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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 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 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 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 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 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 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 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 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 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]
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