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QUESTION 13 (7 marks) A function is defined as f(x) = x(In(x))? x > 0. The graph of the function is shown and has a local maximum at point 4 and a global minimum at point B. The derivative of the function is given by f`(x) = 2 In(x) + (In(x))? x > 0. a) Verify that there 1s a stationary point at x = 1. [2 marks] b) Determine the coordinates of A. [3 marks] The graph of the function has a point of inflection at x = ¢` c) Determine p. [2 marks]
QUESTION 16 (4 marks) Consider the following graph of f(x). Diagram 1 Diagram 2 Diagram 3 Justify your decisions using mathematical reasoning.
QUESTION 19 (6 marks) A horizontal point of inflection is a point of inflection that is also a stationary point. In(x) kx Determine the value/s of £ for which the graph of f(x) = r xrl has only one horizontal point X+ of inflection.
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 10 The approximate value of x where the graph of the function y = X +6x%+Tx— 2 cos(x) changes concavity is (A) -3.26 (B) -2.85 (C) -2.20 (D) —1.89
QUESTION 4 The second derivative of the function f(x) is given by f” (x)=— The interval on which the graph of f (x) 1S concave up is A x<0 (B) x<0 ) x>0 (D) x>0
QUESTION 5 The graph of f (x) is shown. -5 Which of the following could be the graph of / (x)? (A) (B) ©) (D)
QUESTION 20 (7 marks) The population of rabbits (P) on an island in hundreds is given by P(7) = 1In (3t)+6 1> 0 where # is time in years. Determine the intervals of time when the population is increasing and the intervals when it is decreasing.
QUESTION 1 Consider the graph of f”(x) for a <x<b. cooee e - Which statement describes all the local maxima and minima of the graph of f(x) over a <x<b? (A) one local minimum and one local maximum (B) one local minimum and two local maxima (C) one local minimum only (D) one local maximum only
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 16 (3 marks) A section of the graphs of the first and second derivatives of a function are shown. Sketch a possible graph of the function on the same axes over the domain 0 < x < 27. Explain all reasoning used to produce the sketch. y
QUESTION 19 (7 marks) Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion e.g. if AUV W is similar to AXYZ then ur _yw _uw LU=4X 2V =LY and LW = £Z and — =— XY YZ XZ Two parallel walls AB and CD where the northern ends are 4 and C respectively are joined by a fence from B to C. The wall AB is 20 metres long the angle ABC = 30° and the fence BC is 10 metres long. A new fence is being built from A4 to a point P somewhere along CD. The new fence 4P will cross the original fence BC at O. Let OB = x metres where 0 < x <10. Determine the value of x that minimises the total area enclosed by AOBA and AOCP. Verify that this total area is a minimum.
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 The derivative of the function f`(x) is given by f”(x) = sin(x3 ) for the domain —1.8 < x <1.8. The number of points of inflection that the graph of f (x) has on this interval is (A) 1 (B) 3 €) 4 (D) 5
QUESTION 16 (5 marks) Determine the equation of the tangent to the function y = x3 + ax? + bx + 1 at the point of inflection (1 6).
QUESTION 18 (6 marks) Determine the area of the largest rectangle that has one side on the x-axis and two vertices on the curve y=e™”* B Express your answer in simplest form. The curve is sketched below. Not drawn to scale
QUESTION 15 (9 marks) Consider the function: f(x) = x2e(=**4) The first and second derivatives are: f`(x) = x(2 — x)eC*+® f(x) = (2% — 4x + 2)e¥* The graphs of f(x) f`(x) and f (x) are shown below: Not drawn to scale f(x) f(x) f(x) a) Determine the coordinates of the local maximum of f(x). [2 marks] b) Use the second derivative test to verify the nature of the stationary point from 15a). [2 marks] c) Determine the x-coordinates of any point/s of inflection. [2 marks] d) Determine the intervals where f (x) is concave up and concave down. [3 marks]
QUESTION 7 The graph of y = x> — 6x? is concave downwards when (A) x < —2 B) x<0 < x<?2 (D) x <4
QUESTION 17 (6 marks) A chemical is added to the water in a swimming pool at 10:00 am to prevent algae. The amount of chemical absorbed into the water over time ¢ (hours) is represented by A=10t7 —4¢ ostsi Determine the time of day when the rate of absorption of the chemical is at its maximum. Use calculus techniques to verify that your time corresponds to a maximum rate.
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
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