Selected Problems for Final Exam and Solutions - Name Student number 1 Each part of the following question is worth 2 points NO PARTIAL CREDIT will be

Selected Problems for Final Exam and Solutions - Name...

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Unformatted text preview: Name: Student number: 1. () Each part of the following question is worth 2 points. NO PARTIAL CREDIT will be given. (i) Describe the difference between the graphs of y = f (x) and y = f (x + 1). (ii) State the Intermediate Value Theorem. (iii) Find the derivative of the function f (x) = e2x /over cos(x) − 1 (iv) Find limx→∞ ln(x) . x (v) Give an example of a function which is continuous but not differentiable at x = 0. 2 continued on next page Name: Student number: (vi) State the Fundamental Theorem of Calculus part I. (vii) State the Fundamental Theorem of Calculus part II. (viii) Find the average value of the function f (x) = ex on the interval [0, 1]. (ix) Express (x) f (x) = 1 x(x−1) x x−1 . in partial fractions. f ′ (x) = 3 continued on next page Name: Student number: 2. () Find the following integrals. 1 dx. (x + 5)2 (x − 1) (i) 2 (ii) 1 √ x2 − 1 dx x 4 continued on next page Name: Student number: (iii) tan5 (x) dx (iv) x2 sin(2x) dx 5 continued on next page Name: Student number: 1 3. () Consider the function f (x) = x from x = 1 to x = 3. Use the midpoint rule with 3 4 intervals to estimate 1 f (x)dx. With reference to the graph of y = f (x) determine whether your estimate is larger or smaller than the exact value of the integral. Using Simpson’s Rule, how large must n be to calculate the integral correct to within 10−5 ? 6 continued on next page Name: Student number: 4. () Find the volume of the solid whose base is enclosed by the circle x2 + y 2 = 1 and whose cross-section perpendicular to the x axis is always an equilateral triangle. 7 continued on next page Name: Student number: 1 5. () Consider the function f (x) = 1 − 2 x2 + sin(x). (i) f has one critical number. Find it correct to one decimal place. (Hint: you may want to use Newton’s method.) (ii) Determine whether f has a maximum or a minimum at the critical number. Justify your answer. (iii) Sketch the graph of y = f (x). 8 continued on next page Name: Student number: √ 6. () Find the largest area of a rectangle inscribed inside the graph of y = 2 − x so that the left side of the rectangle is on the y-axis, the bottom of the rectangle is on the x-axis and the top right vertex of the rectangle is on the graph. 9 continued on next page Name: 7. (10) Sketch the graph of y = Student number: x2 . x2 − x − 2 10 continued on next page Math 1A03/1ZA3 Examination Practice Problems (Practice Test Set #3)  ,I - ²%³ ~ % b% %  /HW ²%³ ~ c! ! ILQG WKH YDOXH RI - Z    ²!³! ZKHUH  LV WKH IXQFWLRQ ZKRVH JUDSK LV WR WKH ULJKW )LQG WKH LQWHUYDOV LI DQ\ RQ ZKLFK  LV FRQFDYH XS  /HW - ²%³ ~ %  ²!³! ZKHUH  ²!³ ~ B  (YDOXDWH OLP % %¦ % ! j b" " " )LQG - ZZ ² ³ OQ² b ! ³! %  /HW ²%³ ~ ²! b ³! % € c  6NHWFK WKH DUHD UHSUHVHQWHG E\ ²%³À c c % % % FRV %  )LQG WKH GHULYDWLYH RI  ²%³ ~ % j% ! 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(YDOXDWH WKH IROORZLQJ LQWHJUDO OQ²% b ³%  56. (YDOXDWH WKH LQWHJUDO VLQ ²OQ %³ FRV ²OQ %³ % % 57. (YDOXDWH WKH IROORZLQJ LQWHJUDO ° %VLQc % % OQ  58. (YDOXDWH WKH IROORZLQJ LQWHJUDO OQ²° ³ % VLQ ²% ³ % McMaster University Math 1A03/1ZA3 Fall 2013 Final Exam Duration: 3 hours PRACTICE VERSION edited 9 December 2013 Instructors: M. Bays, D. Haskell, E. Harper, C.McLean Name: Student ID Number: This test paper is printed on both sides of the page. There are 30 question on 10 pages. You are responsible for ensuring that your copy of this test is complete. Bring any discrepancies to the attention of the invigilator. ALL answers must be submitted on the Scantron card. The ten questions in Part I are each worth one point. The twenty questions in Part II are each worth 2 points. There is no partial credit. Instructions (1) Only the standard McMaster calculator Casio-fx 991 is allowed. (2) All answers must be entered on the computer card with an HB pencil. Read the marking instructions on the card. (3) Write your name and ID number on the computer card. (4) Fill in your ID number and the version number in the bubbles. (5) No marks will be deducted for wrong answers or blank answers. (6) Any question left blank will receive 0 marks, even if the correct answer is circled on the exam page. You must enter your answers on the computer card. (7) Scratch paper is available for rough work; ask the invigilator. This PRACTICE version of the midterm is intended to give you an idea of the format, approximate length and approximate difficulty of the actual midterm. There is no guarantee as to the actual length and difficulty of the actual exam. In particular, the actual midterm will NOT be “just the same with the numbers changed”. page 1 of 9 Math 1A03/1ZA3 Fall 2013 Final Exam — Practice Part I Each question in this part is worth one point. Mark your answer on the scantron card, in the bubbles numbered 1–10. 1) Find the derivative of f (x) = e−2x . A. f ′ (x) = e−2x C. f ′ (x) = 2e2x B. f ′ (x) = −2e−2x D. f ′ (x) = e−2x−1 E. f ′ (x) = e−2x+1 2) Find the derivative of f (x) = x2 sin(x3 + 1). A. f ′ (x) = 2x sin(x3 + 1) + x2 cos(x3 + 1) B. f ′ (x) = 2x cos(3x2 ) C. f ′ (x) = 2x sin(x3 + 1) + 3x4 cos(x3 + 1) D. f ′ (x) = 2x sin(3x2 ) E. f ′ (x) = 2x sin(x3 + 1) + x2 sin(x3 + 1) 3) Find the indefinite integral A. cos(3x + 2) dx. 1 1 cos(3x + 2) + C B. sin(3x + 2) + C C. 3 sin(3x + 2) + C 3 3 D. 3 cos(3x + 2) + C E. −3 sin(3x + 2) + C √ x x2 + 1 dx. 4) Find the indefinite integral A. 1 1 √ 2 1 2 2 B. (x2 + 1)3/2 + C C. x2 x2 + 1 + x (x2 + 1)3/2 + C x (x + 1)3/2 + C 3 3 2 3 D. (x2 + 1)−1/2 + C E. (x2 + 1)3/2 + C π sin3 (x) dx. 5) Find the definite integral −π A. 2π B. π 2 C. 0 D. 1 E. the function does not have an elementary antiderivative page 2 of 9 Math 1A03/1ZA3 Fall 2013 Final Exam — Practice 3x2 + 4x = ? x→∞ 1 − 5x2 6) lim A. ∞ B. 3 5 C. −3 5 D. 0 E. −∞ 7) Suppose the function f satisfies the following properties: lim f (x) = 0, lim f (x) = 0, f is x→∞ x→−∞ increasing on (−∞, 0) and decreasing on (0, ∞). Which of the following functions could be f ? A. f (x) = ex C. f (x) = ex B. f (x) = e−x 2 D. f (x) = e−x 2 E. −e−x 2 8) Consider the function √ x, √ − −x, f (x) = if x ≥ 0 if x < 0. Which one of the following statements is correct. A. By the Mean Value Theorem, since f (−1) = −1 and f (1) = 1, there is c ∈ [−1, 1] with f ′ (c) = 1. B. By the Intermediate Value Theorem, since f (−1) < 0 and f (1) > 0, there is c ∈ [−1, 1] with f (c) = 0. C.. The Intermediate Value Theorem does not apply to this function because it is not continuous on the interval [−1, 1]. D. This function does not make sense, as the square root is not defined for negative numbers. E. None of the above are true. 9) The region between the function f (x) = 4x − x2 and the x-axis is rotated around the x-axis. Which of the following integrals expresses the volume of the solid formed? 4 A. 0 4 π(f (x) − 2)2 dx 2 0 2 f (x) dx 0 4 2 π(f (x)) dx + 0 C. 4 2 D. 4 πy 2 dy B. π(f (x) − 2) dx page 3 of 9 π(f (x))2 dx E. 0 Math 1A03/1ZA3 Fall 2013 Final Exam — Practice 10) Which of the following integrals will calculate the area bounded by the graphs of y = cosh(x), y = sinh(x) and the vertical lines x = −1, x = 1? 0 A. −1 1 (cosh(x) − sinh(x)) dx + 0 1 (sinh(x) − cosh(x)) dx 1 C. −1 B. −1 (sinh(x) − cosh(x)) dx 1 (cosh(x) − sinh(x)) dx D. −1 0 E. −1 (cosh(x) − sinh(x))2 dx 1 (sinh(x) − cosh(x)) dx + 0 (cosh(x) − sinh(x)) dx Part II Each question in this part is worth two points. Mark your answer on the scantron card, in the bubbles numbered 11–28. There is no partial credit. Only the Scantron cards will be marked. 11) Find the indefinite integral A. 4 1− √ √ y+C B. 1 √ ln |1 − y| + C 2 √ D. 2 y − ln |y| + C √ √ y) + C y| + C 3 2 sec5/2 (x) + C 5 D. E. −2 ln |1 − C. 2 ln(1 − sec 2 (x) tan(x) dx 12) Find the indefinite integral A. 1 √ dy. y(1 − y) 3 2 B. − cos− 2 (x) + C 3 3 2 sec 2 (x) + C 3 3 C. sec 2 (x) + C E. none of these 13) The region enclosed by the graphs of y = ex/2 , y = (x − 1)2 , and the line x = 1 from x = 0 to x = 1 is rotated about the x-axis. What is the volume of the solid generated? A. 11 4 B. 4 C. 9 2 page 4 of 9 6 D. π(e − ) 5 E. 25 4 Math 1A03/1ZA3 Fall 2013 Final Exam — Practice 14) Find lim x3 e−x 2 x→∞ A. 0 B. ∞ C. −∞ 15) Let f (x) = sin(arctan(x)). Find f (2). √ √ −2 5 2 5 A. B. 5 5 C. 3 2 D. π 2 D. 16) For what value of c is the function f continuous? 2 cx + x, x ≥ 2 f (x) = cx − 1, x < 2 A. −3 2 B. 3 2 C. 1 2 E. D. −1 2 2 3 −π 2 E. 1 E. 0 17) What function f has derivative (2 + h)3 − 8 ? h→0 h f ′ (2) = lim A. 2x B. (2 + h)x D. (x + 2)3 C. 2x + 4 18) Let f (x) = x2 e2x . Find f ′ (−2). A. 4 e4 B. 0 C. 12 e4 page 5 of 9 D. −4 e4 E. −8 e4 E. x3 Math 1A03/1ZA3 Fall 2013 Final Exam — Practice 19) Let f (x) = sin(πx) 1 . Find f ′ ( ). cos(−πx) 3 A. −π B. 4 3 C. 4π D. 4π 3 E. 4 20) A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder. 4πr3 B. √ 3 3 A. 4πr3 4πr3 3 C. D. 2πr2 E. 2πr2 3 21) Find the perimeter of a rectangle with area 20m2 whose perimeter is as small as possible. √ √ √ √ A 24 5 B 4 5 C 20 D 2 5 E 8 5 22) lim h→0 cos(ax) cos(ah + b) (cos(ax + b) + sin(ax) sin(ah + b)) − h h A cos(ax) cos b B −cos(ax+b)−sin(ax) sin b D −a sin(ax + b) 23) Find lim x→0 1 1 − 2 2 sin x x A 0 = ?. C cos(ax)−sin(ax) E −b cos(ax + b) . B 1 C 1/3 D 1−∞ E ∞ t 24) The derivative of the function g(t) := f S(u)du is equal to 1 t A f′ t S(u)du . B f ′ (S(t)). C f′ 1 t D f′ S(u)du S(t). 1 t S(u)du S ′ (u). E f′ 1 S(u)du S(t)S ′ (t). 1 page 6 of 9 Math 1A03/1ZA3 Fall 2013 Final Exam — Practice 25) The most general antiderivative of f ′ (g(h(x)))g ′ (h(x))h′ (x)dx is B f (g(h(x)))g ′ (h(x)) + C A f (g(h(x))) + C D f (g(h(x)))g(h(x))h′ (x) + C E f ′ (g(h(x))) + C e2x g(t)dt. Let h(x) := 26) Let f (x) = x2 A g(e) − g(0) C f (g(h(x)))g ′ (h(x)) + C d f (sin x). Then h(0) =? dx C g ′ (1) B 0 D g(0) E 2g(1) kπ xn cos(nx)dx is equal to: 27) For n, k integers greater than 1, 0 kπ A − 0 kπ xn D 0 nxn+1 sin(nx)dx n+1 sin(nx) dx n kπ B − kπ n sin(nx) cos(nx)dx C 0 0 xn+1 cos(nx)dx n+1 kπ E − xn−1 sin(nx)dx 0 π 28) Evaluate y ′ at the point (1, ) if y satisfies the following equation 2 x3 cos(y) + ex = 1 + e. A 29) Find 0 B 3+e √ C e x2 − 9 dx. x3 √ D e−2 √ x2 − 9 A sin ( x2 − 9) + C B sec (x/3) − +C x2 √ 1 1 x2 − 9 +C E √ D sec−1 (x/3) − +C 6 2x2 x2 − 9 −1 −1 page 7 of 9 E DNE C √ x2 − 9 +C 2x2 Math 1A03/1ZA3 Fall 2013 Final Exam — Practice 30) Find x2 − 5x + 16 dx. (2x + 1)(x − 2)2 1 ln |2x + 1| + C A ln |(2x + 1)(x − 2)2 | + C B 2 1 3 ln |2x + 1| − 2 ln |x − 2| + ln |x2 − 4x + 4| − 2(x − 2)−1 + C C 2 2 D 3 ln |2x + 1| − 2 ln |x − 2| + ln |(x − 2)2 | + C E 3 ln |2x + 1| − 2 ln |x − 2| + ln |x2 − 4x + 4| − (x − 2)−1 + C page 8 of 9 Math 1A03/1ZA3 Fall 2013 Final Exam — Practice A few useful formulae d 1 (loga x) = dx x ln a d 1 (arcsin x) = √ dx 1 − x2 1 d (arctan x) = dx 1 + x2 ax dx = 1 d (arccos x) = − √ dx 1 − x2 1 d (arccot x) = − dx 1 + x2 ax +C ln a sec x dx = ln | sec x + tan x | + C sec3 x dx = x2 csc x dx = − ln | csc x + cot x | + C 1 1 sec(x) tan(x) + ln | sec x + tan x | + C 2 2 1 x 1 +C dx = arctan 2 +a a a √ a2 x 1 +C dx = arcsin 2 a −x ANSWER KEY 1) B, 2) C, 3) B, 4) B, 5) C, 6) C, 7) D, 8) B, 9) E, 10) C 11) E, 12) D, 13) D, 14) A, 15) B, 16) A, 17) E, 18) A, 19) C, 20) B 21) E, 22) D, 23) C, 24) C, 25) A, 26) E, 27) E 28) C, 29) D, 30) C page 9 of 9 ...
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