Chapter 5: Introduction to Limits. Chapter 5 Recommendations

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1 Chapter 5: Introduction to Limits Chapter 5 Topics: Inverse and Direct Variation Transformations of Rational Functions Graphing Reciprocals of Functions Introduction to Limits Working With One-Sided Limits Continuit - Formal Definition Piecewise Functions and Limits Review Topics: Simplifing Complicated Rational Epressions Solving Equations with Radicals Chapter 5 Recommendations The Closure Lesson at the end of each chapter in the student tet (on the review da) directs students to look back at the main topics in the chapter and review those concepts. Be clear with our epectations and the students will respond accordingl. Man teachers use this as an opportunit for the students to take their time and do a complete write-up of solutions the wish to submit for grading. These assignments can be graded quickl using rubric grading gives students the opportunit to show ou their best work. It is suggested that ou give a team test b the end of the chapter. Emphasize that students need to review the entire chapter, as well as previous chapters, when studing for the individual eam. You ma want to have students brainstorm as a class, or in teams, all of the main topics and review concepts covered in the chapter. It is alwas best to include a few review problems on each individual assessment. This keeps students up to date on their basic skills and allows students who struggle to be successful on some of the problems on the assessment. The sample team test for this chapter does not contain review problems, so ou ma want to add some. If our class periods are such that ou feel there is not enough time to include the review problems ou would like on an eam, use the review problems as a dail warm-up or quick quiz. You ma be read to give a semester eam at the end of Chapter 5. This is a good time to review all of the concepts that students have studied to date. Pre-Calculus with Trigonometr Assessment Bank Chapter 5 92

2 Chapter 5 Assessment Problems Lesson Inverse and Direct Variation 1. If varies directl as (! 1) and f (4) = 4, find f (!2). 2. If varies directl as ( 2 + 2) and f (5) = 9, find f (!1). 3. If varies inversel as ( + 1) and f (!3) = 5, find f (3) If varies inversel as 3! 1 ( ) and f (!1) = 7 4, find f 1 ( ). 2 Lesson Transformations of Rational Functions 1. Sketch the graph of f () = 1 +2! Sketch the graph of f () = 1!4! Rewrite f () = as a transformation of g() = 1 and sketch the graph of f (). 4. Rewrite f () = 3!7!3 as a transformation of g() = 1 and sketch the graph of f (). 5. Write the equation of a rational function that has a horizontal asmptote at = 13 and a vertical asmptote at =!2. 6. Write the equation of a rational function that has a horizontal asmptote at =!12 and a vertical asmptote at = 9. Pre-Calculus with Trigonometr Assessment Bank Chapter 5 93

3 Lesson Graphing Reciprocals of Functions 1. Given f () shown at right, sketch the graph of 1 f (). f() 2. Given f () shown at right, sketch the graph of 1 f (). f() 3. Given f () shown at right, sketch the graph of 1 f (). f() 4. Give an eample of a function f () such that 1 = 4 and =!3. f () has vertical asmptotes at 5. Give an eample of a function f () such that 1 = 6 and =!5. f () has vertical asmptotes at Pre-Calculus with Trigonometr Assessment Bank Chapter 5 94

4 Lessons and Introduction to Limits 1. Evaluate the following limits. If the limit does not eist, eplain wh. a. lim!" d. lim!" ( ) b. lim !" ( ) e. lim ( ) c. lim 2+1 2#1!" #32!" ( ) f. lim!" ( ) # 3 ( cos ) 2. Sketch the graph of f () = 1! 2. Use it to evaluate the limit statements below. +2 a. lim!" f () b. lim f () c. lim f ()!"2 "!" Sketch the graph of f () = Use it to evaluate the limit statements below.!3 a. lim f () b. lim f () c. lim f ()!"!3 "! Given f () = 5, rewrite the function in the form = a + k. Then evaluate the!2!h following limit statements. a. lim f () b. lim f () c. lim f ()!"!2 "! Given f () = 3!1!1 following limit statements., rewrite the function in the form = a + k. Then evaluate the!h a. lim f () b. lim f () c. lim f ()!"!1 "!1 + Lesson Working With One-Sided Limits 1. Let f () = 4!3. Complete the table of values below to estimate the value of lim 4 "3.! f () Pre-Calculus with Trigonometr Assessment Bank Chapter 5 95

5 2. Let f () = 2 +3!10.!2 Complete the table of values below to estimate the value of lim 2 +3"10.!2 " f () 3. Let f () = 3!8!2. Complete the table of values below to estimate the value of lim 3 "8!2 " f () 4. Use a graph or a table to evaluate the following limits. a. lim b. lim 2 "6!" #2!4 +1 c. lim!5 " 2 " Use a graph or a table to evaluate the following limits. a. lim b. lim 2 +1!" 2#1!"2 +1 c. lim!1 + 3 "1 " 2 Lessons and Continuit, Piecewise Functions, and Limits 1. Determine whether the given function is continuous at the point specified and then determine is the limit eists. Eplain our answers. a. f () = 3 +2! 1,! =!2 b. f () = 2!1!4 + 3,! = 4 # c. f () = 2! 1!!!!!!for! < 3 $,! = 3 d. f () = 2!25 +5 %& 2( + 1)!!!for! " 3 2. Find values for m and n such that f () will be a continuous function. + 1,! =!5 #!3 + m!!!!!!for!!!!! <!1 # 1 a. f () =!2 2 2 % % + m!!!!!for!!!!! < 4 $ + 4!!! for! 1 " " 1 b. f () = $ 2! 8!!!!!for!4 " " 7 & %!!3 + n!!!!!!for!!!!! > 1 % 2 + n!!!!!!for!!!!! > 7 & Pre-Calculus with Trigonometr Assessment Bank Chapter 5 96

6 3. Given the piecewise function f () shown below, evaluate the following epressions. a. f (!6) b. lim!"6 " f () c. lim f ()!3 d. f (7) e. lim!7 f () f. lim!" f () g. lim!"# f () 4. Given the piecewise function f () shown below, evaluate the following epressions. a. f (!6) b. lim!"6 f () c. lim!4 " f () d. lim!4 + f () e. f (4) f. lim!" f () g. lim!"# f () # 5. Let f () = 2! 1!!!! for " 3 $. % 2 + 1!!!!for > 3 a. Find lim!3 f (). b. Find lim!0 f (). # 6. Let f () = 3 + 2! 1!!!!for " 1 $!4 +7 % +1!!!!!!!!!!!!for > 1. a. Find lim!1 f (). b. Find lim!2 f (). Pre-Calculus with Trigonometr Assessment Bank Chapter 5 97

7 Review Problems 1. Simplif. a.! !3!1!!1 b. 4!2 +!3 2!3!!4 c.!1!!1! Solve. a. + 5 =! 1 b. 2 2! 9 = c. 5 2! 1 = Factor completel. (! 4) 3 (2 + 3) + (! 4) 2 (6 + 9) 4. Multipl. 1/2 ( 3/2! 4 ) ( 1/2 + 4 ) 5. Simplif. 2! ! !2 6. Solve. a. 200 ( 3 2 ) +2 = 700 b. 5(2) = 50 c. 3 3!1 = Find the area of!jkl. J 8 cm 16 cm 10 cm K L 8. Find the inverse of f () = 4(5)! Graph f () = 2 log 3 (! 3) Write the sigma notation for finding the area under the curve f () = 2 +! 12 when 4!! 10 using 20 right-endpoint rectangles. Pre-Calculus with Trigonometr Assessment Bank Chapter 5 98

8 Word Problems 1. The maimum weight M that can be supported b a beam in directl proportional to its width w and the square of its height h. It is inversel proportional to its length l. a. Write the general equation of proportionalit for the given situation. b. Determine the constant of proportionalit if a beam 4 inches wide, 6 inches high, and 12 feet long can support 7400 pounds. c. If a beam is 10 feet long, 3 inches wide, and 10 inches high, what is the maimum weight it can support? 2. The electrical current produced b a wind-powered generator varies directl with the square of the wind velocit and inversel with the square root of the height of the generator. A generator that operates at a height of 2500 feet and produces 3000 watts when the wind is blowing 25 mph. How much energ will the generator produce if the wind is blowing at 20 mph? 3. The gravitational attraction between two bodies varies directl with their masses and inversel with the square of the distance between them. B what percent does the force of gravitational attraction change if one mass is increased b 20%, the other mass is decreased b 20%, and the separation between them is reduced b 25%? 4. The heat eperienced b a hiker at a campfire is directl proportional to the amount of wood placed on the fire and inversel proportional to the cube of the distance from the fire. If the hiker is 26 feet from the fire and the amount of wood on the fire is doubled, how far from the fire would the hiker need to stand so the he feels the same amount of heat as before? 5. The frequenc of vibration of a string is directl proportional to the square root of the tension, inversel proportional to the square root of its mass and also inversel proportional to its length. Find the ratio of the frequencies of two strings, one that is 3 times longer and thus has 3 times more mass than the other. It also requires twice the tension to be held in place. Pre-Calculus with Trigonometr Assessment Bank Chapter 5 99

9 Lesson Answers ! ! Lesson Answers f () = f () = 2 +2! f () = a f () = a +2!9! 12 Lesson Answers Pre-Calculus with Trigonometr Assessment Bank Chapter 5 100

10 3. See graph at right. 4. f () = a(! 4)( + 3) 5. f () = a(! 6)( + 5) Lessons and Answers 1. a. 0 b. 2 c. DNE /!" d. 1 e. DNE /!" f. DNE 2. a. 2 b.!" c.! 3. a. 1 b.!" c.! 4. a. 5 b.!" c.! 5. a. 3 b.!" c.! Lesson Answers f () f () f () a. 5 b. 2 c.!" 5. a. 3 b. 5 c.! Pre-Calculus with Trigonometr Assessment Bank Chapter 5 101

11 Lesson and Answers 1. a. Continuous, limit eists. b. Not continuous, limit DNE. c. Continuous, limit eists. d. Not continuous, limit = a. m =!1,!n =!1 b. m = 6,!n = a. 2 b.!" c. 1 d. 1 e. 3 f.! g a. 2 b. 3 c. 1 d. 1 e. 1 f. 1 g.! 5. a. 7 b a. DNE b.! 1 3 Review Problem Answers 1. a (!) 4 b !1 c.! 2. a. = 4 b. = ±3 c. = 1 3. (! 4) 2 (2 + 3)(! 1) 4. 5/ ! 4! 16 1/2 5. (!1) 3(!5) 6. a. = 1.09 b. = 3.32 c. = cm 2 8. f!1 () = log +3 5 ( 4 ) 9. See graph at right. 20 & ( ) 10. " # 0.3 ( ) 2 + ( )! 12 $ % =1 f() Pre-Calculus with Trigonometr Assessment Bank Chapter 5 102

12 Word Problem Answers 1. a. M = kwh2 l b c. 18,500 pounds watts % stronger feet 5. short long = or long short = 6 9 Pre-Calculus with Trigonometr Assessment Bank Chapter 5 103

13 Section 5.1 Quiz 1. If varies directl as (! 2) 2 and f (4) = 5, write the particular equation of variation and 3 find f (!1). 2. Rewrite f () = as a transformation of g() = 1 and sketch the graph. 3. Simplif. 16!!4 4+!2 4. Given f () shown at right, sketch the graph of 1 f (). f() 5. Solve. 6 +! 1 = + 3 Pre-Calculus with Trigonometr Assessment Bank Chapter 5 104

14 Section 5.2 Quiz 1. Evaluate the following limits. If the limit does not eist, eplain wh. a. 5+2 lim!"# 2"1 b. lim!" Given the piecewise function f () shown below, evaluate the following epressions. a. lim!" b. lim!"# f () f () c. lim!9 " f () d. lim!9 + f () e. f (!3) f. lim!"3 f () g. f (3) h. lim!3 f () 3. Graph g() = +1!2 and then find A(g(), 2!! 5) using 6 right-endpoint rectangles. Pre-Calculus with Trigonometr Assessment Bank Chapter 5 105

15 Pre-Calculus Chapter 5 Team Test Names 1. Use what ou know about the graph of = 2 +! 6 to sketch the graph of f () = 1 2 +!6. 2. Use the graph of = h() at right to evaluate the following epressions. a. f (0) b. lim!0 " h() c. lim h() d. lim h()!0 +!0 e. lim!" h() f. lim!"# h() h() 3. Given the graph of = h() above, write a piecewise function for h. You ma assume the following: For!" < < 0, h() is sinusoidal. For 0! < 3, h() is of the form = a!b " 4. For 3 < <!, h() is a rational function. Pre-Calculus with Trigonometr Assessment Bank Chapter 5 106

16 4. If z varies directl as and inversel as the square root of, what happens to z if and are both quadrupled? 5. Use a table or graphing function to evaluate lim find our answer. cos "1!0 tan. Eplain the method ou used to 6. Solve = 8 7. Simplif.!2 +!2!2!!2 Pre-Calculus with Trigonometr Assessment Bank Chapter 5 107

17 Pre-Calculus Name Chapter 5 Test 1. Find the values of j and k that will make the given function continuous. " $ 2! 5!!!! < 1 f () = #!! j!!!!!!!!!! = 1 %$ 2 + k!!!! > 1 2. Use the graph at right to evaluate the following epressions. a. f (0) f() b. lim!0 " f () c. lim!0 + f () d. lim!0 f () 3. Use a table or graphing function to evaluate lim find our answer. 2 ""2!2 2 "4. Eplain the method ou used to 4. The resistance of a wire varies directl as its length and inversel as the square of its radius. A wire that is 2.4m long and 0.008m in diameter has a resistance of 150 ohms. Find the diameter of a wire that has a resistance of 90 ohms and is 1.2m long. Pre-Calculus with Trigonometr Assessment Bank Chapter 5 108

18 5. Solve each equation. a. 7 +! 5 = b. 3! 25 = 0 6. Simplif each epression. a b. log 4 3! log Find the area of a triangle with sides of length 3, 9, and Sketch a function with the following properties: lim!" f () = 2 lim f () = #!"2 " lim f () = "4!"2 + f (0) = 1 Pre-Calculus with Trigonometr Assessment Bank Chapter 5 109

19 Section 5.1 Quiz Answers f () =! ; See graph at right ! = 5 ( = 2 does not check.) Section 5.2 Quiz Answers 1. a. 5 2 b. DNE 2. a.! b. 1 c.!" d. 1 e. 2 f. 1 g. 0 h. DNE 3. See graph at right u 2 Pre-Calculus with Trigonometr Assessment Bank Chapter 5 110

20 Chapter 5 Team Test Answers 1. See graph at right. 2. a. 2 b. 1 c. 2 d. DNE e. 2 f. DNE % cos (! 2 )!!!!!!!!!!!"# < < 0 ' 3. h() = & 2( 3) " 4!!!!0 $ < 3 '" 1 "3 ( + 2!!!!!!!3 $ < # 4. It is doubled. f() 5.! = !+ 2! Chapter 5 Test Answers 1. j =!4, k =!6 2. a. 2 b. 1 c DNE m 5. a. = 9 b. =!5, 0, 5 6. a u 2 8. Answers var. Sample answer graphed at right. b. 2 f() Pre-Calculus with Trigonometr Assessment Bank Chapter 5 111