FINAL EXAM (PRACTICE A) MATH 265

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UNIVERSITY OF CALGARY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS & STATISTICS FINAL EXAM (PRACTICE A) MATH 265 NAME STUDENT ID EXAMINATION RULES 1. This is a closed book eamination. 2. Calculators are not permitted.

1 UNIVERSITY OF CALGARY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS & STATISTICS FINAL EXAM (PRACTICE A) MATH 265 NAME STUDENT ID EXAMINATION RULES 1. This is a closed book eamination. 2. Calculators are not permitted. 3. The use of personal electronic or communication devices is prohibited. 4. The eam has many questions. 5. Scantron sheets must be filled out during the eam time limit. No additional time will be granted to fill in scantron form. 6. A University of Calgary Student ID card is required to write the Test. If adequate ID is not present, the Student may be asked to complete an Identification Form. 7. Students late in arriving will not be permitted to write the eam thirty (3) minutes after the eamination has started. 8. No student will be permitted to leave the eamination room during the first thirty (3) minutes, nor during the last ten (1) minutes of the eamination. Students must stop writing and hand in their eam immediately when time epires. 9. All inquiries and requests must be addressed to the eams Supervisor. 1. Students are strictly cautioned against: (a) communicating to other students; (b) leaving answer papers eposed to view; (c) attempting to read other students eamination papers. 11. If a student becomes ill during the course of the eamination, he/she must report to the Invigilator, hand in the unfinished paper and request that it be cancelled. 12. Once the eamination paper has been handed in for marking, the Student cannot request that the eamination be canceled. 13. Failure to comply with these regulations may result in the rejection of the eamination paper.

2 Part I: True/False questions are worth 2 marks each, and multiple choice questions are worth 4 marks each. For each question, clearly circle your choice on this booklet, and record your answer on the scantron sheet provided. Make sure that you answer all the questions: remember there is no penalty in guessing. 1. True / False. If a function f() iscontinuousat = a, thenf() isalsodi erentiableat = a. 2. True / False. Suppose f() is everywhere continuous and ˆ 1 f(t)dt =sin. Thenf( ) = True / False. For any two functions f() andg(), the derivative of f() g() isequaltof () g (). 4. True / False. Suppose lim!2 f() =3. Thenat least one of the following holds: f() iscontinuousat =2. f() isdi erentiableat =2. f(2) = True / False. Assuming >1, the equation log (2 )= 4 has a unique solution of =16. Page 2

3 6. Find the (natural) domain of the function g() = (a) ( 1, 1). (b) [5, 1). (c) ( 1, 1) [ ( 1, 1) [ (1, 1). (d) ( 1, 1) [ ( 1, 1) [ (1, 3]. (e) ( 1, 1) [ ( 1, 1) [ (1, 5]. p Which of the following is equal to sin 1 sin (a). (b) /5. (c) 2 /5. (d) 3 /5. (e) /5. 3? 5 8. Evaluate the limit L = lim! 1 (a) L =4. (b) L = 2. (c) L = 4. p (d) L = 4 3. (e) L does not eist. Page 3

4 9. Let f be a continuous function on [ 2, 7]. If f( 2) = 1andf(7) = 3, then the Intermediate Value Theorem guarantees that (a) f() =. (b) f (c) = 4 9 for at least one c between 2and7. (c) 1 apple f() apple 3forall between 2and7. (d) f(c) =1foratleastonec between 2and7. (e) f(c) =foratleastonec between 1and3. 1. Which of the following does not describe the derivative of a function f() at = a: f() f(a) (a) lim.!a a f( + h) f(a) (b) lim. h!a h f(a + h) f(a) (c) lim. h! h (d) The slope of the tangent line to the graph y = f() at = a. (e) The instantaneous rate of change of f() at = a. 11. If f() isafunctionsuchthatlim!1 2f() 2f(1) 1 (a) The limit of f() as approaches 1 does not eist. (b) f() isnotdefinedat =2. (c) The derivative of f() at =1isequalto. (d) f() iscontinuousat =. (e) f(1) =. =, which of the following must be true? Page 4

5 12. Find an equation of the tangent line to f() = (a) y =16. (b) y = 16. (c) y = (d) y = at = The Taylor polynomial of degree 2, centred at =,forthefunctiony = e 2 is: (a) (b) (c) 1+2( 2) + 2( 2) 2. (d) Find dy d if 2 +3y 2 =2y. (a) dy d = 1 3. (b) dy d = y 3y. (c) dy d = 3y. (d) dy d = y +3y. (e) dy d = 2y. Page 5

6 15. Find the derivative of the function f() = p sin( ). (a) (b) cos( ) 2 p sin( ). cos( ) 2 p sin( ). (c) cos( ) p sin( ). (d) 1 2 (sin( )) 1 2. (e) p cos( ). 16. The derivative of the function ( +1) +1 is: (a). (b) ( +1) +1 (c) ( +1) +1 (ln ) (d) ( +1) +1 (ln( +1)+1) 17. What is the derivative of f() = 2 cos 1 ()? (a) 2 p 1 2. (b) 2 cos 1 ()+ 2 p 1 2. (c) 2 cos 1 () 2 p 1 2. (d) 2 cos 1 ()+ 2 (cos 2 ()) sin. (e) 2 cos 1 () 2 (cos 2 ()) sin. Page 6

7 18. Which of the following is the di erential of y = cos(2)? (a) dy = sin(2) d. (b) dy = 2sin(2) d. (c) dy =( sin(2) + cos(2)) d. (d) dy =(2sin(2)+cos(2)) d. (e) dy =( 2 sin(2)+cos(2)) d. 19. Answer the following question given the function and its derivative f() = 2 +1, f () = (2 1) ( 2 +1) 2. (a) f has a local minimum at = 1 and a local minimum at =1. (b) f has a local maimum at = 1 and a local maimum at =1. (c) f has a local minimum at = 1 and a local maimum at =1. (d) f has a local maimum at = 1 and a local minimum at =1. (e) f has a local minimum at =. 2. Which of the following is the absolute minimum value of f() = + 4 on the interval [1, 4]? (a) 1. (b) 2. (c) 3. (d) 4. (e) 5. Page 7

8 21. If =2isacriticalpointoff() = (a) 3/2. (b) 2. (c) 3. (d) 6. (e) 12. a ,whatisthevalueoftheconstanta? 22. Suppose we want to estimate the value of p 3 by applying Newton s method to the equation 2 3=(i.e.,usingf() = 2 3). If we start with the first estimate 1 =2,whatisthe value of the second estimate 2? (a) 3/2. (b) 1/4. (c) 5/4. (d) 7/4. (e) 13/ Acompanywantstomanufactureacookietinwithasquarebaseandtopeachofsidelength, andrectangularsides.thematerialforthesidescosts$3/cm 2, and for the top and bottom the cost is $4/cm 2.Thetinistohaveavolumeof5/cm 3.Thedimensionsofthecheapestsuch container can be found by: (a) minimizing the function C() =4 2 +6/ 2 on (, 1). (b) minimizing the function C() =4 2 +6/ on (, 1). (c) minimizing the function C() =8 2 +6/ on (, 1). (d) minimizing the function C() =8 2 +4/ on (, 1). (e) minimizing the function C() =8 2 +4/ 2 on (, 1). Page 8

9 24. Determine the concavity of the function f() = (a) Concave upward on ( 1, ) and on (2, 1), concave downward on (, 2). (b) Concave downward on ( 1, ) and on (2, 1), concave upward on (, 2). (c) Concave upward on ( 1, ) and on (, 3) and concave downward on (3, 1). (d) Concave downward on ( 1, ) and on (, 3) and concave upward on (3, 1). 25. Suppose that the graph of f passes through the point (1, 3) and that the slope of its tangent line at (, f()) is 2. What is the value of f(3)? (a) 7. (b) 8. (c) 9. (d) 1. (e) If ˆ 4 f() d =2and ˆ 4 (a) 6. (b) 3. (c). (d) 3. (e) 6. g() d = 3, what is the value of ˆ 4 (f()+g()+1)d? Page 9

10 27. Find the function F (), given that F () = 3 9 and F (1) =. 3 7 (a) F () = (b) F () = (c) F () = (d) F () = (e) F () = Evaluate (a) e. (b) e. ˆ /2 cos e sin (c) 1 e. (d) e 1. d. 29. Evaluate (a). (b) 1/3. (c) 1/4. (d) 1/12. ˆ 1 2 (1 ) d. Page 1

11 3. Calculate ˆ e 1 ln d. (a) 1 2. (b) e (c). (d) 1 e. 31. If h() = ˆ 9 ln(t 2 +1)dt, then (a) h () =ln( 2 +1). (b) h () = ln( 2 +1). (c) h () =ln(82) ln( 2 +1). (d) h () =ln( 2 +1) ln(82). (e) h () =2 ln( 2 +1). 32. If F () = (a). (b) 1. (c) e. (d) 1/e. ˆ (e) It does not eist. 1 t 2 e t dt, thenwhatisthevalueof(f 1 ) ()? Page 11

12 33. Determine the area enclosed by the curves = y and = y 3. (a). (b) 1/2. (c) 1/3. (d) 1/ Which of the following is NOT an improper integral? (a) (b) (c) (d) (e) ˆ 1 1 ˆ 1 ˆ 3 ˆ ˆ 1 sin d. e 3 cos() d. ( 3) 2 d d. 4 d. 35. The improper integral (a) converges to 1. (b) converges to e 1. (c) converges to e. (d) converges to e +1. (e) diverges. ˆ 1 e d? Page 12

Math 295: Exam 3 Name: Caleb M c Whorter Solutions Fall /16/ Minutes

Math 295: Exam 3 Name: Caleb M c Whorter Solutions Fall /16/ Minutes Math 295: Eam 3 Name: Caleb M c Whorter Solutions Fall 2018 11/16/2018 50 Minutes Write your name on the appropriate line on the eam cover sheet. This eam contains 10 pages (including this cover page)