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QUESTION 5 The equation of the tangent to the curve f(t) =te` att=11is (A) y=et (B) y=2et—e ©) y=et—e® +1 (D) y=2et—2¢*+1
QUESTION 11 (3 marks) Determine the derivative of each of the following with respect to x. 1 D 7 b) y=x`xe* Express your answer in factorised form. [1 mark] [2 marks]
QUESTION 17 (6 marks) The volume of water in a tank is represented by a function of the form Vt)= Aekt where V` is in litres and ¢ 1s in minutes. Initially the volume is 100 litres and it is decreasing by 50 litres per minute. : : : : : 50 .. : Determine the time at which the volume is decreasing at the rate of 3 litres per minute. Express your answer in the form In(a).
QUESTION 1 The limit of 12 -1 as h approaches 0 is closest to A) 0.0 ! (B) 1.0 C) 2.5 (D) 3.0
QUESTION 9 The displacement of a particle (in metres) at time ¢ (in seconds) is represented by the function s@)=tln@t)—t 0<t<4 Determine the approximate acceleration of the particle at time ¢ = 3. (A) 0.66ms > (B) 0.33ms> (C) -033ms™ (D) —0.66m s
QUESTION 11 (5 marks) Determine the derivative with respect to x of the following functions. a) y= (e“` + l)3 b y= &g\) (Give your answer in simplest form.) X [2 marks] [3 marks]
QUESTION 14 (4 marks) Consider the function f(x)=In(3x+4) for x > _?4 a) Determine f`(x). [1 mark] b) Determine the x-intercept of the graph of f/(x). [2 marks] c) Determine the gradient of the tangent to the graph of f(x) at the x-intercept. [1 mark]
QUESTION 2 A substance is being heated such that its temperature 7 in °C after # minutes is given by the function 7` = 2 The first integer value of 7 for which the instantaneous rate of change of temperature is greater than 100 °C per minute is (A) t=10 (B) =9 C) t=8 (D) t=7
QUESTION 8 The displacement (in metres) of a particle is given by s(‘t) =-3cos(7)+2sin (t) where 7 1s in seconds. : : : T : The instantaneous velocity of the particle at time 7 = . seconds is et (A) 3ms! (B) 2ms”! (C) 2ms”! (D) 3ms`
QUESTION 10 An object travels in a straight line so that its velocity at time # seconds is given by v(#)=2¢+sin(2¢). Determine the expression for acceleration as a function of time. (A) a(t)=2+2cos(2r) (B) a(t):2—%cos(2f) (©) a(t)=1 +2cos(2¢) (D) a(t)ztz— cos(27) b | —
QUESTION 17 (4 marks) Rabbits and foxes are among two species of mammals that live on an isolated island. Rabbits represent a significant food source for the foxes. The populations of rabbits and foxes were monitored each month for two years. The graph shows the population of foxes (in thousands) and the population of rabbits (in thousands) at any time ¢ (in months) over the two years. The two populations can be modelled using trigonometric functions. (0 14.5) (12 14.5) - o NN (15 11) Population (thousands) 0 3 10 15 Time (¢) in months since 1 January 20 Foxes ------- Rabbits Jane believes that there were periods of time over the two years when the total population of foxes and rabbits on the 1sland exceeded 25 000. Evaluate the reasonableness of Jane`s claim.
QUESTION 18 (3 marks) The number of animals in a population (in thousands) is modelled by the function P such that . 100 P(1)= 1+4e Determine the number of animals in the population when the population is growing the fastest. — where 7 18 in years.
QUESTION 13 (9 marks) a) Determine the derivative of f`(x)= 32 [1 mark] : ln(x) : : b) Given that g(x)=——= determine the simplest value of g`(e). [3 marks] X ¢) Determine the second derivative of 4(x) = xsin(x). (Give your answer in simplest form.) /5 marks]
QUESTION 15 (4 marks) The derivative of a function 1s given by f `(x) =e* (x — 4). Determine the interval on which the graph of f (x) is both decreasing and concave up.
QUESTION 1 The position (in cm) of a particle is given by x = cos(4t) where ¢ 1s time (in seconds). The velocity of the particle when ¢ =5 is (A) 1.6323 cms ! (B) 0.4081 cm s (C) —0.9129 cms ™ (D) —3.6518cms !
QUESTION 2 Identify the correct features of the function f(x)= x (A) f`(-1)=0 f (-1)<0 B f(=1)=0/(-1)> ©) f`(-1)<0 f (-1)<0 (-1)<0 ./ (-1) (D) f 1)>0 1 1)> 1)<0
QUESTION 13 (4 marks) A sandy beach has a fence on one side and ocean on the other. The width of the beach is the distance (in metres) from the fence to the water`s edge. The width w(t) at a certain point is given by w(t) =a +bsin(£t —zj 0<r<24 6 3 where ¢ is time (in hours) since 6 am. The width of the beach is 8 metres at 8 am and 3 metres at 5 pm. a) Determine a and b. [2 marks] b) Determine the rate of change of the width of the beach at 8 am and the first time after this when this rate of change is repeated. [2 marks]
QUESTION 3 If y = cos?(3x) then Z—i` = (A) —65sin(3x) cos(3x) (B) —2sin(3x) cos(3x) (C) 2 sin(3x) cos(3x) (D) 6 sin(3x) cos(3x)
QUESTION 5 The slope of the tangent to the graph of y = In(x?) atx = e? is 1 (A) 2 2 (B) 2 4 ©) i 1 (D) —
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 19 (6 marks) Determine the equation of the tangent to the curve y = x In(x) that passes through the point (0 —e).
QUESTION 2 A population of bacteria after t hours is given by P(t) = 5000e°%18¢ The rate of increase of the population (to the nearest unit) at 15 minutes is (A) 74 399 bacteria/hour. (B) 13 392 bacteria/hour. (C) 5230 bacteria/hour. (D) 941 bacteria/hour.
QUESTION 5 Two particles move along the x-axis. For 0 < t < 5 their respective position functions are given by: x ; = sin(2t) and x = e~ For how many values of t do the particles have the same position? A) 4 B) 3 €) 1 (D) 0
QUESTION 8 A particle travels along a straight line with a velocity of v(t) = 3e(_7t) X sin(2t) ms™. Determine the acceleration of the particle at t = 2 seconds. (A) —1.03ms™? (B) —0.01 ms™? (C) 2.49ms? (D) 3.51ms~?
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 18 (5 marks) The amount of a certain drug in the bloodstream M (mg) at any time ¢t (hours) is modelled closely by M(t) = Ate™t where A and b are parameters. a) Determine the exact values of A and b if the maximum amount of the drug in the [3 marks] bloodstream was 120 mg at t = 2 hours. [2 marks] b) Evaluate the reasonableness of your solution.
QUESTION 1 el (*) is equivalent to (A) 0 (B) 1 (C) x (D) —
QUESTION 2 If f(x)=e°** determine the value of / (2). (A) e? (B) 2e (C) -e? (D) —2e?
QUESTION 6 qa —1 Substitutions for / are used to estimate the limit of as h — 0. Which sequence is the most appropriate? (A) —4 -2 -1 -0.5 -0.25 -0.125 ... (B) -0.05 -0.1 -0.2 -0.4 -0.8 ... (C) 2 1 0 -1 -2 -3... (D) 1 2 3 4 5 6...
QUESTION 16 (5 marks) Solve for x in the equation 4+ 7e?* = 3e”.
QUESTION 18 (4 marks) A person enters the lowest carriage of a miniature Ferris wheel with a six-metre diameter. The bottom carriage is one metre off the ground. When top speed is reached it takes three seconds for a carriage to travel from the lowest to the highest point of the ride. It is claimed that: The vertical motion of the Ferris wheel produces a maximum vertical acceleration on each rider that is more than half the acceleration of free fall. Free fall occurs when gravity is the only force acting resulting in an acceleration of 9.8 ms”. Evaluate the reasonableness of the claim.
QUESTION 19 (7 marks) Jaxon and Shari each own a shop and have recorded the number of customers entering their shop on two consecutive days. The number of daily customers for each shop can be modelled by the equation y = A In( Bx) where x is the day and y is the number of customers. The constants A and B are different for each shop. Determine algebraically whether the total number of customers for Jaxon and Shari’s shops will be the same every day in the future.
QUESTION 1 If f(x) = sin(3x) determine the value of f {Z). (A) 2.772 (B) 1.148 (C) 0.929 (D) 0.383
QUESTION 4 The displacement (m) of a moving particle is given by d = e°°! _1 where ¢ is time (s). The acceleration (ms~”) of the particle when t= 4 is (A) 7.3891 (B) 6.3891 (C) 3.6945 (D) 1.8473
QUESTION 6 Sa | ; | dx a#0is a x+a (A) 1.7918 (B) 1.6094 (C) 1.3863 (D) 1.0986
QUESTION 8 The number of koalas in a conservation park is modelled by N= 15 In(7t+ 1) t = 1 where ¢ represents the time (years) since the park opened. There were 20 koalas in the park when it opened. Determine the approximate rate of change in the number of koalas when f= 3. (A) 46 (B) 26 (C) 25 (D) 5
QUESTION 9 If f(x) = e?*(x+1) and f(x) = ae**(x + 1) determine the expression for a. (A) 3x45 (B) 3x+3 (C) S5x+5 (D) 5x+3
QUESTION 15 (4 marks) Determine the derivative of f (x) =Inx? + In(x —35 y. Express the derivative as a single fraction in its simplest and factorised form.
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