Functions and Their Applications MCD2130 Sample Examination 2 Question and Answer Booklet READING TIME: 10 minutes WRITING TIME: 3 hours Structure of Examination Section Number of Questions to be answered Total Possible Marks Actual Marks % Section A ALL (20) 80 Section B ALL (4) 50 Total Marks 130 During an examination, you must not have in your possession a book, notes, paper, calculator, pencil case, mobile phone, electronic devices, smart watch, electronic pens or other material/item which has not been authorised for the exam or specifically permitted as noted below. Any material on desk, chair or person will be deemed to be in your possession. You are reminded that possession of unauthorised materials in an examination is a discipline offence under Monash College regulations. AUTHORISED MATERIALS ELECTRONIC DEVICES/CALCULATORS NO OPEN BOOKS NO SPECIFICALLY PERMITTED ITEMS NO INSTRUCTIONS TO CANDIDATES (1) A formulae sheet is included (pages 38-39). (2) Answer all questions in the space provided in this Question Booklet. (3) Write clearly. Marks cannot be awarded if your handwriting cannot be read. Candidates must complete this section STUDENT ID DESK NUMBER THIS ENTIRE EXAMINATION PAPER MUST BE HANDED IN AT THE END OF THE EXAMINATION. DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO. Examiner Use Only Section A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mark 15 16 17 18 19 20 Section B 1 2 3 4 Mark MCD2130 SECTION A Section A (1) Find the slope-intercept form, y = mx+ b, of the line which passes through the point (x, y) = ( 1, 5) and is perpendicular to the line y = 5 2x. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 3 of 39 MCD2130 SECTION A (2) Algebraically find all values of x which satisfy the inequality 4 x ≤ 3 x 2 . Write your answer in interval notation. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 4 of 39 MCD2130 SECTION A (3) Find the domain and range of the function f(x) = 5√ 9 x2 . Write your answers in interval notation. Support your answer with clear explanations or working. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 5 of 39 MCD2130 SECTION A (4) Consider the functions f(x) = √ x 1 and g(x) = x2 + 2. Find the composition function g f(x) and state the domain of g f . Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 6 of 39 MCD2130 SECTION A (5) Consider the functions f(x) = 3×2 12x+ 6. By applying the method of “completing the square” write f in the form f(x) = a (x b)2 +B and then determine the Cartesian coordinates for the turning point. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 7 of 39 MCD2130 SECTION A (6) Consider the cubic function f(x) = 3×3 + 4×2 13x+ 6 (a) Show that x = 1 is a zero of f(x). (b) Completely factorise the cubic function f . Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 8 of 39 MCD2130 SECTION A (7) Consider the rational function f(x) = ( 9 x2) (x+ 5) x2 4x+ 3 . (a) Identify the Cartesian coordinates of any holes of y = f(x). (b) Identify and characterise any asymptotes of y = f(x). Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 9 of 39 MCD2130 SECTION A (8) (a) On the first grid provided, sketch the graph of y = 1 x . All asymptotes, holes, intercepts and axes intercepts should be easily determined from your sketch or clearly labelled. Furthermore, clearly label the point (x, y) = (1, 1). (b) Two transformations in the x-direction are applied to y = 1 x which result in the graph y = 1 2x+ 4 . List, in words, the two transformations in the correct order. (c) On the second grid provided, sketch the graph of y = 1 2x+ 4 . All asymptotes, holes, intercepts and axes intercepts should be easily determined from your sketch or clearly labelled. Furthermore, clearly label the point where (x, y) = (1, 1) on y = 1 x ends up on y = 1 2x+ 4 after the two transformations. Grid for question 8(a) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 10 of 39 MCD2130 SECTION A Grid for question 8(c) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 11 of 39 MCD2130 SECTION A Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 12 of 39 MCD2130 SECTION A (9) Find the exact solution(s) for the equation loge(x) + loge(x 1) = 2 loge (√ 2 ) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 13 of 39 MCD2130 SECTION A (10) Solve for x in the following equation 8x 2 (4x) 8 (2x) = 0 Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 14 of 39 MCD2130 SECTION A (11) Find all solutions for x ∈ [0, 2pi) of the trigonometric equation: 1 cos(3x) = cos(3x) . Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 15 of 39 MCD2130 SECTION A (12) Prove the following identity sin(x) cos2(x) sin(x) = sin3(x) . Make sure you show all working and give clear explanations. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 16 of 39 MCD2130 SECTION A (13) If sin(x) = √ 5 3 for pi 2 < x < pi then determine the values for cos(x) and tan(x). Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 17 of 39 MCD2130 SECTION A (14) Evaluate the limit lim x → 3 ( x2 + x 6 x+ 3 ) . Be sure to apply appropriate mathematical rigour to the limit(s). Simply writing numbers without clear explanation will result in zero marks. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 18 of 39 MCD2130 SECTION A (15) Use first principles to find the first derivative of the function: f(x) = 1√ x . Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 19 of 39 MCD2130 SECTION A (16) Find the first derivative of the function f(x) = e x 2 sin(2 3x) . Do not use first principles. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 20 of 39 MCD2130 SECTION A (17) Given that 3 x = eloge(3 x) show that d dx ( 3 x ) = loge(3) 3 x Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 21 of 39 MCD2130 SECTION A (18) Find the indefinite integral∫ ( e2x + 3√ 1 5x pi cos ( 2pix+ pi 4 )) dx Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 22 of 39 MCD2130 SECTION A (19) Given that d dx ( x2 loge(x) ) = 2x loge(x) + x find the indefinite integral ∫ ( x loge(x) ) dx. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 23 of 39 MCD2130 SECTION A (20) Using an appropriate substitution evaluate the definite integral∫ 1 0 ( 5x√ 4 x2 ) dx. (Note: A mark will be deducted if the derivative is treated like a fraction.) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 24 of 39 MCD2130 SECTION B Section B (1) Consider two lines in the xy-plane. (a) Find the equation of the straight line, L1(x) = ax + b, which passes through the points (x, y) = (0, 1) and (x, y) = (32 , 0). Show all working. (b) Find the equation of the straight line, L2(x) = cx + d, which passes through the points (x, y) = ( 1, 9) and (x, y) = (1, 4). Show all working. (c) Explain why the two lines y = L1(x) and y = L2(x) are not parallel. (d) Are the two lines y = L1(x) and y = L2(x) perpendicular Explain your answer. (e) Find the Cartesian coordinates for the point of intersection, (x, y) = (X,Y ), between the two lines y = L1(x) and y = L2(x). (f) Sketch the two lines on the grid provided. Clearly label each line. The point of intersection and any axes intercepts should be easily determined from your sketch. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 25 of 39 MCD2130 SECTION B Grid for Section B question 1(f) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 26 of 39 MCD2130 SECTION B (2) Consider the function f(t) = 100 1 + 2 t for t ≥ 0 (a) Find any t- and y-intercepts. (b) Find the value of the limit lim t →∞(f(t)) . Therefore, what is the equation for horizontal asymptotes for y = f(t) (c) State the domain and range for f . (d) Explain why the inverse function f 1 exists and then determine the domain and range of f 1. (e) Find the equation of the inverse function f 1. Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 27 of 39 MCD2130 SECTION B Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 28 of 39 MCD2130 SECTION B Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 29 of 39 MCD2130 SECTION B (3) Beginning with the function f (x) = sin(x) sketch the graph for each of the transformations indicated, show at least one period. Furthermore, for each graph indicate clearly where the point (x, y) = ( pi 2 , 1 ) is mapped to under each transformation. (a) y = sin(x) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 30 of 39 MCD2130 SECTION B (b) y = sin(x) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 31 of 39 MCD2130 SECTION B (c) y = sin(3x) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 32 of 39 MCD2130 SECTION B (d) y = sin(3x pi) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 33 of 39 MCD2130 SECTION B (e) y = sin(3x pi) 12 Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 34 of 39 MCD2130 SECTION B (4) Consider the two functions f(x) = 5 + 4x x2 and g(x) = x+ 5. (a) Find the x- and y-intercepts for y = f(x). (b) Find the Cartesian coordinates of the turning point of the graph of y = g(x). (c) Find the Cartesian coordinates of the two points where the graphs of y = f(x) and y = g(x) intersect. (d) Sketch the graphs of y = f(x) and y = g(x) on the grid provided. Shade the region(s) bounded by the graphs of y = f(x) and y = g(x). (e) Using definite integrals, find the area of the shaded region(s) bounded by the graphs of y = f(x) and y = g(x). Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 35 of 39 MCD2130 SECTION B Grid for Section B question 4(d) Sample Examination 2 QUESTIONS CONTINUE NEXT PAGE Page: 36 of 39 MCD2130 SECTION B Sample Examination 2 END OF QUESTIONS Page: 37 of 39 MCD2130 FORMULAE SHEET (Page 1 of 2) The quadratic formula: The quadratic equation ax2 + bx+ c = 0 has solutions x = b±√b2 4ac 2a General Transformation Form: Given y = f(x) can be graphed then the graph of y = ±Af ( ±1 k (x b) ) +B can be determined using reflections, dilations/contractions and translations. Absolute value: The absolute value function |x| = { x if x ≥ 0 x if x < 0 Rules of Exponents: a b = 1 ab a1/n = n √ a a0 = 1, provided a 6= 0 cacb = ca+b cab = (ca)b (ab)c = acbc ca ÷ cb = c a cb = ca b ca/b = ( b √ c) a = b √ ca Rules of Logarithms: ln(x) = loge(x) y = logb(x) is equivalent to b y = x logb(b) = 1 logb(1) = 0 logb(xy) = logb(x) + logb(y) logb ( x y ) = logb(x) logb(y) logb ( 1 x ) = logb(x) logb(xa) = a logb(x) blogb(x) = x Trigonometric identities: sin(x) = opp hyp cos(x) = adj hyp tan(x) = sin(x) cos (x) = opp adj cos2(x) + sin2(x) = 1 sin(x) and tan(x) are odd functions cos(x) is an even function sin(α± β) = sin(α) cos(β)± cos(α) sin(β) cos(α± β) = cos(α) cos(β) sin(β) sin(α) Exact trigonometric values: θ 0 pi 6 pi 4 pi 3 pi 2 pi sin(θ) 0 1 2 1√ 2 √ 3 2 1 0 cos(θ) 1 √ 3 2 1√ 2 1 2 0 1 tan(θ) 0 1√ 3 1 √ 3 undefined 0 Page: 38 of 39 MCD2130 FORMULAE SHEET (Page 2 of 2) Limits: lim x →c(k) = k for constant k lim x →c(x) = c lim x →c ( xk ) = ( lim x →c(x) )k for constant k lim x →c(f(x)± g(x)) = limx →c(f(x))± limx →c(g(x)) lim x →c(f(x) g(x)) = ( lim x →c(f(x)) )( lim x →c(g(x)) ) lim x →c ( f(x) g(x) ) = lim x →c(f(x)) lim x →c(g(x)) provided lim x →c(g(x)) 6= 0 If lim x →c(g(x)) = L and f is continuous at L then limx →c(f(g(x))) = f(L) Derivatives: f ′(x) = lim h →0 ( f (x+ h) f(x) h ) equivalently df dx = lim x →0 ( f (x+ x) f(x) x ) d dx ( f(x) g(x) ) = df dx g(x) + f(x) dg dx d dx ( f(x) g(x) ) = df dx g(x) f(x) dg dx( g(x) )2 d dx ( f g(x) ) = d dx ( f ( g(x) )) = df dg dg dx Graph Sketching: The function f (1) has a critical point at x = c ∈ dom(f) if either f ′(c) = 0 or is undefined. (2) has a turning point at x = c ∈ dom(f) if it is a local maximum or minimum. (3) has an inflection point at x = c if f ′′(x) changes sign at the point. (4) is increasing on an interval I if f ′(x) > 0 for all x ∈ I (5) is decreasing on an interval I if f ′(x) < 0 for all x ∈ I (6) is concave down on an interval I if f ′′(x) < 0 for all x ∈ I (7) is concave up on an interval I if f ′′(x) > 0 for all x ∈ I Integration: Indefinite integral: ∫ ( f(x) ) dx = F (x) + C (anti-derivative), such that dF dx = f(x). Fundamental theorem of calculus: If f(x) is continuous on [a, b] and F (x) is any anti- derivative of f(x), then ∫ b a ( f(x) ) dx = F (b) F (a). End of Sample Examination 2 Page: 39 of 39
