Answers (Lesson 1-3)

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A7 1-3 Study Guide and Intervention 1-3 Study Guide and Intervention(continued) Continuity, End Behavior, and Limits Continuity, End Behavior, and Limits

ContinuityA functionf (x) is continuous at x = c if it satisfies the End Behavior The end behavior of a function describes how the function behaves at following conditions. either end of the graph, or what happens to the valuef(x) of as x increases or decreases without bound. You can use the concept of a limit to describe end behavior. (1) f(x) is defined atc ; in other words,f( c) exists.

Left-End Behavior(as x becomes more and more negative): lim f(x) (2) f (x) approaches the same function value to the left and rightc; in of other x → -∞ words,lim f(x) exists. x → c Right-End Behavior(as x becomes more and more positive): lim f(x)

x → ∞ (3) The function value thatf(x ) approaches from each side cof is f(c); in The f x values may approach negative infinity, positive infinity, or a specific value. other words,lim f x f c . ( ) ( ) = ( ) x → c Functions that are not continuous arediscontinuous. Graphs that are Example Use the graphf of(x) x3 2 to describe y discontinuous can exhibitinfinite discontinuity, jump discontinuity, = + 8 its end behavior. Support the conjecture numerically. or removable discontinuity (also calledpoint discontinuity). 4 3 Answers As x decreases without bound, they-values also f(x) =x +2 decrease without bound. It appears the limit is negative 4 2 24x Example Determine whether each function is continuous at the given − − 0 infinity: lim f(x) = -∞. −4 x-value. Justify using the continuity test. If discontinuous, identify the type of x → -∞ discontinuity asinfinite , jump, or removable. As x increases without bound, they-values increase −8 2x a. f(x) = 2|x| + 3; x = 2 b.f(x) = ; x = 1 without bound. It appears the limit is positive infinity: −2 x - 1 lim f(x) = ∞. (1) f(2) = 7, so f(2) exists. The function is not defined xat = 1 x → ∞ (Lesson1-3) (2) Construct a table that shows values for because it results in a denominator of 0. Construct a table of values to investigate function valuesx| as increases. | f( x) for x-values approaching 2 from the The tables show that for valuesx of left and from the right. approaching 1 from the left,f(x ) x -1000 -100 -10 0 10 100 1000 becomes increasingly more negative. For Copyright ©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. f(x) -999,999,998 -999,998 -998 2 1002 1,000,002 1,000,000,002 x y = f(x) x y = f(x) values approaching 1 from the right, 1.9 6.8 2.1 7.2 f(x) becomes increasingly more positive. 1.99 6.98 2.01 7.02 As x −∞, f(x) -∞. As x ∞, f(x)∞. This supports the conjecture. x y = f(x) x y = f(x) 1.999 6.998 2.001 7.002 0.9 -9.5 1.1 10.5 The tables show thaty approaches 7 0.99 -99.5 1.01 100.5 Exercises

as x approaches 2 from both sides. 0.999 -999.5 1.001 1000.5 Lesson 1-3 Use the graph of each function to describe its end behavior. Support It appears thatlim f(x) = 7. the conjecture numerically. x → 2 The function has infinite discontinuity

(3) lim f(x) = 7 andf (2) = 7. at x = 1. y y x → 2 1. 8 2. 8

PDF 2nd The function is continuousx at= 2. 4 4 5x f(x) x 4 2x f(x) = Exercises = - - x -2 −4 −2 0 24x −16 −8 0 816x Determine whether each function is continuous at the givenx-value. −4 −4 Justify your answer using the continuity test. If discontinuous, −8 8 identify the type of discontinuityinfinite as , jump, or removable. − Glencoe Precalculus

⎧ 2x + 1 if x > 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. lim f(x)=-∞; lim f(x)=-∞ lim f(x)= 5; lim f(x)= 5 1. f(x) = ⎨ ; x = 2 2. f(x) = x2 + 5x + 3; x = 4 f(4) = 39 x → -∞ x → ∞ x → -∞ x → ∞ x - 1 if x ≤ 2 ⎩ lim f(x) = 1 and lim f(x) = 5 , lim f(x) = 39 and lim f(x) = 39, See students’ work. See students’ work.

– + - + → → → x → x 2 x 2 4 x 4 so the function is not continuous; so the function is continuous. it has jump discontinuity. 110/16/09 10:38:02 AM

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2 Answers

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18 18 19 19 Chapter 1 Chapter Glencoe Precalculus Glencoe Chapter 1 Chapter Glencoe Precalculus Glencoe

Resistance decreases and approaches zero. approaches and decreases Resistance the resistance? resistance? the − 16

constant but the current keeps increasing in the circuit, what happens to to happens what circuit, the in increasing keeps current the but constant

− 8

0 − −

16 8 x 8 16 − =

voltage voltage R as circuit a in I current and , E . If the voltage remains remains voltage the If .

ELECTRONICS Ohm’s Law gives the relationship between resistance , R resistance between relationship the gives Law Ohm’s 9. 9.

Glencoe Precalculus

y PDF Pass

See students’ work. students’ See See students’ work. students’ See

See students’ work. students’ See

conclusion from part . part from conclusion b

-∞ → ∞ → -∞ → ∞ →

x x x x

∞ = ) ( ∞ = ) ( - = ) ( - = ) ( lim lim lim lim

; 2 2;

x f x f x f x f

your verify to function the Graph c. c.

∞ →

-∞ →

lim )=∞ ( )=-∞ ( ; ; x f x f lim

= 0; infinite. 0; x

−

− 4 conjecture numerically. numerically. conjecture is discontinuous at at discontinuous is x f ) ( 8

behavior of the function. Support your your Support function. the of behavior

does not exist, exist, not does 0 because No; f ) (

− − 2 4

Use the graph to describe the end end the describe to graph the Use

− − . removable or , jump , infinite as

816 0 x − − 8 16 48 x 4 8

and identify the type of discontinuity discontinuity of type the identify and

Day

- - =

5 x 4 x ) x ( f discontinuous, explain your reasoning reasoning your explain discontinuous, 4

5 x 3

Lesson 1-3 246

) x ( f

answer using the continuity test. If If test. continuity the using answer

x 6

Price perShare($) 4

8. 8. 7.

y Is the function continuous? Justify the the Justify continuous? function the Is b. b. 6

numerically. conjecture the

5 x → 12

= = Use the graph of each function to describe its end behavior. Support Support behavior. end its describe to function each of graph the Use lim 10. 5, x f x ) (

function is defined when when defined is function

= 10, the the 10, 5 because Yes; f ) (

answer using the continuity test. continuity the using answer

Stock

= 5. Justify the the Justify 5. x at continuous

- - + = ) ( - - + = ) (

3, 2] 3, [ 6; x 10 x x g 2] 6, [ 4; x 5 x x f 6. 6. 5. 5.

4 2 3

+ - + )= - )= ( 15.29. x 1.8 x 1.4 x 0.15 x f

Determine whether the function is is function the whether Determine a. a.

2 3

restructuring is modeled by modeled is restructuring

(Lesson 1-3) (Lesson interval. given the on located are function each

side of the base. the of side

days after a company company a after days x stock certain a

Determine between which consecutive integers the real zeros of of zeros real the integers consecutive which between Determine

−

STOCK The average price of a share of of share a of price average The 4. 4.

)= ( is the length of one one of length the is x where , x f

250

of 250 cubic units can be modeled by modeled be can units cubic 250 of

= ) ( both sides; sides; both 1. 1 f

prism with a square base and a volume volume a and base square a with prism

people is 0. is people

- = 1 as as 1 2. at discontinuity from 1 approaches x x

GEOMETRY The height of a rectangular rectangular a of height The 2. 2.

because the fewest number of of number fewest the because

- = - =

1 and infinite and 1 at discontinuity approaches function the 1, x x

will not be negative negative be not will No, x

Yes; the function is defined at No; the function has a removable a has function the No; at defined is function the Yes;

∞ →

Answers x

−

in the relevant domain? Explain. domain? relevant the in

)=-∞ ( lim x f + +

2 x 3 x

= + - = ) ( - = - = = ) (

1 x at 2; x 2 x x f 2 x and 1 x at ; x f

4. 4. 3.

Are there any points of discontinuity discontinuity of points any there Are b. b.

3 + 1 x

-∞ →

)=-∞ (

lim ; ; x f

25 discontinuity at at discontinuity x

- - = ) (- - . 1 sides; both f −

infinite infinite

discontinuity. discontinuity.

end behavior. end - - -

1 from from 1 approaches as x −

and describe any points of of points any describe and

on a graphing calculator. Describe the the Describe calculator. graphing a on

- = - = 1, the function approaches discontinuous at at discontinuous approaches function the 1, 4. x x

calculator. Use the graph to identify identify to graph the Use calculator.

decades since 1900. Graph the function function the Graph 1900. since decades

Yes; the function is defined at No; the function is infinitely is function the No; at defined is function the Yes;

Graph the function using a graphing graphing a using function the Graph a. a.

+ 47.37, where where 47.37, x 4.12 is the number of of number the is x

x 3

+ 4 x

−

2 −

- + - )=- ( x 1.54 x 0.09 x 0.0009 x h

- = - = = ) ( - - = ) ( 1 x at ; 4 x at ; x f x f 2. 2. 1. 1.

4 2 3 - 2 2 x people on the trip. the on people

from 1900 to 2000 can be modeled by by modeled be can 2000 to 1900 from

+ − 25 x

)= ( is the number of of number the is x where , x f

of Americans who owned a home home a owned who Americans of identify the type of discontinuity as as discontinuity of type the identify . removable or , jump , infinite 600

Census Bureau, the approximate percent percent approximate the Bureau, Census climbing expedition can be modeled by by modeled be can expedition climbing -value(s). Justify using the continuity test. If discontinuous, discontinuous, If test. continuity the using Justify -value(s). x

HOUSING TRIP According to the U.S. U.S. the to According The per-person cost of a guided guided a of cost per-person The 1. 1. 3. 3. Determine whether each function is continuous at the given given the at continuous is function each whether Determine

Continuity, End Behavior, and Limits and Behavior, End Continuity, Continuity, End Behavior, and Limits and Behavior, End Continuity, d d

2 Practice 1-3 Word Problem Practice Problem Word 1-3 A8 0 8 3 9 8

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A9 1-3 Enrichment 1-4 Study Guide and Intervention Extrema and Average Rates of Change Reading Mathematics Increasing and Decreasing BehaviorFunctions can increase, decrease, or remain The following selection gives a definition of a continuous function as constant over a given interval. The points at which a function changes its increasing or decreasing behavior are called critical points. A critical point canrelative be a minimum, it might be defined in a college-level mathematics textbook. absolute minimum, relative maximum, or absolute maximum. The general term for Notice that the writer begins by explaining the notation to be used for minimum or maximum is extremumextrema or . various types of intervals. Although a great deal of the notation is standard, it is a common practice for college authors to explain their

notations. Each author usually chooses the notation he or she wishes to use. Answers Example Estimate to the nearest 0.5 unit and classify the extrema for the graph off (x). Support the answers numerically. Throughout this book, the setS, called the domain of definition of a function, will usually be 5 3 an interval. An interval is a set of numbers satisfying one of the four inequalitiesa < x < b, Analyze Graphically g(x) =x -4x +2x -3 a ≤ x < b, a < x ≤ b, or a ≤ x ≤ b. In these inequalities,a ≤ b. The usual notations for the intervals It appears thatf( x) has a relative maximum of 0 at y corresponding to the four inequalities area, b (), [a, b), (a, b], and a[ , b], respectively. 8 x = -1.5, a relative minimum -of3.5 at x = -0.5, An interval of the forma ,( b) is called open, an interval of the form 4 a relative maximum of-2.5 at x = 0.5, and a relative (Lesson1-3andLesson1-4) [a, b) or (a, b] is called half-open or half-closed, and an interval of the form minimum of- 6 at x = 1.5. It also appears that −4 −2 0 4 x [a, b] is called closed. lim f(x)=-∞ and lim f(x)=∞, so there appears to SupposeI is an interval that is either open, closed, or half-open. Supposeƒ(x) is a function defined x → -∞ x → ∞ −8 on I and x is a point inI. We say that the functionƒ(x ) is continuous at the pointx if the quantity 0 0 be no absolute extrema.

⎪ƒ(x) - ƒ(x0)⎥ becomes small asx ∈ I approachesx 0. Support Numerically Use the selection above to answer these questions. Choosex -values in half-unit intervals on either side of the estimatedx-value for each extremum, as well as one very small and one very large valuex. for 1. What happens to the four inequalities in the first paragraph awhen = b? x -100 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 100 Only the last inequality can be satisfied. Copyright ©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. f(x) -1 × 1010 -7 -0.09 -2 -3.5 -3 -2.47 -4 -5.91 1 1 × 1010 2. What happens to the four intervals in the first paragraph awhen = b? The first interval is∅ and the others reduce to the pointa = b. Becausef (-1.5)>f(-2) and f(-1.5)>f(-1), there is a relative maximum in the interval (-2, -1) near -1.5. 3. What mathematical term makes sense in this sentence? Becausef (-0.5)( ) ( )>( )

0, 1 near 0.5. contained in the intervalI? ( ) x ∈ I Becausef (1.5)

PDF Pass (1, 2) near 1.5. 5. In the space at the right, sketch the graph f(-100)f(1.5), which supports the conjecture that of the functionf( x) defined as follows. f has no absolute extrema. ⎧ 1 1 − if x ∈ 0, − Lesson 1-4 2 [ 2) Exercises f(x) = f (x) ⎨ 1 Use a graphing calculator to approximate to the nearest hundredth the Glencoe Precalculus 1 if x ∈ − , 1 ⎩ 2 1 relative or absolute extrema of each function. Statex-value(s) the where Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 6. Is the function given in Exercise 5 continuous on the they occur. 1 interval [0, 1]? If not, where is the function discontinuous? 2 1. f(x)= 2x6 + 2x4 - 9x2 2.f(x )=x3 + 9x2 1 No; it is discontinuous xat = − . abs. min. of- 5.03 atx =-0.97 and rel. min. of 0 xat= 0; 2 0 1 1 x 2 at x = 0.97; rel. max. of 0 atx = 0 rel. max. of 108 xat=- 6

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8 Answers

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23 23 22 22 Chapter 1 Chapter Chapter 1 Chapter Glencoe Precalculus Glencoe Glencoe Precalculus Glencoe

16.5 ft 16.5 feet. Find the maximum height of the rocket. rocket. the of height maximum the Find feet.

) ( + + - = ) ( is in in is t h where 0.5, t 32 t 16 t h function the by modeled be can

PHYSICS seconds after a toy rocket is launched straight up up straight launched is rocket toy a after seconds t height The 8. 8.

- - 132 160 132 - 56 7 56

= ) ( - - = ) ( - - - + x x g ; [2, 6] [2, ; x 4 x 3 x g 2] 4, [ 5; x 2 6. 6. 7. 7. = ) ( - + - = ) ( - - + x x f 3; [0, 1] [0, 3; x 8 x x f 0] 4, [ 3; x 8 5. 5. 6. 6.

3 2 4 4 4

Find the average rate of change of each function on the given interval. given the on function each of change of rate average the Find Glencoe Precalculus

Pdf Pass

) - ( ) (- 4.02 1.05, min. rel. ; 6.02 1.05, max. rel.

Lesson 1-4

14 26 14

where they occur. they where

= ) ( - - + = ) ( - - - - +

x x f 4; [1, 3] [1, 4; x 7 x 5 x x f 1] 3, [ 4; x 7 x 5 3. 3. 4. 4.

+ - = ) ( -values -values x the State 1. x 6 x x of h extrema absolute or relative

3 2 3 2 5

GRAPHING CALCULATOR GRAPHING Approximate to the nearest hundredth the the hundredth nearest the to Approximate 5. 5.

See students’ work. students’ See

- 28 0 28 = - 1; See students’ work. students’ See 1; at 6 of min. rel. x

= = - 0.5; 0.5; at 0 of min. rel. 0; at 5 of max. rel. x x

- - - + = ) ( - - - - + = ) (

1, 0] 1, [ 1; x x 2 x x f 2] 3, [ 1; x x 2 x x f 2. 2. 1. 1.

4 3 4 3 - = - = - 1; at 1 of max. rel. 1.5; at 8.5 of min. rel. x x

Find the average rate of change of each function on the given interval. given the on function each of change of rate average the Find

Exercises

− 4

0 − 0 4 x 4 x

−

) (- -

1 1

= − or

Evaluate and simplify. and Evaluate

) (- -

1 1

1 2

- y x x

2 1

- . for 1 and for 1 Substitute x x

2 1

) (- - ) ( 1 f 1 f - + - =

- + = ( - ) ( ) 4. 4. 3. 3.

x f x f 5 x x 3 x ) x ( f

x x x ) x ( f

2 4 2 3

(Lesson 1-4) (Lesson -

1, 1] 1, [ b.

graph of each function. Support the answers numerically. numerically. answers the Support function. each of graph −

) (- - -

3 1

Estimate to the nearest 0.5 unit and classify the extrema for the the for extrema the classify and unit 0.5 nearest the to Estimate

−

= =

Simplify.

17 ) (- - 19.5 –2.5 −−−

See students’ work. students’ See

) (- - -

3 1

= =

- - 3). ( and 1) ( Evaluate f f

∞) ( ; ; 1.5, on increasing

) (- + ) (- - ) (- + ) (- ] 3 2 3 [0.5 ] 1 2 1 [0.5

3 3

∞) ( ) ( − ; See students’ work. students’ See ; 0, on ; 1.5 0, on decreasing −

) (- - -

3 1

1 2

- x x

2 1

- - . for 1 and for 3 Substitute x x ) (-∞ ) (-∞ ; decreasing decreasing ; 0 , on decreasing ; 0 , on increasing

1 2 ) (- - ) (- 3 f 1 f ) ( - ) ( x f x f

Answers

- - 1] 3, [ a.

each interval. each

+ = on on x 2 Find the average rate of change of of change of rate average the Find x 0.5 ) x ( f

Example

0 −

1 2

- x x

sec

1 2

) ( - ) ( x f x f

y x 5

sec

. m line, ) x ( f

+ - =

2. 2. 1. 1.

x 2 x 2 x ) x ( g

2 1

2 3 5 The average rate of change on the interval [ interval the on change of rate average The ] is the slope of the secant secant the of slope the is ] x , x y

line through any two points on a curve is called a a called is curve a on points two any through line . secant line secant Support the answer numerically. numerically. answer the Support

is the slope of the line through those points. The The points. those through line the of slope the is of f graph the on points two 0.5 unit on which the function is increasing, decreasing, or constant. constant. or decreasing, increasing, is function the which on unit 0.5

Average Rate of Change Change of Rate Average The The between any any between average rate of change of rate average 0 Use the graph of each function to estimate intervals to the nearest nearest the to intervals estimate to function each of graph the Use

Extrema and Average Rates of Change of Rates Average and Extrema Extrema and Average Rates of Change of Rates Average and Extrema d d

Practice Study Guide and Intervention Intervention and Guide Study 1-4 1-4 (continued) A10 0 8 3 9 8

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A11 1-4 Word Problem Practice 1-4 Enrichment Extrema and Average Rates of Change 1. FLARE A lost boater shoots a flare 3. RECREATION For the function in “Unreal” Equations straight up into the air. The height of the Exercise 2, find the average rate of There are some equations that cannot be graphed on the real-number flare, in meters, can be modeled by change for each time interval. coordinate system. One example is the equationx2 - 2x + 2y2 + 8y + 14 = 0. h(t)=-4.9t2 + 20t + 4, wheret is the Completing the squares xin and y gives the equation(x - 1) 2 + 2( y + 2) 2 = -5. time in seconds since the flare was a. Day 2 to Day 6 2 2 launched. 1395 For any real numbersx and y, the values of( x - 1) and 2( y + 2) are nonnegative. So, their sum cannot- be5. Thus, no real values xof and y a. Graph the function. b. Day 13 to Day 15 satisfy the equation; only imaginary values can be solutions. h(t) 24 19 Determine whether each equation can be graphed on the 18 real-number plane. Writeyes or no. c. Day 18 to Day 20 12 -921 1. (x + 3) 2 + ( y - 2) 2 = -4 2.x2 - 3x + y2 + 4y = -7 6 no no Answers 246t 4. BOXES A box with no top and a square 0 3. (x + 2) 2 + y2 - 6y + 8 = 0 4.x2 + 16 = 0 base is to be made by taking a piece of b. Estimate the greatest height reached cardboard, cutting equal-sized squares yes no by the flare. Support the answer from the corners and folding up each numerically. 5. x4 + 4y2 + 4 = 0 x6.2 + 4y2 + 4xy + 16 = 0 side. Suppose the cardboard piece is 24.4 m; See students’ work. square and measures 18 inches on no no

2. RECREATION The daily attendance at a each side. In Exercises 7 and 8, for what valuesk of: (Lesson1-4) state fair is modeled byg( x)=-x4 + 48x3 a. Write a functionv( x) wherev is the a. will the solutions of the equation be imaginary? - 822x2 + 5795x - 7455, wherex is the volume of the box andx is the length b. will the graph be a point? number of days since opening. Estimate of the side of a square that was cut

to the nearest unit the relative or Copyright ©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. c. will the graph be a curve? from each corner of the cardboard. absolute extrema and thex-values where d. Choose a value ofk for which the graph is a curve. Then sketch they occur. v(x)= 4x3 - 72x2 + 324x the curve on the axes provided.

2 2 2 2 8000 b. What value ofx maximizes the 7. x - 4x + y + 8y + k = 0 8.x + 4x + y - 6y - k = 0 4 3 2 volume? What is the maximum g(x) = -x +48x -822x +5795x -7455 a. k > 20; b.k = 20; a.k < -13; b.k = -13; 7000 volume? c. k < 20; k c. > -13; 3 in.; 432 in3 6000 d. y d. y

5000 c. What is the relative minimum of the function? Explain what this k 12

Pdf 3rd =− 4000 minimum means in the context of

Attendance the problem. 0 x 0 x 3000 (9, 0); When squares with k = 19 Lesson 1-4 2000 9-inch sides are cut from each corner, the volume of the box Glencoe Precalculus 1000 is 0 because no material Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. remains. 9. Why would it make no sense to discuss extrema and average rate

0 2 4 6 8 10 12 14 16 18 20 22 24 26 of change for the graphs in Exercises 7 and 8? Day Number Sample answer: They are not functions. rel. max.( 7, 6897); rel. min.( 13, 5857); rel. max.( 16, 5909) 110/23/09 5:09:07 PM

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