATEY is an acronym for the following:

R is for Roots (zeros) and Removable discontinuities. Factor the numerator and denominator. Identify common factors and label them as removable discontinuities(holesinthegraph).Simplifytheexpression,setthenumeratorequalto0,andsolve.Graphthesepoint(s)onthex-axis.

A is for Asymptotes. (Vertical only)Usingthesimplifiedexpression,setthedenominatorequalto0andsolve.Sketch the vertical asymptotes as dotted lines.

T is for Two and means Two things.DoanyofthefactorsofRandAhaveanevenexponent(amultipleof2)suchas (x 1)2 or (x 1)4 ?In the numerator, this means “T”angency at the root.In the denominator, this means “T”ogetherness about the asymptote.

E is for End Behavior.Determine the limit of the function that models the end behavior as x ± . Compare the degree of the numerator and the denominator.

● Ifthenumeratoranddenominatorhavethesamedegree,dividethecoefficientofthelargestdegreeterminthenumeratorbythecoefficientofthelargestdegreeterminthedenominatortodetermine the horizontal asymptote.

● If the degree of the denominator is larger than the degree of the numerator, the end behavior asymptote is y = 0 .

● If the degree of the numerator is larger than the degree of the denominator, perform long division to determine the oblique asymptote.

Sketch the asymptotes for the end behavior as dotted lines. Y is for Y-Intercept.

Let x equal zero and solve for y.Graph this point on the y-axis.

9

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Mathematics—RATEY – A Graphing Organizer

For questions 1-8, identify the RATEY characteristics and sketch the graph:

1. f (x) =

x 4x + 2

2. f (x) =

(x +1)(x 3)x + 2

3. f (x) =

x2 + x 6(x +1)2

10



4. f (x) =

2x(x 1)2

(x +1)3

5. f (x) =

3(x 1)2

(x +5)2

6. f (x) =

9 xx2 4

11



7. f (x) =

(x2 + 4x +8)(x 4)4(x2 4x)

8. f (x) =

2x2 3x 9x

9. Describe the similarities and differences between the graphs of f (x) =

6x 62x 3

and g(x) =

2x 36x 6

.

12



10. Write a possible equation for each of the functions in these graphs. Compare a calculator’s graph of your equation to the given graph.a.

b.

c.

1

1x��1

f �x�

2

�1

�1

x��2

f �x�

13



11. Given f (x) =

ax + bb ax

:

a. What is the root of f (x) ?

b. What is the equation of the function’s vertical asymptote?

c. What is the end behavior of the function?

d. What is its y-intercept?

e. If a > b > 0, sketch f (x) .

f. On what intervals does f (x) appear to be increasing?

g. On what intervals does f (x) appear to be concave up?

14



Use a graphing calculator to assist in answering questions 12 – 16. For questions 15 – 16, set the calculator to radian mode.

12. If f (x) =

x 2x + 2

and g(x) = x , describe how the graph of the composition

g( f (x)) =

x 2x + 2

differs

from the graph of f (x). Support your answer with mathematical reasoning.

13. Let f (x) =

4x 4x +1

. Does this function exist for all real numbers? If not, where does it fail to exist?

Let

g(x) =

4x 4x +1 . Does the square root affect the domain of this function compared to the domain

of f (x) ? If so, how is it affected? State the domain of g(x) and support your answer with mathematical reasoning.

14. Why are the domains of f (x) =

ln(x)x

and g(x) =

ln(x2 )x2 different? Sketch the graphs of f and g.

15



15. Determine the end behavior asymptote of f (x) =

sin(x)x

and state the domain of f (x). The function f

crossesitshorizontalasymptoteinfinitelymanytimes.Explainwhythisistrue.

16. The functions f (x) =

sin(x)x

and g(x) =

cos(x)x

are both discontinuous when the value of x is 0.

Determine which has an asymptote and which has a hole and describe the similarities and differences of the graphs of f and g. Sketch each function on the axes provided.

16

Mathematics

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Average Rate of Change versus Instantaneous Rate of Change

1. Beginning 1

23

miles from home, Jonathan drove away from home at a constant rate for 20

minutes. If his constant rate is 35 miles per hour, how far is he from home at the end of the

20 minutes? Draw a graph to model his distance from home during the 20 minute time

period.

2. Susan, Jonathan’s sister, also drove away from home beginning 1

23

miles from home and

following the same path as Jonathan. Susan kept varying her velocity by frequently speeding

up and slowing down. She arrived at the same location as Jonathan at the end of 20 minutes.

To model Susan’s distance from home during the 20 minute time period, draw a smooth

curve without any sharp corners.

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

0t in minutes

d in miles

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

0t in minutes

d in miles

17

Student Activity – Average Rate of Change versus Instantaneous Rate of Change


3. Calculate the average velocity for both drivers by calculating the change in position divided

by the change in time. These two calculations have the same value; explain why this makes

sense. Compare your answers to the rate given in question 1.

4. On the graph showing Susan’s position, draw the line segment connecting the point at t = 0

and the point at t = 20. What is the slope of this line segment and what are the units for the

slope? How does this slope compare to the slope of line that modeled Jonathan’s distance

from home?

5. If the speed limit over the entire path is 35 miles per hour, did Susan ever drive over the

speed limit? Explain your answer by referring to Susan’s graph.

6. For non-linear position functions, the exact velocity at a particular time, called instantaneous

rate of change or instantaneous velocity, cannot be calculated precisely without the tools of

calculus. However, the velocity can be estimated by approximating the slope of a short line

segment drawn tangent to the curve at the particular time. On the graph showing Susan’s

distance, locate at least one time when Susan’s instantaneous velocity has the same value as

the average velocity.

(Position a straightedge on the graph so that it is parallel to the line segment drawn on the

curve in question 4. Move the straightedge around on the graph keeping the slope of the

straightedge fixed. When the straightedge appears to be tangent to the curve, mark the

point(s) and sketch a short segment tangent to the curve. At these point(s), the instantaneous

velocity is the same as the average velocity.)

18



7. Using the function g(x) shown in the graph, draw a small tangent line segment at each

labeled point. A small tangent line is drawn at point P as a sample. There is not enough

information to draw the segments perfectly, so sketches may vary slightly. Match the slope at

each labeled point on the curve with an approximate rate of change value in the table. The

slope at each point is called the instantaneous rate of change at a point because it is the rate

of change at that one instant in time.

Hint: The slope may be the same at different places along the graph.

Rate of Change at Point Letter

–5

–2

0

2

3

4

3

3 P

15

-4 -3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

0

P

E

B

H

J

C W

L

19



8. The graph represents the position x(t) in inches of an object that is moving along a line

extending perpendicularly from a wall at a given time, t, measured in seconds. The distance

between the object and the wall is indicated on the vertical axis, while time is measured on

the horizontal axis.

a. What do the coordinates of B (0.13, 27.5) and D (2, 19) represent in the context of this

situation?

b. Mark small tangent line segments on each of the points that are named. Using these

tangent segments, for which of the point(s) is the instantaneous rate of change negative?

What do you know about the motion of the object if the instantaneous rate of change is

negative?

c. Observing the tangent segments, over which time intervals is the object moving away from the wall? What do the slopes of these line segments mean in the context of the position function?

d. At which point(s) has the object stopped moving? Describe the slope of the tangent

line(s) at the point(s).

e. Speed indicates how fast an object is moving without regard to direction. Order the

speeds at the following points from least to greatest: D, G, Z. Explain your reasoning.

Points t x(t)

B 0.13 27.5

G 0.45 30

Z 0.87 32

V 1.2 31

D 2.0 19

K 3.02 0

A 3.25 2

C 4.1 70

20



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.

21


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?

22



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?

23



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.

24



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?

25



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.

26



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.

27



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 _______________________________

28



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

≈ .

29

Student Activity


Slope of the Secant Line and Limits For f(x) = x2, find the slope of the line containing the following points.

1. (x, f(x)) and (3, f(3))

2. (x, f(x)) and (1, f(1))

3. (x + 1, f(x + 1)) and (x – 1, f(x – 1))

4. (x, f(x)) and (x + h, f(x + h))

5. (x, f(x)) and (x + 2, f(x + 2))

Find the following limits. Fill in answers in table.

6. 39lim

2

3 −−

→ xx

x

7. 11lim

2

1 −−

→ xx

x

8. 2

)1()1(lim22

1

−−+−→

xxx

9. h

xhxh

22

0

)(lim −+→

Can you generalize about the slope of the secant line connecting two very, very, very close points on x2? Use the generalization to answer question 10.

10. a

aa

22

0

2)2(lim −+→

Problem answer

6 lim as 3x →

7 lim as 1x →

8 lim as 1x →−

9 lim as 0h →

30

Student Activity


The following points lie on f(x) = x2. Find the slope of the line containing the two points. Use the table function on the graphing calculator. For problems 12 through 23, let

y1 = (x + .001)2

y2 = x2 y3 = (y1–y2)/.001

11. (3, 9) and (3.001, f(3.001))

12. (1, 1) and (1.001, f(1.001))

13. (0, 0) and (0.001, f(0.001))

14. (x, f(x)) and (x + .001, f(x + .001))

For 15 through 18, let y1 = (x + .0001)2 and y3 = (y1–y2)/.0001

15. (3, 9) and (3.0001, f(3.0001))

16. (1, 1) and (1.0001, f(1.0001))

17. (0, 0) and (0.0001, f(0.0001))

18. (x, f(x)) and (x + .001, f(x + .001))

For 19 through 22, let y1 = (x + .00001)2 and y3 = (y1 – y2)/.00001

19. (3, 9) and (3.00001, f(3.00001))

20. (1, 1) and (1.00001, f(1.00001))

21. (0, 0) and (0.00001, f(0.00001))

22. (x, f(x)) and (x + .00001, f(x + .00001))

31

Student Activity


23. Fill in the table below:

x 001.

sin)001.sin( xx −+ cos x

0

6π

4π

3π

2π

What conclusion can you make from the table above?

Use the given x values and the function to answer the following.

24. f(x) = 2x2 – x + 1 for x = 2 and x = 2.001.

a) Find the slope of the secant line.

b) Draw a picture of the function and the secant line.

c) Find the equation of the secant line.

25. f(x) = 11

+−

xx for x = 1 and x = 1.001.




32

Student Activity


26. f(x) = 2+x for x = 2 and x = 2.001.




27. f(x) = ex for x = 1 and x = 1.001.




28. f(x) = cos x for x = 1.56 and x = 1.57.




33



Slopes of Curves

1. For 3 24 2 8 13

y x x x= − − + , the slope function m for any point on the curve is given by 24 4 8m x x= − − .a. Determine the points where the slope is zero. Show the algebraic steps.

b. Write the equation(s) of the horizontal tangent line(s) to the curve at the point(s).

c. Determine if the curve has a maximum or minimum value or neither at the point(s). Explain your answer using the values of m.

2. For 23 2 4 10y x y− + = , the slope function m for any point on the curve is given by 1

3 2m

y=

+.

a. Is this slope ever zero? Show why or why not.

b. What does part (a) indicate is true about the graph?

c. When would the slope be undefined? Show the process used to determine the answer.

34


Mathematics—Slopes of Curves

d. What is x when 23

y = − ?

e. Describe the tangent line to the curve where the slope is undefined.

f. When would the slope be positive? Describe the curve when the slope is positive.

g. On a TI-84, solve the original equation for x in terms of y and then graph the curve using parametric

equations. Let x(t) = equation for x in terms of t instead of y, and let ( )y t t= . On the TI-Nspire™, solve the original equation for x in terms of y and then use the scratchpad to graph ( )x f y= List the equations used and sketch the graph.

h. Verify your answers to part (f) by examining the graph.

3. Consider the curve 3 2 22 6 12 6 1y x y x y+ − + = . The slope function m of the curve at all points is given by

2 2

4 21

x xymx y

−=+ +

.

a. Determine the slope of the curve at the point 1 1,2 2

.

b. Is the curve increasing or decreasing at the point 1 1,2 2

? Explain.

35



c. Determine the values of x and y where m = 0.

d. Using your answers to part (c), determine the point(s) on the curve where the slope is zero.

e. Write the equations of the horizontal tangent lines to the curve.

f. On the TI-84, solve the equation of the curve for x in terms of y then use parametric equations to graph the curve. Let ( )x t = equation for x in terms of t instead of y, and let ( )y t t= . On the TI-Nspire™, solve the equation of the curve for x in terms of y and then use the scratchpad to graph ( )x f y= List the equations used and sketch the graph.

g. What is the horizontal asymptote of the curve?

36



4. Consider the curve given by 2 3 6xy x y− = . The slope function m of the curve at any point

is 2 2

3

32x y ymxy x

−=−

.

a. Determine all points on the curve whose x-coordinate is 1.

b. Write an equation for the tangent line at each of the points where the x-coordinate is 1.

c. Determine the x-coordinate of each point on the curve where the tangent line is vertical.

d. Use the quadratic formula to solve for y in terms of x.

e. On the TI-84, use parametric equations to graph the curve. On the TI-Nspire™, use the scratchpad to graph the curve, using the equations from part (d). List the equations used and sketch the graph.

37

Pre-Calculusimages.pcmac.org/SiSFiles/Schools/AL/StClairCounty/...Rational Functions with Removable...

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