more revision for integral calculus sac...more revision for integral calculus sacquestion 1 tasmania...
TRANSCRIPT
MORE REVISION FOR INTEGRAL CALCULUS SAC
Question 1
Tasmania Jones is painting the outside of his house. He is painting at a rate of:
𝑑𝐴
𝑑𝑡= 2loge(𝑡 + 1) m2/min where A is the area of the wall in square metres
and t is the time in minutes he spends painting. He began this task yesterday,
so today when he starts, there is already 10 m2 which has been painted.
a. Find 𝐴 in terms of t.
b. Hence or otherwise, calculate the area of wall that he paints in the first
half hour. Give your answer in m2, correct to one decimal place.
c. How much longer does it take him to paint a further 50m2? Give your
answer to the nearest minute.
MATHMBTH EXAM 2 8
Question2
Let/ be the function f D ➔ R, f(x) = loge<3 - 4x), where D is the largest possible domain over which
/is defined.
a. Find the exact coordinates of the interc.epts of the graph of y = f(x) with the x- and y-axes.
b. Find D, the largest possible domain over which/is defined.
e. Use calculus to show that the rate of change off(x) with respect tox is always negative.
2marks
1 mark
2marks
Question 3 - continued
VCE Mathematical Methods Units 3&4 Trial Examination 2 Question and Answer Booklet
Copyright © 2017 Neap MMU34EX2_QA_2017.FM 17
Question 3 (7 marks)
Consider the function f(x) = x2 and .
Parts of the graphs of f and g are shown below, as is the line with equation y = a2, where . Points A and B are the intersection points of the line with the curve as indicated.
a. Complete the correct sequence of transformations required to transform the graph ofy = f(x) to the graph of y = g(x). 2 marks
• dilation factor of __________ from the x-axis or alternatively a dilation factor
of __________ from the y-axis
• translation of ________________________________________
b. Show that the point B has the coordinates . 1 mark
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
g x( ) 14---x
25–=
a 0>
A (a, a2)
x
y
B
x = a
y = a2
f (x) = x2
g x( ) 14---x
25–=
2 a2
5+ a2,
VCE Mathematical Methods Units 3&4 Trial Examination 2 Question and Answer Booklet
18 MMU34EX2_QA_2017.FM Copyright © 2017 Neap
c. i. Find an expression in terms of a for the shaded area which is defined as the region bounded by the curve y = g(x) and the lines y = a2 and x = a. 2 marks
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
ii. Hence, find the coordinates of the point A that gives the minimum area of the region bounded by the curve y = g(x) and the lines y = a
2 and x = a. 2 marks
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
2017 MAV Mathematical Methods Trial Exam 2 21
© The Mathematical Association of Victoria, 2017
Question 4 (15 marks) Due to a disease, the rate at which a rabbit population is changing in a particular area is given by
32 (1 )( ) 3(1 ) tg t t e −′ = − − , where t is the time in months, t > 0, and g′ the number of rabbits in hundreds per month.
a. Sketch the graph of ( )y g t′= on the set of axes below. Label any asymptotes with their equations and the stationary points and endpoint(s) with their coordinates correct to the nearest hundred rabbits.
3 marks
b. By how many rabbits did the population decline in the first two months? Give your answer to the nearest integer. 2 marks _____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
SECTION B - Question 4 - continued TURN OVER
2017 MAV Mathematical Methods Trial Exam 2 22
© The Mathematical Association of Victoria, 2017
Assume (1) 2g = . c. Find a rule for g. 2 marks
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
d. How many rabbits were there initially? Give your answer to the nearest integer. 1 mark _____________________________________________________________________________
_____________________________________________________________________________
e. If conditions do not change, how many rabbits will there be in the future? Give your answer to the nearest integer. 1 mark _____________________________________________________________________________
_____________________________________________________________________________
f. Sketch the graph of ( )y g t= on the set of axes below. Label any asymptotes with their equations and the stationary points and endpoints with their coordinates correct to the nearest hundred rabbits.
2 marks
SECTION B - Question 4 - continued
2017 MAV Mathematical Methods Trial Exam 2 23
© The Mathematical Association of Victoria, 2017
Rabbits run across paddocks at fast speeds to escape from predators. The velocity, v m/s, of a particular rabbit,
which was travelling in a straight line, was recorded and is modelled by the rule 3( ) ( 1)( )( 4)2
v t t t t b t= − − − − ,
where t is the time in seconds and b is a real constant, 1 4b< < .
g. Find the value of b if the rabbit first stopped after running 2.2 m. 2 marks _____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
h. Find the maximum speed of the rabbit in m/s correct to one decimal place, and the total distance
the rabbit ran, in m, during the first four seconds. 2 marks _____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
END OF QUESTION AND ANSWER BOOK
23 2017 MATHMETH EXAM 2
SECTION B – Question 4 – continued
Copyright © Insight Publications 2017 TURN OVER
3: , ( ) ( 4) 1f R R f x x x
Question 5 (14 marks)
Consider the function .
a. If 2( ) ( 1)( 4)f x a x x , where a is a constant, show that 4a .
2 marks
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
b. The coordinates of the stationary points of the graph of ( )y f x are ( , 26)u
and (4, ).v Find the values of u and v.
2 marks
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
2017 MATHMETH EXAM 2 24
Copyright © Insight Publications 2017 SECTION B – Question 4 – continued
c. Find the values of x, correct to three decimal places, for which both ( ) 0f x and
( ) 0f x .
2 marks
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
d. Find the real values of k, correct to three decimal places, for which both solutions to the
equation ( ) 2f x k are positive.
1 mark
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
25 2017 MATHMETH EXAM 2
SECTION B – continued
Copyright © Insight Publications 2017 TURN OVER
The function ( )f x can also be written as 4 3 2( ) 12 48 64 1f x x x x x .
e. Show that the area bounded by the graph of ( ) 1y f x and the x-axis is
51.2 square units.
3 marks
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
f. i. Describe a sequence of transformations that maps the graph of ( )y f x
on to the graph of (2 ) 1y f x .
2 marks
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________
ii. Hence, find the x-axis intercepts of the graph of (2 ) 1y f x .
1 mark
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________
iii. Use the answers to part e. and part f. to find the area of the region bounded by
the graph of (2 ) 1y f x and the x-axis.
1 mark
_________________________________________________________________
_________________________________________________________________
_________________________________________________________________
Question 6
Tasmania Jones, prospector and adventurer, is working a small mining site shown below. The site is 40 metres long and W metres wide. Tasmania has surveyed the site completely so he knows that it has a constant cross-section all along its 40 metre length. He also knows that a gold seam (layer of gold) runs through the site and that it is underneath some granite rock.
A
:.io metres
.....____�/ I1+----W metres -----�
The diagram below shows the cross-section, AE, of the site and shows the depths and locations of the rock and the gold seam, where x is the horizontal distance (in metres) from A and y is the vertical distance (in metres) above the line AE.
y
B
X
-1
-2
-3
granite
-4
The equation y = sin(;�} 0 '> x '> W represents the surface of the rock (ABCDE).
The equation y = co{ 7� )-3, 0 $ x $ W represents the top (upper surface) of the gold seam (FGHJJ).
Question 4 - continued
Write down an expression in terms of x which represents the vertical distance
from the surface to the top of gold seam.
Use calculus to determine the exact value of the minimum vertical distance from the surfaceto the top of the gold seam. x for which this occurs.
1 mark
distance you find is a minimum.You may assume that the required value You do not need to justify that the
State the exact value of
Tasmania decides to remove all the granite from above the gold seam. Use calculus to determine the exact cross-sectional area of granite that he will remove.
4 marks
Tasmania Jones decides that it is too expensive to remove all the granite from above the gold seam at his site. He decides to excavate all the granite vertically over the 5 metresbetween D and E. Calculate the percentage of the total amount of gold which he is nowable to mine.Give your answer to the nearest per cent.