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Semester One Final Examinations, 2015 MATH1051 Calculus and Linear Algebra I Venue ____________________ Seat Number ________ Student Number |__|__|__|__|__|__|__|__| Family Name _____________________ This exam paper must not be removed from the venue First Name _____________________ School of Mathematics & Physics EXAMINATION Semester One Final Examinations, 2015 MATH1051 Calculus and Linear Algebra I This paper is for St Lucia Campus and St Lucia Campus (External) students. Examination Duration: 120 minutes Reading Time: 10 minutes For Examiner Use Only Exam Conditions: Question Mark This is a Central Examination 1 This is a Closed Book Examination - specified materials permitted 2 During reading time - writing is not permitted at all 3 This examination paper will be released to the Library 4 Materials Permitted In The Exam Venue: 5 (No electronic aids are permitted e.g. laptops, phones) An unmarked Bilingual dictionary is permitted 6 Calculators - No calculators permitted 7 Materials To Be Supplied To Students: 8 none 9 Instructions To Students: 10 Additional exam materials (eg. answer booklets, rough paper) will be provided upon request. 11 • Answer all questions in the space provided. Use the blank pages or backs 12 of pages if space is insufficient. • Questions carry the number of marks shown. 13 • This exam contains 120 marks in total. • Show all working. Solutions without justification may not receive full marks. • The last page contains a formula sheet. Total _______ / 120 Marks Page 1 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 2/19 Question 1. Determine whether the following sequences {an } converge or diverge as n → ∞. cos(n2 ) (a) an = √ (b) an = arctan(2n) n 6 Marks Page 2 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 3/19 Question 2. Determine the following limits or show that they do not exist. u2 + 2u − 3 sin(x) cos(x) sin(t) − log(1 − t) (a) lim (b) lim (c) lim u→−3 u2 − u − 12 x→0 x t→0 t 10 Marks Page 3 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 4/19 Question 3. Define f : R → R by    x · sin 1 ,  if x 6= 0 f (x) = x if x = 0.  0, (a) Is f continuous at x = 0? (b) Use the definition of derivative to determine whether f is differentiable at x = 0. 10 Marks Page 4 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 5/19 Question 4. Define a function f : [−5, 5] → R by x+3 f (x) = . x2 + 7 (a) Find the critical points of f and classify them. (b) Find the range of f . 8 Marks Page 5 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 6/19 Question 5. Let f : R → R be defined by f (x) = sin(x) − 2x + 2. (a) Is f a 1-to-1 function? (b) Use the Intermediate Value Theorem to show that the equation sin(x) = 2x − 2 has at least one real solution. (c) Does the equation in (b) have more than one real solution? 10 Marks Page 6 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 7/19 Question 6. Determine whether each of the following series are convergent or divergent. ∞ ∞ ∞ X 2n X (−1)n X 22n+1 (a) (b) √ (c) . 1 + n2 3 n+2 3n−1 n=1 n=1 n=0 10 Marks Page 7 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 8/19 [This page intentionally left for your working.] Page 8 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 9/19 Question 7. Find the interval of convergence of the series given below. ∞ X (x + 3)n n·32n+1 n=1 14 Marks Page 9 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 10/19 [This page intentionally left for your working.] Page 10 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 11/19 Question 8. It is known that the Taylor series expansion for sin(x2 ) about x = 0 is given by 1 6 1 1 sin(x2 ) = x2 − x + x10 − x14 + · · · . 3! 5! 7! Find the Taylor series expansion for 2x cos(x ) about x = 0, for terms up to and including x13 . 2 (Do not worry about any convergence issues.) 8 Marks Page 11 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 12/19 Question 9. Evaluate the following integrals. √ Z Z Z 2 5x + 7 (a) x ln(x) dx (b) dx (c) dx 1 + 4x2 x2 + 3x + 2 14 Marks Page 12 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 13/19 [This page intentionally left for your working.] Page 13 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 14/19 1 Question 10. The region bounded by the curve y = √ above the interval [e, e2 ] is rotated x ln x about the x-axis. Find the volume of the resulting solid. 7 Marks Page 14 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 15/19 Z x √ Question 11. Define a function S by S(x) = sin(t2 ) dt. Find the derivative of S( x ) with respect 0 to x at x = π/2. 6 Marks Page 15 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 16/19 Question 12. Let A and B be n × n matrices with real entries satisfying det(AB 2 ) = 4, det(A2 B) = 2. Find det(A) and det(B). 5 Marks Page 16 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 17/19 Question 13. Let     a    V =  b  ∈ R3 2a − b + 3c = 0 .     c   Is X a vector space? If so, find a basis for X. 12 Marks Page 17 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 18/19 [This page intentionally left for your working.] Page 18 of 19 Semester One Final Examinations, 2015 Math 1051 Calculus & Linear Algebra 1 19/19 Formula Sheet tan θ = sin θ/ cos θ, cot θ = cos θ/ sin θ, sec θ = 1/ cos θ, csc θ = 1/ sin θ sin2 θ + cos2 θ = 1, tan2 θ + 1 = sec2 θ, cot2 θ + 1 = csc2 θ cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ. sin 2θ = 2 sin θ cos θ. n−1 X a(1 − rn ) For a geometric series arj = , if r 6= 1. 1−r j=0 Differentiation rules (appropriate domains assumed): d d d dx sin x = cos x dx cos x = − sin x dx tan x = sec2 x d x d d a dx e = ex dx ln x = 1/x dx x = axa−1 d √ 1 d 1 d 1 dx arcsin x = 1−x2 dx arccos x = − √1−x 2 dx arctan x = 1+x2 If the graph of y = f (x) above the interval [a, b] is rotated about the x-axis, the resulting solid has Z b volume V = π [f (x)]2 dx. a ∞ X f (n) (a) The Taylor series of a function f about x = a is given by (x − a)n , provided this exists. n! n=0 A set of vectors V is a vector space if: (a) V is non-empty (b) u, v ∈ V =⇒ u + v ∈ V (c) v ∈ V and α ∈ R =⇒ αv ∈ V . Page 19 of 19 Semester Two Final Examinations, 2015 MATH1051 Calculus and Linear Algebra I Venue ____________________ Seat Number ________ Student Number |__|__|__|__|__|__|__|__| Family Name _____________________ This exam paper must not be removed from the venue First Name _____________________ School of Mathematics & Physics EXAMINATION Semester Two Final Examinations, 2015 MATH1051 Calculus and Linear Algebra I This paper is for St Lucia Campus students. Examination Duration: 120 minutes Reading Time: 10 minutes For Examiner Use Only Exam Conditions: Question Mark This is a Central Examination 1 This is a Closed Book Examination - specified materials permitted 2 During reading time - writing is not permitted at all 3 This examination paper will be released to the Library 4 Materials Permitted In The Exam Venue: 5 (No electronic aids are permitted e.g. laptops, phones) 6 An unmarked Bilingual dictionary is permitted 7 Calculators - No calculators permitted 8 Materials To Be Supplied To Students: none 9 Instructions To Students: 10 Additional exam materials (eg. answer booklets, rough paper) will be 11 provided upon request. 12  Answer all questions in the space provided.  Use the back of pages if space is insufficient. 13  Each question carries the number of marks shown.  The total number of marks available is 100.  Answers given without justification may not receive full marks. Total ______ / 100 Page 1 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 2/18 Question 1. Determine whether the following sequences {an } converge or diverge as n → ∞. s   n2 + n (a) an = sin (3n + 12 )π (b) an = 4 Marks 4n2 + 1 Page 2 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 3/18 Question 2. Determine the following limits or show that they do not exist. u2 + 4u − 12 x + sin x et + e−t − 2 (a) lim (b) lim (c) lim u→2 u2 − 5u + 6 x→∞ 3x t→0 t2 9 Marks Page 3 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 4/18 Question 3. Define f : R → R by    x2 sin 1  if x 6= 0 f (x) = x if x = 0.  0 (a) Is f continuous at x = 0? (b) Use the definition of derivative to determine whether f is differentiable at x = 0. 9 Marks Page 4 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 5/18 Question 4. Let f (x) = arctan(2 − 2x + x2 ). (a) Classify the critical points of f . (b) Find the range of f . (c) Is f a 1-to-1 function? 9 Marks Page 5 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 6/18 Question 5.Determine whether each of the following series are convergent or divergent. ∞ ∞ ∞ r r ! X n X n+1 n+2 X ln n (a) 3 (b) − (c) (−1)n 2 3+n n n+1 n n=1 n=2 n=3 12 Marks Page 6 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 7/18 [This page intentionally left for your working.] Page 7 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 8/18 Question 6. Find the interval of convergence of the series given below. ∞ X (x − 3)2n n·2n−1 n=1 9 Marks Page 8 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 9/18 [This page intentionally left for your working.] Page 9 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 10/18 Question 7. Find the Taylor series expansion of the function f (x) = ln x about x = 1. 7 Marks Page 10 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 11/18 Question 8. The water level in a dam at time t ≥ 0 is given by Z t 2
x− x W (t) = e 12 dx 0 (a) Is there any time at which the water level is decreasing? (b) When is the water level increasing most rapidly? 8 Marks Page 11 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 12/18 Question 9. Evaluate the following integrals. Z Z Z 1 1 x (a) dx (b) dx (c) dx 4 − x2 4 + x2 4 + x2 9 Marks Page 12 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 13/18 [This page intentionally left for your working.] Page 13 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 14/18 √ Question 10. The curve y = xe−x above the interval [0, 2] is rotated about the x-axis. Find the volume of the resulting solid. 6 Marks Page 14 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 15/18 Question 11. Is the following set of vectors linearly dependent or linearly independent?          1 2 1              2   0   6  v1 =  , v2 =     , v3 =  5  .        1   −2       3 7 2   6 Marks Page 15 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 16/18 Question 12. Let A be a matrix. Prove that NS(A) is a vector space. 6 Marks Page 16 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 17/18   4 Question 13. Let X be the set of all vectors in R3 that are orthogonal to  1 . It is known that   −3 X is a vector space. Find a basis for X. 6 Marks Page 17 of 18 Semester Two Final Examinations, 2015 MATH 1051 Calculus & Linear Algebra 1 18/18 Formula Sheet sin θ cos θ 1 1 tan θ = cos θ cot θ = sin θ sec θ = cos θ csc θ = sin θ sin2 θ + cos2 θ = 1 sec2 θ = tan2 θ + 1 csc2 θ = cot2 θ + 1 sin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ Differentiation rules (appropriate domains assumed): d d d dx sin x = cos x dx cos x = − sin x dx tan x = sec2 x d x d 1 d a dx e = ex dx ln x = x dx x = axa−1 d √1 d dx arcsin x = dx arccos x = − √ 1 d dx arctan x = 1 1−x2 1−x2 1+x2 If f is continuous and non-negative on [a, b] and the graph of y = f (x) above the interval [a, b] is Z b  2 rotated about the x-axis, the resulting solid has volume V = π f (x) dx. a n−1 X a(1 − rn ) arj = if r 6= 1. 1−r j=0 ∞ X f (n) (a) The Taylor series of a function f about x = a is given by (x − a)n provided this exists. n! n=0 A set of vectors V is a vector space if: (a) V is non-empty (b) u, v ∈ V =⇒ u + v ∈ V (c) v ∈ V and α ∈ R =⇒ αv ∈ V . Page 18 of 18 Semester One Final Examinations, 2016 MATH1051 Calculus and Linear Algebra I Venue ____________________ Seat Number ________ Student Number |__|__|__|__|__|__|__|__| Family Name _____________________ This exam paper must not be removed from the venue First Name _____________________ School of Mathematics & Physics EXAMINATION Semester One Final Examinations, 2016 MATH1051 Calculus and Linear Algebra I This paper is for St Lucia Campus and St Lucia Campus (External) students. Examination Duration: 120 minutes Reading Time: 10 minutes For Examiner Use Only Exam Conditions: Question Mark This is a Central Examination 1 This is a Closed Book Examination - specified materials permitted 2 During reading time - write only on the rough paper provided 3 This examination paper will be released to the Library 4 Materials Permitted In The Exam Venue: (No electronic aids are permitted e.g. laptops, phones) 5 Calculators - No calculators permitted 6 Unmarked Bilingual dictionary is permitted 7 Materials To Be Supplied To Students: 8 None 9 Instructions To Students: Additional exam materials (eg. answer booklets, rough paper) will be 10 provided upon request. 11 • Answer all questions in the space provided. Use the back of pages if required. 12 • Answers given without justification may not receive full marks. 13 • Each question carries the number of marks shown. • The total number of marks on this exam is 100. Total _____ / 100 Page 1 of 18 Semester One Final Examinations, 2016 MATH1051 Calculus and Linear Algebra 1 2/18 Question 1. Determine whether the following sequences {an } converge or diverge as n → ∞. 2n2 an = 1 + (−1)n · ln n   (a) an = (b) 3 Marks n2 + 3 Page 2 of 18 Semester One Final Examinations, 2016 MATH1051 Calculus and Linear Algebra 1 3/18 Question 2. Determine the following limits or show that they do not exist. p u2 + u − 6 1 + sin x 1 + 6t − cos t (a) lim (b) lim (c) lim u→2 u2 − 4 x→∞ x t→0 t 8 Marks Page 3 of 18