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Paper size Letter 1. Fill in the blanks so that the resulting statement is true. A function f is continuous at a when three conditions are satisfied. . f is defined at so that is a real number. . lim exists. . lim A function f is continuous at a when three conditions are satisfied. . f is defined at (1) so that (2) is a real number. . lim (3) exists. (4) . lim (5) = (6) (7) . (1) O f(a), (2) O f(a) (3) 0 a (4) 0 a-f(x) (5) O f(x) (6) O f(a) (7) Ox-a O f(x), O a Of( x) O a-x O f(a) O a O a-x O a, O f( x ) O f(a ) O x-f(a) O a Of( x) O x-f(a) Ox-a O a-f(x)2. Fill in the blanks so that the resulting statement is true. 1-x ifx 1 We find lim f(x) using f(x) = We find lim f(x) using f(x) = The function's definition indicates that f(1) = If we X-1 x-1 determine that lim f(x) = lim f(x) =f(1), we can conclude that x-1 x- 1* We find lim f(x) using f(x) = (1) We find lim f(x) using f(x) = (2) The function's definition indicates that X-1 x-1 f(1) = (3) If we determine that lim f(x) = lim f(x) =f(1), we can conclude that (4) X-1 X- 1* (1) 0 x2- 1. (2) 0 1-X. (3) 0 1-X. (4) O f is continuous at 1. O 1 - X. O x 2 - 1. O O. O f is discontinuous at 1. O O. O O. O x2 - 1. 3. Determine whether the following statement is true or false. Every polynomial function is continuous at every real number. Choose the correct answer below. O False O True4. Use the definition of continuity to determine whether f is continuous at a = 5. f(x) = 4x + 7 Is f continuous at 5? Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. O A. No, because f(5) = and lim f(x) = and f(5) # lim f(x). X -5 X-+5 O B. No, because lim f(x) does not exist. X-+5 O C. Yes, because f(5) = and lim f(x) = and f(5) = lim f(x). X-5 X-+5 O D. Yes, because lim f(x) exists. X-5 5. x2 + 2 Use the definition of continuity to determine whether f(x) = X - 6 IS continuous at 7. Is f continuous at 7? Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. O A. Yes. Because f(7) = and lim f(x) = X-7 O B. No. Because f(7) = and lim f(x) = and f(7) # lim f(x). X-7 X-7 O C. No. Because f(7) does not exist.6. Use the definition of continuity to determine whether f is continuous at a. x- - 100 if x # 10 f(x) = x - 10 if x = 10 a = 10 Is f continuous at 10? Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. O A. Yes, because both f(10) and lim f(x) exist. x-10 O B. Yes, because f(10) = and lim f(x) = and f(10) = lim f(x). X-10 X-10 O C. No, because f(10) = and lim f(x) = and f(10) # lim f(x). X-10 X-10 O D. No, because f(10) does not exist. 7. Use the definition of continuity to determine whether f is continuous at a. 12 - X if x 12 a = 12 Is f continuous at 12? Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. O A. No, because f(12) = and lim f(x) = and f(12) # lim f(x). X-12 X-12 O B. Yes, because f(12) = and lim f(x) = and f(12) = lim f(x). X-12 X-+12 O C. No, because f(12) does not exist. O D. Yes, because f(12) and lim f(x) exist. X-128. Determine for what numbers, if any, the given function is discontinuous. f(x) =x4+6x+2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function f is discontinuous at (Use a comma to separate answers as needed.) O B. The function f is continuous everywhere. 9. Determine for what numbers, if any, the given function is discontinuous. x - 5 f(x ) = = (x - 5)(x +9) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function f is discontinuous at (Use a comma to separate answers as needed.) O B. The function is continuous everywhere. 10. Determine for what numbers, if any, the given function is discontinuous. X - 4 if x $ 4 f(x ) = if x > 4 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function f is discontinuous at (Use a comma to separate answers as needed.) O B. The function f is continuous everywhere.Determine for what numbers, if any, the given function is discontinuous. X+9 if x= 0 f ( x ) = 9 if 0 7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function is discontinuous at x = (Use a comma to separate answers as needed.) O B. The function is continuous everywhere. 12. Determine for what numbers, if any, the given function is discontinuous. 21x if x 20 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function f is discontinuous at (Use a comma to separate answers as needed.) O B. The function is continuous everywhere. 13. Explain how to determine whether a function is continuous at a number. A function is continuous at a number when the following conditions are satisfied. Select all that apply. A. The function has a real number value at that number. OB. The limit of the function at that number is not equal to the function value at that number. O) C. The values of the function do not approach a single limit as the domain values approach that number. OD. The limit of the function at that number is equal to the function value at that number. OE. The values of the function approach a single limit as the domain values approach that number. OF. The function is not defined at that number.14. Define g(2) for the given function so that it is continuous at x = 2. 5x2 - 20 g(x)= 5x - 10 Define g(2) as (Simplify your answer.) 15. Find A so that the function is continuous at 5. f ( x ) = x 2 X 7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The function f is discontinuous at (Use a comma to separate answers as needed.) O B. The function is continuous everywhere.1 . Fill in the blanks so that the resulting statement is true. The principle of mathematical (1) . states that a statement involving positive integers is true for all positive integers when the following two conditions have been satisfied. The first condition states that the statement is true for the positive integer (2) The second condition states that if the statement is true for some positive integer k, it is also true for the next positive integer (3) (1) O counting (2) O n. (3) On + k. O induction O 1 . OK + 1. 2. Fill in the blanks so that the resulting statement is true. Consider the statement 2 + 4 + 6 + ... + 2n = n(n + 1). If n = 1, the statement is 2 = 1(1 + 1). If n = 2, the statement is 2 + 4 = 2(2 + 1). If n = 3, the statement is If n = k + 1, the statement is Choose the correct terms below. Consider the statement 2 + 4 + 6 + ... + 2n = n(n + 1). If n = 1, the statement is 2 = 1(1 + 1). If n = 2, the statement is 2 + 4 = 2(2 + 1). If n =3, the statement is (1) If n= k + 1, the statement is (2) (1) O 2+4+6=4(2+1). (2) O 2+4+6 + ...+2(k + 1) = (k+1)(k + 2). O 2+4+6+...+2(k+1) =k(k+1). O 2+4+2=2(3+1). O 2+4+6 + ...+ (k+1) = (k+1)(k+2). 2+4+6=3(3 +1). O 2+4+6 + ...+ 2k=k(k + 1). O 2(3) = 3(1 + 1). O 2+4+6 =(k +1)(k + 2).3. Fill in the blanks so that the resulting statement is true. Consider the statement 3 + 7 + 11 + ... + (4n -1) = n(2n + 1). If n = 1, the statement is 3 = 1(2 + 1). If n = 2, the statement is 3 + 7 = 2(4 + 1). If n = 3, the statement is If n = k + 1, the statement before it is simplified is If n =k + 1, the statement after it is simplified is Choose the correct terms below. Consider the statement 3 + 7 + 11 + ... + (4n -1) = n(2n + 1). If n = 1, the statement is 3 = 1(2 + 1). If n = 2, the statement is 3 + 7 = 2(4 + 1). If n = 3, the statement is (1) If n =k + 1, the statement before it is simplified is (2) If n = k + 1, the statement after it is simplified is (3) (1) O 3+7+ 11 + ...+(12-1)=3(6+1). O 3+ (12 -1) =2(6+1). O 3+7+ 11 + ... =3(6+1). O 3+7 +11 =3(6 +1). (2) O 3+7 +11 + ...+ [4(k+1)- 1] = k[2(k + 1)]. O 3+7+ 11+ [4(k + 1) - 1] =(k+ 1)[2(k + 1) + 1]. O 3+7+11 + ...+[4(k + 1) - 1] = (k + 1)[2k + 1]. O 3+7 +11 + [4(k + 1)- 1] = (k+ 1)[2k + 1]. O 3+7 + 11 + ... + [4(k + 1) - 1] = (k + 1)[2(k + 1) + 1]. (3) 0 3+7+ 11 +(4k+ 3) = (k+ 1)[2k+ 1]. O 3+7 + 11 + ... + (4k + 3) = (k + 1)[2k + 3]. 3+7 +11 + ... + (4k+ 3) = k[2(k + 1)]. O 3+7 + 11 + (4k + 3) = (k+ 1)[2k + 3]. O 3+7 + 11 + ... + (4k+3) = (k+ 1)[2k + 1].Fill in the blanks so that the resulting statement is true. Consider the statement "2 is a factor of n2 + 3n." If n =1, the statement is "2 is a factor of If n =2, the statement is "2 is a factor of If n = 3, the statement is "2 is a factor of . If n = k+ 1, the statement before it is simplified is "2 is a factor of If n = K+ 1, the statement after it is simplified is "2 is a factor of Choose the correct terms below. Consider the statement "2 is a factor of n2 + 3n." If n =1, the statement is "2 is a factor of (1 ) If n =2, the statement is "2 is a factor of (2) If n = 3, the statement is "2 is a factor of (3) If n = k+ 1, the statement before it is simplified is "2 is a factor of (4} If n = K+ 1, the statement after it is simplified is "2 is a factor of {5) (1) Q 3." Q 12." Q 20." (2) Q 12." Q 3." 0 1o" (3) Q 3." Q 11." Q 6." Q a." 0 1e." 0 4." Q 10." Q 18." Q 4." Q 8." 0 1e." 0 4." Q 8." Q 11." 0 2o." 0 1o" (4) O K2+3k." Q (k+1)2+3k." (5) O k2+4k+1.\" O k2+3k." O 2+(k+1)2+3[k+1)." O k2+5k+4.\" Q [k+1)2+3(k+1).\" O k2+5k+6.\" O [k+1)+3(k+1)." O 4k+2." 0 2o." 0 11." O 8." O 6." Q 12." 5. Fill in the blank so that the resulting statement is true. k2+Tk+12=[k2+4k]+3( ) k2+rk+12=(k2+4k)+3( J 6. A statement S about the positive integers is given below. Write statements Sk and Sk + 1. Sn: 3 +7 + 11 + ... + (4n - 1) = n(2n +1) SK: 3 +7+ 11 + ...+(4k-1)= SK + 1: 3+7+11 + ...+ [4(k +1) -1]= (Simplify your answer. Type your answer in factored form.) 7. A statement S, about positive integers is given below. Write statements Sk and Sk + 1, Simplifying Sk + 1 completely. n(3n + 1) S : 2+5 +8 + ... + (3n - 1) = 2 Write the statement Sk. S : 2+5 +8 + ...+ (Simplify your answer.) Write the statement Sk + 1- SK + 1: 2+5+8+...+ ([ (Simplify your answer.) 8. A statement S about the positive integers is given. Write statements Sk and Sk + 1, Simplifying Sk + 1 completely. Sn : 2 is a factor of n - 7n + 4 SK : 2 is a factor of n2 - 7n + 4 (Simplify your answer. Do not factor.) Sk + 1: 2 is a factor of (Simplify your answer. Do not factor.)9. Use mathematical induction to prove that the statement is true for every positive integer n. r+21 +35+ +{14n ?)=Tn2 What is the first step in a mathematical induction proof? O A. Show that SKH is true. 0 B. Show that Sk is true. O C. Show that 31 is true. O D. Show that 50 is true. Show that 7' + 21+ 35 + + {Mn 7') = 7n2 for n = i . Write the statement 31. : :I {Do not simplify.) Simplify 81 on the right. Is 81 a true statement\"? O Yes O No What is the next step of the proof? 0 A. Show that Sk is true. O B. Show that SK\" is true. 0 C. Show that if8k+1 is true, then Sk is true. O D. Show that ifSk is true. then Slut is true. Write 8k. {Do not simplify.) Write Sk + 1- Paper size Letter Print 7+ 21+35 + ... + (Do not simplify.) What do we do with statements Sk and Sk + 1? O A. Use Sk + 1 to prove that Sk is true. O B. Use Sk and Sk + 1 to prove that S, is true. O C. Use Sk to prove that Sk + 1 is true. O D. Use Sk and S, to prove that Sk + 1 is true. To continue the proof by induction, begin with 7+ 21 + 35+ ... + (14k -7) = 7k which is assumed to be true, and add the k + 1 term to both sides of the formula. What is this term? O A. 7 O B. 14(n + 1)-7 O C. 14K - 7 O D. 14(k + 1) - 7 Add this term to both sides of the equation. 7 + 21+ 35 + ... +(14k - 7) = 7k2 7 +21 +35 + ... + (14k - 7) + = 7k2 + (Do not simplify.) Simplify the right side. 7+ 21 +35 + ... + (14k - 7) + [14(k + 1) - 7]= (Do not factor.) Factor the right side. 7 +21 +35 + ... + (14k - 7) + [14(k + 1) - 7] = Factor completely.)The right side of the above equation is Sky. Have we finished proving the given statement for all positive integers n? O A. No, since we must show that S satisfies the statement. O B. No, since we must show that S , satisfies the statement. O C. No, since we have only shown that Sky satisfies the statement. O D. Yes.10. Use mathematical induction to prove that the following statement is true for every positive integer n. lgntn+l) 19+38+5Y+m+19n= 2 Check whether 31 is true. '? S1319: {Type an integer or a simplified fraction} Is the statement 81 true? 0 Yes O No Assume that SK is true. Using SK, oheclr whether 8k\" is also true,. 9 SKH:19+38+57+---+19(k+1);|:| {Simplify your answer.) Is the statement SK +1 true\"? 0 Yes O No Is the given statement true for every positive integer n? O No 0 Yes