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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 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 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 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 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 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 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 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 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 1 If f(x) = sin(3x) determine the value of f {Z). (A) 2.772 (B) 1.148 (C) 0.929 (D) 0.383
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