2018 Semester 1 Mid-sem PDF

Preview

Summary

Semester One Mid-Semester Examinations, 2018 MATH1051 Calculus & Linear Algebra 1Page 1 of 21This exam paper must not be removed from the venueSchool of Mathematics & PhysicsEXAMINATIONSemester One Mid-Semester Examinations, 201 8MATH1051 Calculus & Linear Algebra 1This paper...

Description

Semester One Mid-Semester Examinations, 2018

MATH1051 Calculus & Linear Algebra 1

This exam paper must not be removed from the venue

Venue

____________________

Seat Number

________

Student Number

|__|__|__|__|__|__|__|__|

Family Name

_____________________

First Name

_____________________

Tutorial Group

________

School of Mathematics & Physics EXAMINATION Semester One Mid-Semester Examinations, 2018

MATH1051 Calculus & Linear Algebra 1 This paper is for St Lucia Campus and St Lucia Campus (External) students. Examination Duration:

90 minutes

For Examiner Use Only

Reading Time:

10 minutes

Question

Mark

Part A

Exam Conditions: This is a School Examination

Total

This is a Closed Book Examination

10

Part B

During reading time - write only on the rough paper provided This examination paper will NOT be released to the Library

1

12

Materials Permitted In The Exam Venue:

2 (No electronic aids are permitted e.g. laptops, phones) Calculators - No calculators permitted

3

Materials To Be Supplied To Students:

4

Rough paper Instructions To Students:

5

Additional exam materials (eg. answer booklets, rough paper) will be provided upon request. • • • • • •

Answer all questions in the space provided. Credit can only be given for work written on the examination script. Use the back of pages if space is insufficient. Each question carries the number of marks shown. The total number of marks in the exam is 80. You must justify your answers in Part B. Solutions without justification will not receive full marks.

Page 1 of 21

6 7

12 10 8 10 8 10

Part B Total

70

Total 80

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 PART A (10 marks) For each of the following 10 multiple-choice questions, enter the letter corresponding to the correct answer in the corresponding box. There is no need to show any working. Each question is worth 1 mark. √ 2x − 4 is 1. The domain of g(x) = 2 x −x (A) R\{0, 1} (B) (−∞, 2]\{0, 1} (C) (−∞, 2] (D) [2, ∞) (E) None of the above

Answer to Question 1:

2. Let g(x) = |2 sin x− 1|. The maximum value attained by g on the interval [0, 2π] occurs when x is equal to (A) −1 (B) π 3π (C) 2 (D) 2 (E) None of the above

Answer to Question 2:

Page 2 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018

3. Let f (x) =

(2 − x)(x + 3) . Which of the curves best resembles the graph of f (x)? (x + 3)

Answer to Question 3:

4. For which of the following does lim f (x) exist? x→4

(A) (1) and (2) (B) (1) and (3) (C) (2) and (3) (D) (2) only (E) None of the above

Answer to Question 4:

Page 3 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018

5. Determine lim

x→0

x . |x|

(A) 1 (B) −1 (C) 0 (D) ∞ (E) Does not exist

Answer to Question 5:

6. If f is a function such that lim x→2

f (x) − f (2) = 0, which of the following must be true? x−2

(A) The limit of f (x) as x approaches 2 does not exist. (B) f is not defined at x = 2. (C) The derivative of f at x = 2 is 0. (D) f is continuous at x = 0. (E) f (2) = 0

Answer to Question 6:

Page 4 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018

7. If the function f is continuous for all real numbers and if f (x) = f (−2) =

x2 − 4 when x 6= −2, then x+2

(A) −4 (B) -2 (C) −1 (D) 0 (E) 2

Answer to Question 7:

8. Let

At x = 0, f (x) is

 x2 + 1, x < 0 f (x) = . cos(x), x > 0

(A) undefined. (B) continuous but not differentiable. (C) differentiable but not continuous. (D) neither continuous nor differentiable. (E) both continuous and differentiable.

Answer to Question 8:

Page 5 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 9. Suppose that the function f is continuous everywhere and that f (−2) = −1, f (−1) = 1,

f (0) = −1, f (1) = −2 and f (2) = −4. Which of the following intervals must have a root of f (x) = 0?

(A) [0, 1] (B) [1, 2] (C) [0, 2] (D) [−2, −1] (E) [2, 3]

Answer to Question 9:

10. If lim f (x) = 7,which of the following must be true? x→3

I. f is continuous at x = 3. II. f is differentiable at x = 3. III. f (3) = 7. (A) II only (B) III only (C) I and III only (D) I, II and III (E) None

Answer to Question 10:

Page 6 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 PART B (70 marks) Each of the following questions carries the stated number of marks. Write your answers in the space provided. Show full working. 1. (12 marks) Determine f ′ (x) in each of the following cases: (a) f (x) = (ln x)(cos x) √ (b) f (x) = 4etan x (c) f (x) = sin2 3x + cos2 3x ex − e−x (d) f (x) = x e + e−x √ (e) f (x) = arcsin(x) − 1 − x2

continued... Page 7 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 Working space for Question 1

continued... Page 8 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018   2 − x, x≤1 2. Let f (x) =  x2 − 2x + 2, x > 1. (a) (2 marks) Is f continuous at x = 1? Justify your answer. (b) (7 marks) Is f differentiable at x = 1? Use the definition of the derivative to justify your answer. (c) (3 marks) Sketch the graph of f .

continued... Page 9 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 Working space for Question 2

continued... Page 10 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 3. Determine the limits of the following sequences, or show that they do not exist. (a) (2 marks) an = 2 + cos(nπ ) arctan(n + 1) (b) (4 marks) an = n3 √ ln n (c) (4 marks) an = n

continued... Page 11 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 Working space for Question 3

continued... Page 12 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 4. Determine the following limits or show that they do not exist. x2 + 5x + 4 x→∞ x2 + 3x − 4 (2 − x)ex − x − 2 (b) (5 marks) lim x→0 x3

(a) (3 marks) lim

continued... Page 13 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 Working space for Question 4

continued... Page 14 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 5. Let f (x) = tan(x). (a) (2 marks) State the domain and range of the inverse function, f −1 (x) = arctan(x). Justify your answer. (b) (2 marks) Sketch the graph of f −1 (x). (c) (6 marks) Prove that d  −1  1 . f (x) = 1 + x2 dx

continued... Page 15 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 Working space for Question 5

continued... Page 16 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 6. Consider the equation 3x + 2 cos x + 5 = 0. (a) (4 marks) Use the Intermediate Value Theorem to show that the equation has at least one real solution for x. (b) (4 marks) Are there any more real solutions? Explain.

continued... Page 17 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 Working space for Question 6

continued... Page 18 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 7. Let f (x) = ex (x3 − 4x + 4). (a) (6 marks) Find and classify the critical points of f . (b) (4 marks) What are the global maximum and global minimum values of f on [−1, 2]?

continued... Page 19 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018 Working space for Question 7

continued... Page 20 of 21

MATH1051 - Calculus and Linear Algebra I Mid-semester Examinations, Semester 1, 2018

Formula Sheet tan θ =

sin θ cos θ

sin2 θ + cos2 θ = 1

cot θ =

cos θ sin θ

sec θ =

sec2 θ = tan2 θ + 1

1 cos θ

csc θ =

csc2 θ = cot2 θ + 1

cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1

1 sin θ

sin 2θ = 2 sin θ cos θ

= 1 − 2 sin2 θ

Differentiation rules (appropriate domains assumed): d dx

d ex dx d dx

d dx

sin x = cos x

d dx

= ex

arcsin x = √ 1

1−x2

d dx

d dx

cos x = − sin x ln x =

1 x

tan x = 1 + tan2 x d a x dx

arccos x = − √ 1

1−x2

Page 21 of 21

d dx

= axa−1 , a 6= 0

arctan x =

1 1+x2...