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QUESTION 1 The graphs of f(x) = ¢* and g(x) =x~— 1 are shown. The area of the shaded section bounded by these graphs between the linesx=0and x =1 is A) 1-e (B) e-2 5 ©) B 1 (D) e=3
QUESTION 2 X i e +1 Determine I e (A) (B) ©) (D) x—e “+c x+te¥+c l+xe*+c x+xe*+c
QUESTION 3 Determine 2 j (4x+ 6)3 dx (A) 16(4x+6)* +¢ (B) 8(4x+6)*+c (4x +6)* 2 (4x + 6)* 8 ©) +C T~ (D)
QUESTION 9 Determine I X—H dx x2 +2x (A) ln[2x+2j+c (B) In(2x+2)+c (C) %ln(xz +2x)+¢ (D) 2In(x? +2x)+c
QUESTION 12 (5 marks) An object is moving in a straight line from a fixed point. The object is at the origin initially. The acceleration a (in m s ™) of the object is given by a(f) =m cos(mt) t=>0 where ¢ is time in seconds. The velocity at =115 0.5 m s ! a) Determine the initial acceleration. [1 mark] b) Determine the initial velocity. [2 marks] c) Determine the displacement after one second. [2 marks]
QUESTION 20 (6 marks) At the end of the first stage of its growth cycle a species of tree has a height of 5 metres and a trunk radius of 15 cm. In the second stage of its growth cycle the tree stays at this height for the next 10 years. However the growth rate of the trunk radius (in cm per year) varies over the 10 years and is given by the function 1 r(t)=1.5+ sm(?j Assume the density (mass per unit volume) of the tree trunk is approximately 1 g/(:m3 and the tree trunk is in the shape of a cylinder. Determine the ratio of the trunk`s mass at the end of the second stage to its mass at the end of the first stage.
QUESTION 3 Let R be the region enclosed by the graph of y = xe” the x-axis and the lines x =—1 and x = 1. The area of R is closest to A) 0.74 (B) 1.26 C) 2.35 (D) 3.09
QUESTION 5 An object moves in a straight line with a velocity v given by v(t)=40(e ` - g )ms ` where £ >0 The object is at the origin initially. The displacement—time graph in the first 6 seconds is (A) ------ ©)
QUESTION 6 Oil is leaking from a tanker at the rate of »(¢) = 9000¢ % litres per hour where ¢ is in hours. Determine how much oil leaks from the tanker (to the nearest litre) from time # = 0 to time ¢ = 10. (A) 38910 litres (B) 8756 litres (C) 7782 litres (D) 1556 litres
QUESTION 12 (7 marks) The rates of change in population for two cities are given by 45 City A: A`() = — y ® t+1 City B: B`(¢) = 105¢ %% where ¢ is the number of years since 2018 and both 4`(f) and B`(f) are measured in people per year. At the beginning of 2018 City A had a population of 5000 and City B had a population of 3500. a) Determine the population models for both cities. [3 marks] b) Use the information in 12a) to predict the population of City B at the beginning of 2028. /I mark] c) Use the information in 12a) to predict the year in which the population of both cities will be the same. [3 marks]
QUESTION 15 (3 marks) A field is divided into five sections as shown. The width of each section is 1 metre. The perpendicular height in metres of each section is given in the diagram. The area of the field was approximated using the trapezoidal rule and found to be 11.12 m` Not drawn to scale a) Determine the height marked x on the diagram. [2 marks] b) Determine the area of the field given the shape of the field is modelled by the function X L 0<x<5 x? +1 [1 mark] J(x)=
QUESTION 19 (7 marks) Consider the following information when completing this question. 2 The length of a curve y = f(x) over the interval [a b] = Ib 1+ (%j dx a \f 67 You are driving along a road with a vertical distance above sea level D (in metres) given by the function D) =300 + In(x> -3x + ¢) where x 1s the horizontal distance from an initial point of measurement (in kilometres) at sea level. Assume that if x 1s positive you are east of the initial point of measurement and if x is negative you are west of the 1nitial point of measurement. You start your drive along the road at a horizontal distance of 10 kilometres west of the initial point of measurement and drive until you are at a horizontal distance of 10 kilometres east of the initial point. Determine the time you spend driving downhill if you drive downhill at an average speed of 40 km/h.
QUESTION 3 Determine JlOe‘“dx` ] 0643{‘4`] 4x +1 (A) +c (B) 40e* +c (C) %e“-“ +c (D) 2™ +¢
QUESTION 7 Determine J (2x+3)dx | (A) 2 (B) 4 C) 14 (D) 16
QUESTION 13 (5 marks) Consider the functions f(x)=x”and g(x)= 4x. a) Determine the x-coordinates of the points of intersection of the graphs of the two functions. [2 marks] b) Use the results from Question 13a) to calculate the area enclosed by the graphs of f(‘c‘) and g(‘c) [3 marks]
QUESTION 16 (4 marks) A tangent is drawn at the point (1 e) on the graph of the function y = ¢` as shown. Not to scale Determine the area of the shaded triangle.
QUESTION 18 (4 marks) The graph of y = f(x) where f(x) is the quadratic function f(x)= ax” + bx +4 is shown. Two regions of the area between the graph of y = /(x) and the x-axis are shaded. Not to scale : .2 43 . 5 Region P has an area of %umts“ and Region Q has an area of ?) units”. Determine the values of a and b.
QUESTION 4 3 Using the trapezoidal rule with an interval size of 1 the approximate value of the integral J 0.5%dx is 0 A) 1.25 (B) 1.26 C) 131 (D) 1.88
QUESTION 7 : 11 : Determine f(x) given f`(x)= 6x% +—+— and f (1)=5. ~ 27 (A) f(\) =2x + %-4— ln(_r) -1 X B) f(x)=2x - L (x)+4 X ©) f(x) =2 —l+i}+2 X x- “ 3 2 D) f(x)=2x+=+5-2 X X~
QUESTION 11 (5 marks) Consider the function f(x)=¢ sin(x) 0<x<27 a) State the exact values of the x-intercepts of the graph of f`(x). [2 marks] b) Write an expression for the area enclosed between the graph of f (r) and the x-axis. [2 marks] ¢) Determine the area enclosed between the graph of f(x) and the x-axis to the nearest square unit. [1 mark]
QUESTION 12 (4 marks) The velocity function of an object in m s is given by v(t) = cos[6r + %] +2 0<1<S. Initially the object is at the origin. a) Determine the displacement function. [2 marks] b) What is the displacement of the object from the origin in metres (m) after three seconds? [2 marks]
QUESTION 3 The area between the curve y =9 — x” and the x-axis is (A) 12 units® (B) 18 units® (C) 36 units? (D) 54 units?
QUESTION 9 The approximate area under the curve f (x) =+/2x +1 between x = 0 and x = 4 using the trapezoidal rule with four strips is (A) 2+-B+5+47 B) 2+2(V3+5+7) () 4+2(\B+£+\fi) (D) 4+3+5+47
QUESTION 14 (6 marks) The rate that water fills an empty vessel is given by L 0.25¢ * (in litres per hour) 0 <t <8In(6) where ¢ is time (in hours). dt a) Determine the function that represents the volume of water in the vessel (in litres). [2 marks] The vessel is full when 7 = 81n(6). b) Determine the volume of water to the nearest litre the vessel can hold when full. [2 marks] The table shows the approximate rate the water flows into the vessel at certain times. c) Use information from the table and the trapezoidal rule to determine the approximate volume of water in the vessel after three hours. [2 marks]
QUESTION 17 (4 marks) . . b b— Determine the value of b given I 3x%dx =117 and I l3x2a`x =56 forb>1. a a
QUESTION 18 (4 marks) A percentile 1s a measure in statistics showing the value below which a given percentage of observations occur. The continuous random variable X has the probability density function 2x—2 1<x<2 f(x)={ 0 otherwise Determine the 36th percentile of X.
QUESTION 7 A marble moves in one direction in a straight line with velocity v = 21n(t + 1) (in metres per second) where 7 is time (in seconds) since the marble passed through the origin. Determine the distance from the origin the marble has rolled after 4 seconds. (A) 040 m (B) 1.60m (C) 322m (D) 8.09 m
QUESTION 17 (4 marks) A snail is travelling along a straight path from point 4. The snail`s velocity (cm min ) is modelled by v(t)= 1.4ln(1 e ) where ¢ is time (in minutes) for 0 <z <15. An ant passes point 4 12 minutes after the snail and follows the snail`s path. The ant moves with a constant acceleration of 2 cm min~2 and passes the snail at # =15 minutes. Determine the ant`s velocity at point A4.
QUESTION 19 (4 marks) Flying foxes enter and leave a fruit-growing region every evening. The rate at which the flying foxes enter the region is modelled by the function A(t)= 42sin[0.03t —%) +71 0<1<240 The rate at which the flying foxes leave the region is modelled by the function L(t)= 42sin(0.04t —gj +42 0<1<240 Both A(¢) and L(¢) are measured in animals per minute and ¢ is measured in minutes after 7 pm. There are 100 flying foxes in the region at 7 pm. Determine the maximum number of flying foxes in the region and the time that this occurs.
QUESTION 2 The approximate area under the curve y = 2x? between x = 1 and x = 4 is found using rectangles (of width equal to one unit) as shown in the diagram. -1 0 1 2 3 4 5 What is the approximate area found using these rectangles? (A) 58 units` (B) 45 units` (C) 42 units® (D) 28 units?
QUESTION 4 0 3 ; . i If [ f(x)dx = 4 and [ f(x)dx = —10 which of the following is true? (A [3 fC)dx = —14 B) [° fe)dx =6 © [P fx)dx=6 D) [ f(x)dx =14
QUESTION 6 7 dx Calculate f_ 1 TG0 (A 1 4 B) 3 € 2 (D) 4
QUESTION 11 (5 marks) Determine: 4 3In(5x) 2) <= 3In(5Y) d b) — (x3cos(x* + 1)) dx c) [3e*dx [1 mark] (You do not need to simplify.) /3 marks] [1 mark]
QUESTION 14 (5 marks) 1 Letv(t) = = + 3sin(t) t = 0 represent the velocity of an object moving in a straight line. T Att = 3` the position of the object is 4. a) Determine the acceleration function. [2 marks] b) Determine the displacement function. [3 marks]
QUESTION 3 Using the trapezoidal rule with four sections the approximate area under the curve y = x? + 5 between x = 0 and x = 2 is (A) 12.67 units® (B) 12.75 units® (C) 13.00 units? (D) 13.75 units?
QUESTION 4 The area of the region enclosed by the graphs of y = xvVx + 1 and y = 2x 1s (A) 1.267 units® (B) 0.467 units® (C) 0.200 units` (D) 0.029 units®
QUESTION 13 (6 marks) 1 11643t [2 marks] —3 Let N(t) = [ 1800t(16 + 3t2)=2 dt a) Determine Z—A: it M(t) = b) Using the result from 13a) determine N (t) [1 mark] ¢) Determine N`(4) [1 mark] The function N (t) models the total number of customers N served by staff after ¢t hours during an 8-hour workday (0 <t < 8). At time t = 0 no customers had been served. d) Determine N(4) [2 marks]
QUESTION 17 (3 marks) During one 30-day period the rate at which pollution passes into a nearby lake is measured every six days and results are given in the table below. Rate of pollution in units per day p(t) This information can be modelled as a quadratic function. Determine the total amount of pollution entering the lake during this 30-day period.
QUESTION 20 (5 marks) A farmer has a paddock with straight fences on two sides (fence A and fence B) perpendicular to each other and bounded by a creek on the other side. Fence B is 4 kilometres long. The creek boundary can be modelled using d=In(5x+¢e)—1 where d is the perpendicular distance in kilometres from fence A to the creek and x is the distance in kilometres along fence A from the point O. The farmer wants to divide his paddock area in half with a straight fence parallel to fence B. Not drawn to scale creek Fence B O Fence A Determine where the farmer should locate the fence.
QUESTION 4 20 ; If the gradient of the function f(x) is given by — then f(x) is equal to x? 60 (A) —7+e xX (B) 2 +¢ Xx 10 (C) -y te X (D) he x
QUESTION 5 Determine } aI 1 2x (A) lin 2 (B) tins 2 1 1 D) =In3 (D) In
QUESTION 7 Determine the mean of the continuous random variable X with the probability density function 1 fO)=48 0 otherwise O<x<4 (A) (B) (C) (D) WloO wplRe WlWw CwleH
QUESTION 9 Determine fj zsin Exax (A) 3 (B) 6 (Cc) -3 (D) -6
QUESTION 10 The continuous random variable Y has the probability density function l+y 0<y<¥ 3-1 f(y) = 0 otherwise ] Determine P(O< y< 5 (ay 1 5 (iB) 3 8 (Cc) > 8 3 D) = (D) 1
QUESTION 12 (4 marks) The region bounded by the x-axis and the curve of f(x) =x(4—x) represents the plan of a garden bed. All measurements are in metres. f(x) a) Estimate the area of the garden bed using sums of the form > f (x; )Ox; where x = 1 Xy = 2 x3=3 and x 4=4. [1 mark] b) Use a definite integral to determine the area of the garden bed. [3 marks]
QUESTION 14 (6 marks) The rate of change in the number of bacteria in a science experiment is represented by < =e 420 where ¢ represents the time (hours) since starting the experiment and P represents the number of bacteria present (thousands). Initially there are 60 000 bacteria present i.e. P(O) = 60. a) Determine the equation for P(t). [2 marks] b) Determine the change in the number of bacteria during the third hour. Express your answer in terms of e. [2 marks] c) Determine how long it will take for the number of bacteria present to double after starting the experiment. [2 marks]
QUESTION 6 Sa | ; | dx a#0is a x+a (A) 1.7918 (B) 1.6094 (C) 1.3863 (D) 1.0986
QUESTION 12 (4 marks) The graph shows the water level under a bridge over a 12-hour period. a) b) Water level (metres) 12:00 am 4:00 am 8:00 am 12:00 pm Time Determine the equation of the cosine function that models the water level as a function of time after 12:00 am. [1 mark] How long in the 12-hour period shown is the rate of change of water level more than 0.55 metres per hour? [3 marks]
QUESTION 13 (4 marks) The curved lines represent graphs of the equations y = x7 —4x+8 and y= 10cos(x +10). y a) Determine the coordinates of the points of intersection A and B. [1 mark] b) State an integral expression representing the area enclosed by the two graphs. [2 marks] c) Determine the area enclosed by the two graphs. [1 mark]
QUESTION 16 (6 marks) A particle is moving in a straight line. The velocity (ms!) of the particle is given by 20sin (2t v(t) =e) 6 —5cos(2r) The initial position of the particle is +6.0 m from the origin. t > 0 where ¢ is time (s) after moving from its initial position. a) Use calculus methods to determine an equation for the position of the particle from the origin at any time ¢. b) Determine the position of the particle relative to the origin when it first reaches maximum velocity. [3 marks] [3 marks]
QUESTION 18 (5 marks) A company makes windows using glass that has a mass of 5.6 kg per square metre. A customer orders an unusual window in a partial parabolic shape as shown. Not to scale 8m Determine the mass of the window.
QUESTION 19 (6 marks) Over a suitable domain a hill has a cross-sectional area given by fr (x) dx = a +c where: ¢ a b and c are constants b+0 ¢ h(x) represents vertical distance (m) x represents horizontal distance (m). It is known that 4(0) = 1.22 and h(40) = 25. Where the gradient of the hill is 0.86 there is a tree stump. A second tree stump is located further up the hill. The difference in hill gradient between the two tree stumps is 0.44. A surveyor predicts that the vertical distance separating the two tree stumps is between 7.5 m and 8.5 m. Evaluate the reasonableness of this prediction.
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