Mathematical Methods Unit 4 2016

School Assessed Coursework:Problem Solving Task 1

Name: _______________________________

This task is based on concepts from the Differential and Integral Calculus areas of study of the course.

It consists of two parts, to be completed over 2 periods.

Reading Time: 5 minutes

Task Time: 65minutes

Each part must be completed within a period and submitted at the conclusion of that period.

Students can refer to their bound reference material and calculator use is encouraged throughout the task, but

no discussion is permitted between students during the task. Bound reference material may be checked by

staff at any stage. You can have access to this sheet throughout the task.

You will be provided with a sheet of miscellaneous formulae.

Answers must be given exactly unless otherwise specified.

Calculators will be cleared at the conclusion of each period and no work related to the task should be taken

from the classroom.

You will need to show your teacher that your calculator has been RESET.

This SAC contributes 8.5% towards the overall study score for Mathematical Methods.

Students will be required to demonstrate the achievement of the following outcomes:

Outcome 1 to define & explain key concepts and apply a range of related mathematical routines

and procedures

Outcome 2 to apply mathematical processes in non-routine contexts and analyse and discuss these

applications of mathematics

Outcome 3 to select and appropriately use technology to develop mathematical ideas, produce

results and carry out analysis

The overall result for this task will be comprised of marks for each of the above outcomes in the

approximate proportions:

OUTCOME 1: 32%

OUTCOME 2: 40%

OUTCOME 3: 28%

NOTE: The mark you receive for this SAC is subject to moderation following your exam results at the end

of the year. A criteria sheet will be provided in the SAC with further details about the marking of the SAC.

Mathematical Methods Unit 4 2016

Calculus Problem Solving Task: Lesson 1

Name:________________________________________

Question 1

Jess is scuba diving on a reef in the Pacific Ocean, the reef is bordered by two straight boundaries (northern

and eastern boundary) and a coast line. The shaded area on the diagram below shows the cross-section of the

reef. The y- axis runs in a north-south line and in relation to the set of axes the northern boundary of the reef

is described by the rule y = a. The eastern boundary is described by the rule x = 5.

The coast line boundary is described by the function

𝑓(𝑥) =1 − 𝑙𝑜𝑔𝑒(𝑥 + 1)

(𝑥 + 1)2, 𝑥 ∈ [0,5].

where 1 unit represents 1 kilometre. The north-west corner of the reef is located at the point (0, a).

a) Show that a =1

(1 mark)

b) Show that the function f(x) crosses the x-axis at x = e – 1.

(2 marks)

reef

c) Find the length of the eastern boundary of the reef. Express your answer in kilometres correct to four

decimal places.

(2 marks)

d) Find the coordinates of the southernmost point on the reef’s boundary. Express the coordinates

correct to two decimal places.

(1 mark)

e) (i) Evaluate ∫ 𝑓(𝑥)𝑑𝑥5

𝑒−1, giving your answer correct to 3 decimal places.

(1 mark)

(ii) Explain why the value of this definite integral is negative.

(1 mark)

f) Write down a single integral that would give the area of the reef and hence find this area, to the

nearest square kilometre.

(2 marks)

Question 2

Harry loves to ride on rollercoasters, his favourite is the ‘Screamer’. The height, for a section of the track,

h(x) metres, of the ‘Screamer’ above ground level is given by

ℎ1(𝑥) = 3 −1

4𝑥2, 0 ≤ 𝑥 ≤ 1

and

ℎ2(𝑥) = 𝐴𝑒−𝑘(𝑥−1) − 0.5, 1 < 𝑥 ≤ 7

where x is the horizontal distance in metres from the entrance point at the top of the ‘Screamer’ and A and k

are real constants.

a) Given that the functions ℎ1(𝑥) and ℎ2(𝑥) intersect at x = 1, show that A = 3.25

(2 marks)

Use A = 3.25 for the remainder of the question.

b) (i) Find ℎ1′ (𝑥), the derivative of ℎ1(𝑥).

(1 mark)

(ii) Find ℎ2′ (𝑥), the derivative of ℎ2(𝑥) .

(1 mark)

(iii) The rollercoaster track is continuous and has equal gradient at the intersection between the

functions ℎ1(𝑥) and ℎ2(𝑥). Use this information to show that k = 2

13.

(2 marks)

c) At what height, h, would Harry be if he is 2 metres horizontally from the start of the ride? Write your

answer correct to 2 decimal places.

(1 mark)

d) What horizontal distance would Harry travel after having vertically descended by 2 metres? Write

your answer correct to 2 decimal places.

(2 marks)

e) On the one set of axes, draw the graphs of the functions ℎ1(𝑥) and ℎ2(𝑥), showing key features,

correct to 3 decimal places where appropriate.

x m

(2 marks)

ℎ(𝑥)

f) As the rollercoaster gets to the end of the track, at x = 7, it derails causing Harry to continue in a

straight line tangentially. Find the equation of this line, giving the constants correct to 2 decimal

places.

(3 marks)

Question 3

The velocity v km/h of a train which moves along a straight track from station A to station B is given by

( ) 1 sin( )v t kt t , where t (hours) is the time measured from when the train left A and k is a positive

constant.

Assume that the train began at station A from rest and did not stop until it reached station B.

a) Show that the time the train takes to travel from A to B is equal to half an hour.

(2 marks)

b) i) Show that the derivative of cost t is cos(𝜋𝑡) − 𝜋𝑡 sin (𝜋𝑡).

(1 mark)

ii) Hence show that an antiderivative of 𝑡𝑠𝑖𝑛(𝜋𝑡) is 1

𝜋2 sin(𝜋𝑡) −𝑡𝑐𝑜𝑠(𝜋𝑡)

𝜋

(2 marks)

c) Find the value of k to the nearest integer if the distance from A to B is 20km.

(2 marks)

End of Part 1

Mathematical Methods Unit 4 2016

Calculus Problem Solving Task: Lesson 2

Name:________________________________________

Question 4

The acceleration, a(t) m/s2, of a particle travelling in a straight line is given by the rule:

𝑎(𝑡) = 6 −2

(𝑡 + 1)2, 𝑡 ≥ 0

where t is the time in seconds from the start.

Initially the particle is at rest.

a) Show that the expression for the velocity, 𝑣(𝑡) m/s, of the particle at time, t seconds, can be given by

the rule:

𝑣(𝑡) = 6𝑡 +2

𝑡 + 1− 2

(3 marks)

b) Sketch a graph of 𝑣(𝑡) against t for the first 5 seconds, showing coordinates of endpoints exactly.

(2 marks)

𝑡 (sec )

𝑣(𝑡)

m/s

c) Find the average velocity in the first 3 seconds. State your answer correct to 2 decimal places.

(2 marks)

d) Find the average rate of change of the velocity in the first 3 seconds.

(2 marks)

e) (i) Explain why the distance the particle travels in the first 4 seconds and the displacement of the

particle in the first 4 seconds are the same.

(1 mark)

(ii) Find the distance the particle travels in the first 4 seconds, correct to 2 decimal places.

(2 marks)

Question 5

a) State the coordinates of the points E and F.

(1 mark)

b) Express the length of line segment

(i) BF in terms of b

(1 mark)

(ii) DE in terms of d

(1 mark)

c) Explain, including a diagram, why triangle DEC and triangle CFB are similar. Hence show that

𝑑 =28

𝑏−7+ 4.

(4 marks)

C (7, 4)

A line BD goes through the point (7, 4), cuts the x-axis

at B (b, 0) and the y-axis at D (0, d), as indicated in the

diagram.

B (b, 0)

D (0, d)

d) Show that the area, A square units, of triangle OBD is modelled by the function 𝐴 =2𝑏2

𝑏−7.

(2 marks)

e) By hand find the value of b for which the area, A, is a minimum and show that it is a minimum.

Hence find the least area of triangle OBD.

(5 marks)

f) Suppose that point C also lies on a curve with rule 𝑦 = 𝑓(𝑥), so that the line BD is tangential to the

curve 𝑦 = 𝑓(𝑥) at C.

The equation of the normal to the curve 𝑦 = 𝑓(𝑥) at C is 2𝑦 = 𝑥 + 1.

Find the coordinates of points B and D in this case.

(3 marks)

Question 6

It is observed that for some time after planting in suitable conditions, the area (A) covered by a particular

species of ground-cover plant has a rate of increase of x cm2/month given by 3 25 6 , 0x t t t t

where t is the number of months after planting

and dA

xdt

.

a) Assume that the model is accurate for the first 8 months after planting. Calculate when during this

period:

(i) A, the area covered by the plant has a maximum value.

(ii) x , the rate of increase in area is a maximum, to the nearest month.

(2+ 2 = 4 marks)

b) If the plant covered 50 cm2 when originally planted, find A in terms of t .

(3 marks)

c) Hence find the area that will be covered at the beginning of May, if the ground cover is planted in

suitable conditions at the beginning of January, giving your answer correct to 2 decimal places.

(2 marks)

End of Task