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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 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 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 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 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 19 (6 marks) Determine the equation of the tangent to the curve y = x In(x) that passes through the point (0 —e).
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 2 If f(x)=e°** determine the value of / (2). (A) e? (B) 2e (C) -e? (D) —2e?
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 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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