Mathematical Methods Problem Solving Task

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Mathematical Methods (CAS) Unit 4, 2013 SAC Analysis Task 1 Page 1

Mathematical Methods Problem Solving Task

Practice

Calculus and Algebra of Functions -: Lesson 1

Name: _______________________________ / 44 marks

Question 1:

The number of mosquitoes, 𝑁, around a pond on a particular night can be modelled by the equation 𝑁 = 100log

₎

2𝑡 + 1 + 5𝑡 + 1000, where 𝑡 is the hours after sunset.

a) Find the number of mosquitos at sunset.

(1 mark)

b) Find the average rate of change over the first 4 hours (to 2 decimal places).

(2 marks)

c) Sketch the graph of N against t from sunset of the first to sunset of the next day. Show any

endpoints in co-ordinate form.

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(2 marks)

d) How long would it take, to the nearest minute, for the population to increase from 1200 to 1300?

(2 marks)

e) Use calculus to explain why the rate of change in mosquito numbers is always positive.

(2 mark)

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f) A scientist proposes another model for the mosquito population using the function:

𝑀 𝑡 =

1650 𝑡 + 1

3 0 ≤ 𝑡 ≤ 3

𝑥

³

− 6𝑡 + 1109 3 < 𝑡 ≤ 12

where t is the number of hours after sunset. Is the function differentiable at t = 3? Explain with reasons.

(2 marks) (Total 11 marks)

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Question 2:

The amount of water in the Thomson reservoir (𝑊) is described as a percentage of the total capacity.

• At the beginning of December 2011 (start of summer), the water level was 34% of the total capacity.

• At the beginning of April 2012 the water level reached 9% of total capacity

• At the beginning of November 2012, it had reached a level of 51% of total capacity.

The water level at the start of April 2012 was the lowest in the reservoir for the year 2012.

Use 𝑡 as the number of months after the beginning of December 2011 so that 𝑡 = 1 represents the beginning of January 2012.

a) Plot and label these points on the axes provided:

(1 mark)

b) For this part, you must make use of the point (11, 51). Write a quadratic relationship of the form 𝑊

₌

𝑡 = 𝑎(𝑡 − 𝑏)

³

+ 𝑐 that models the capacity of the reservoir against time measured in months for 𝑡 ∈ [0, 11]. (Use exact values throughout).

(2 marks)

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c) Sketch your quadratic function on the axes provided in part a) showing any key features.

(1 mark)

Subsequent measuring during 2012 showed that after December 2011 a better model for 𝑊

₃

(𝑡) is a cubic function. There is still a local minimum of 9% at the beginning of April 2012 and a local maximum of 51% at the beginning of November 2012.

d) (i) Explain why 𝑊

₃

′ 𝑡 = 𝑘(𝑡 − 4)(𝑡 − 11) could be the rule of the gradient function, for 𝑡 ∈ 0, 11 and 𝑘 ∈ 𝑅

^G

.

(1 mark)

(ii) What happens to the graph as k increases? What does this mean in terms of the water levels?

(2 marks)

(iii) A cubic function of the form 𝑊

_H

𝑡 = 𝑘 𝑑𝑡

^H

+ 𝑒𝑡

³

+ 𝑓𝑡 + 𝑔 where

𝑑, 𝑒, 𝑓, 𝑔, 𝑘 ∈ 𝑅 and 𝑡 ∈ [0, 11] was used to model the capacity of the reservoir between December 2011 and November 2012. Use integration, or otherwise to find values for 𝑑, 𝑒, 𝑓, 𝑔 and 𝑘, and hence state the equation of the cubic function.

(4 marks)

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d) (iv) Using this cubic model find the average water level, to the nearest whole percentage of total capacity, in the reservoir between the beginning of April 2012 and September 2012.

(2 marks)

e) The Upper Yarra Reservoir can be modelled by a hemisphere. When the depth of the water is h m, the volume V in ML of water in the reservoir is given by

10

2

(50 ) 3

V

π

h h

= − .

(i) If water is being released from the reservoir at a rate of 20 ML/hour, find the rate at which the depth of water is decreasing when the depth is 30 m

(3 marks)

(ii) Find, using the linear approximation rule: 𝑓 𝑥 + ℎ ≈ 𝑓 𝑥 + ℎ𝑓

^O

𝑥 , the

approximate change in the volume of water in the reservoir when the depth of water changes from 32 to 30 m.

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Question 3

On June 30 this year the Phillip Island Tip was closed and, together with the Phillip Island Nature Park, local Council will work to restore this former landfill site into a reserve which will help protect the Rhyll Inlet and attract wildlife back to the area.

The land will be levelled flat and covered with clean fill in order to create parkland. A straight path is constructed and runs from one of the entry points to the park to a seat at a point A which is a horizontal distance of 50 metres from the entry point and is the highest point in the park. A cross-sectional view of this path is shown on the graph in Figure 1 below.

With respect to the axes shown, the path follows a curve with the rule 1 log ( 1)

50

^e

y x

+

x

= +

a) Write down the coordinates of point A, giving the y value to 2 decimal places.

(1 mark)

b) Find

dy

dx

at 𝑥 = 25, correct to 3 decimal places and explain what this represents.

(2 marks)

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c) Determine whether the gradient of the path ever exceeds 0.1. Show your reasoning.

(2 marks)

d) (i) Find an approximation for the area of the cross-section between 𝑥 = 0 and 𝑥 = 50 using right endpoint rectangles of width 10 metres. Give your answer to 2 decimal places.

(2 marks)

(ii) Find the actual area under the graph in Figure 1 between 𝑥 = 0 and 𝑥 = 50.

Give your answer to 2 decimal places

(2 marks)

(iii) Is your approximation from part (i) an under or over approximation of the area?

Explain why.

^{(1 mark)}

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(iv) Assuming that the path is 2 m wide and that there is no change in the slope of the path from one side to the other, find the volume of clean fill lying directly beneath the path to two decimal places.

(1 mark)

e) (i) Find the derivative of 𝑦 =

^(QG=)_=SS^R

log

₎

(𝑥 + 1)

(ii) Hence find 1 log ( 1)

50

^e

x

+

x dx

∫ +

(1+3= 4 marks)

(Total: 15 marks)

END OF LESSON 1

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Mathematical Methods

Calculus and Algebra of Functions : Lesson 2 Name: _________________________________ / 44

Question 4

The Acme seed company is about to bring out a new product called the ‘All Temperatures Dwarf Bean’. Each packet of beans will contain 100 bean seeds:

• 50 of type A (a bean said to grow well at temperatures between 0

^{C and 20}

C) and

• 50 of type B (a bean said to grow well at temperatures above 30 C).

Vicky, a research assistant at the company has been conducting tests on the two types of bean.

She set up a series of experiments where she attempted to grow 50 beans of one type - either A or B.

Vicky found that the number of plants that grew was approximated by the following models:

Type A ( ) 25 25sin

40

A T

π

T

= +

0 ≤ T ≤ 60

Type B B T

( ) =

a

log (2

_e T

+ 1)

0 ≤ T ≤ 60

Where A(T) and B(T) were the number of seeds of type A and B respectively that produced plants. T was the temperature in degrees Celsius and ‘a’ was a positive constant.

a) Vicky observed that the 50 bean seeds of type B grew into plants at 60 C. Show that 25

log 11

_e a =

(2 marks)

b) How many beans seeds of type B grew at 30°C?

(1 mark)

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c) Use Calculus to write an equation, that gives the temperature at which the greatest number of seeds of type A produced plants.

(2 marks)

d) On the axes below, sketch the graphs of A(T) and B(T). Label any endpoints and intercepts with coordinates.

(4 marks)

e) Determine the range of temperatures for which type B plants produce more plants than type A. Give answers to one decimal place.

(1 mark)

f) For bean seeds of type B, determine the temperature, to one decimal place, at which the rate of change in the number of seeds that produce plants is

⁼₃

.

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Question 5

Let 4

: , ( ) 5

f R R f x x

x

+

→ = + −

a) State the interval for which the graph of v(t) is strictly increasing.

(1 mark)

b) Determine the absolute minimum value of v(t) and explain what this means in terms of the velocity of the particle.

(2 mark)

c) Particle B is moving along a path which is given by the equation of the tangent to the graph of v(t) at the point where 𝑡 = 16. Find this equation and, hence, the initial position of particle B.

(3 marks)

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d) Particle C’s velocity is given by the rule 𝑔 𝑡 = 𝑣(𝑡)

Sketch the graph of 𝑦 = 𝑔(𝑡) on the same set of axes as the graph of 𝑣 𝑡 , clearly indicating the coordinates of any local maxima and local minima.

(3 marks)

e) (i) Write an integral expression, in terms of 𝑣(𝑡), that would find the area enclosed by the graph of 𝑦 = 𝑔(𝑡) and the t-axis.

(ii) Calculate this area and state what it represents.

(2 + 2 = 4 marks)

f) The point (𝑎, 0), where 0 < 𝑎 < 16, is such that the area enclosed by the graph of 𝑦 = 𝑔(𝑡), the t-axis and the line 𝑡 = 𝑎 is half the area found above. Write an equation using a suitable integral that could be used to find a and also find the value of a.

(2 marks)

g) Particle D’s displacement is given ℎ 𝑡 = 𝑡

³

𝑓𝑜𝑟 𝑡 ≥ 1

(i) Given that 𝑔(ℎ 𝑡 ) exists, state its domain.

(ii) Find the rule of 𝑔(ℎ 𝑡 ) and find the value(s) of t for which it has an absolute minimum.

(1+2 = 3 marks) (Total 18 marks)

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Question 6

Given the function ^: ^, ( ) ^{3log 1}

e

₂ ^,

f D R f x

⎛

x

⎞

→ = ⎜ + ⎟

⎝ ⎠

a) Show that D, the maximal domain of the function f, is −2, ∞

(1 mark)

b) State a sequence of three transformations which takes the graph of 𝑔 𝑥 = log

₎

𝑥 to the graph of f(x).

(3 marks)

c) Find the inverse function

f⁻¹

.

(2 marks)

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d) Sketch the graphs of f and f

⁻¹

on the axes below, clearly labelling the graphs, stating any axial intercepts and giving the equations of all asymptotes.

(3 marks)

e) Show that 𝑓 𝑓

^Y=

𝑥 = 𝑥

(2 marks)

f) One of the coordinates of the points of intersection between the graphs of f and f

⁻¹

, is given by (

p p where

^, )

p > .

0 i. Find the value of p, correct to three decimal places.

ii. Hence, write down an integral which would give the approximate area bound by f(x), the

y-axis and the line 𝑦 = 𝑝

(1 +2 =3 marks) Total 14 marks