PHYS1034 NOVEMBER EXAM PREP (MATHEMATICAL MODELLING)
Sources: Phys1034 Lecture Notes & Tutorials, APPM Notes, Mathematical Methods & Modelling
Notes
CHAPTER 1: (Dimensional Analysis & Others) ✓
CHAPTER 2: (Recurrence Relationships and Difference Eqn Sols) ✓
CHAPTER 3: (Continuous Change)
QN 1. Solve the following continuous equations
𝑑𝑦
i) − 2𝑦 2 = 0
𝑑𝑥
𝑑𝑦
ii) 𝑑𝑥 − 2𝑦 = 0
𝑑𝑦
iii) 𝑑𝑥 − 5 = 10
𝑑𝑦 5
iv) 𝑑𝑥 − 𝑥 𝑦 = 0
𝑑𝑥
QN 2. Suppose the function 𝑥(𝑡) satisfies the equation, − 𝑡 + 𝑡𝑥 = 0, and that 𝑥(0) = 2,
𝑑𝑡
find the value of 𝑥(1) rounded to two decimal places.
QN 3. Suppose that South Africa’s Population is increasing at a rate which is proportional to
the number of people living at any time. Let 𝑃 = number of people at any time 𝑡. When the
calculation starts, there are 55.5million people in South Africa.
i) Which mathematical model below best describes the population growth of South Africa
with time?
𝑑𝑃
A. = 𝑒 𝜆𝑃(𝑡)
𝑑𝑡
𝑑𝑃
B. = −𝜆𝑃(𝑡)
𝑑𝑡
𝑑𝑃
C. = 𝜆𝑃(𝑡)
𝑑𝑡
𝑑𝑃
D. 𝑑𝑡 = (55.5 × 106 )𝑒 𝜆𝑡
E. None of the above
ii) Which of the below states the initial condition mathematically?
A. 𝑃(0) = 55.5
B. 𝑃(0) = 0
C. 𝑃(0) = 55.5 × 106
D. 𝑃(𝑡) = 55.5 × 106
E. None of the above
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, PHYS1034 NOVEMBER EXAM PREP (MATHEMATICAL MODELLING)
QN 4. The rate of decay of any given radioactive substance is directly proportional to the
amount of sample 𝑁 available at any given time 𝑡.
i)Derive the equation relating the amount of sample that remains after a certain time 𝑡. Take
𝜆 to be the decay constant.
ii) How long will it take for the half of the original amount of substance to decay? Take this
time to be 𝑡1/2.
QN 5. The growth rate of bacteria in a culture medium is directly proportional to the difference
̅.
between the number of bacteria and a certain number 𝑵
i)An expression for the number of bacteria 𝒏 as a function of the time 𝒕, given that the initial number
of bacteria placed in the culture medium is 𝑁0 and the constant of proportionality is 𝝀, is given by:
A. 𝑛(𝑡) = 𝑁0 + (𝑁̅ − 𝑁0 )𝑒 𝜆𝑡
B. ̅ + (𝑁
𝑛(𝑡) = 𝑁 ̅ − 𝑁0 )𝑒 𝜆𝑡
C. 𝑛(𝑡) = 𝑁0 + (𝑁0 − 𝑁 ̅)𝑒 𝜆𝑡
D. ̅ + (𝑁0 − 𝑁
𝑛(𝑡) = 𝑁 ̅)𝑒 𝜆𝑡
E. None of these options
̅, the time 𝑡 ∗ taken for the bacteria to vanish in the
ii) If the initial number of bacteria 𝑁0 is less than 𝑁
culture medium is given by:
A. None of these options
̅
𝑁
B. 𝑡 ∗ = 𝜆ln (𝑁 ̅
)
0 −𝑁
1 ̅
𝑁
C. 𝑡 ∗ = ln ( ̅
)
𝜆 𝑁0 −𝑁
̅
𝑁
D. 𝑡 ∗ = 𝜆ln ( ̅ )
𝑁−𝑁0
̅−𝑁0
𝑁
E. 𝑡 ∗ = 𝜆ln ( ̅
𝑁
)
QN 6. (Qn5 Look-alike) If the growth rate of bacteria in a culture medium is directly proportional to
the difference between the number of bacteria and a certain optimal number 𝑃̅ .
i)Derive an expression for the number of bacteria P as a function of the time 𝑡 given that the initial
number of bacteria placed in the culture medium is 𝑃0 and that the constant of proportionality is 𝜆
ii) How long does it take for the difference in the number of bacteria and the optimal number to reduce
to ¼ of the initial difference?
MSM Tutoring (Pty) Ltd | © 2022. All Rights Reserved.
Sources: Phys1034 Lecture Notes & Tutorials, APPM Notes, Mathematical Methods & Modelling
Notes
CHAPTER 1: (Dimensional Analysis & Others) ✓
CHAPTER 2: (Recurrence Relationships and Difference Eqn Sols) ✓
CHAPTER 3: (Continuous Change)
QN 1. Solve the following continuous equations
𝑑𝑦
i) − 2𝑦 2 = 0
𝑑𝑥
𝑑𝑦
ii) 𝑑𝑥 − 2𝑦 = 0
𝑑𝑦
iii) 𝑑𝑥 − 5 = 10
𝑑𝑦 5
iv) 𝑑𝑥 − 𝑥 𝑦 = 0
𝑑𝑥
QN 2. Suppose the function 𝑥(𝑡) satisfies the equation, − 𝑡 + 𝑡𝑥 = 0, and that 𝑥(0) = 2,
𝑑𝑡
find the value of 𝑥(1) rounded to two decimal places.
QN 3. Suppose that South Africa’s Population is increasing at a rate which is proportional to
the number of people living at any time. Let 𝑃 = number of people at any time 𝑡. When the
calculation starts, there are 55.5million people in South Africa.
i) Which mathematical model below best describes the population growth of South Africa
with time?
𝑑𝑃
A. = 𝑒 𝜆𝑃(𝑡)
𝑑𝑡
𝑑𝑃
B. = −𝜆𝑃(𝑡)
𝑑𝑡
𝑑𝑃
C. = 𝜆𝑃(𝑡)
𝑑𝑡
𝑑𝑃
D. 𝑑𝑡 = (55.5 × 106 )𝑒 𝜆𝑡
E. None of the above
ii) Which of the below states the initial condition mathematically?
A. 𝑃(0) = 55.5
B. 𝑃(0) = 0
C. 𝑃(0) = 55.5 × 106
D. 𝑃(𝑡) = 55.5 × 106
E. None of the above
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, PHYS1034 NOVEMBER EXAM PREP (MATHEMATICAL MODELLING)
QN 4. The rate of decay of any given radioactive substance is directly proportional to the
amount of sample 𝑁 available at any given time 𝑡.
i)Derive the equation relating the amount of sample that remains after a certain time 𝑡. Take
𝜆 to be the decay constant.
ii) How long will it take for the half of the original amount of substance to decay? Take this
time to be 𝑡1/2.
QN 5. The growth rate of bacteria in a culture medium is directly proportional to the difference
̅.
between the number of bacteria and a certain number 𝑵
i)An expression for the number of bacteria 𝒏 as a function of the time 𝒕, given that the initial number
of bacteria placed in the culture medium is 𝑁0 and the constant of proportionality is 𝝀, is given by:
A. 𝑛(𝑡) = 𝑁0 + (𝑁̅ − 𝑁0 )𝑒 𝜆𝑡
B. ̅ + (𝑁
𝑛(𝑡) = 𝑁 ̅ − 𝑁0 )𝑒 𝜆𝑡
C. 𝑛(𝑡) = 𝑁0 + (𝑁0 − 𝑁 ̅)𝑒 𝜆𝑡
D. ̅ + (𝑁0 − 𝑁
𝑛(𝑡) = 𝑁 ̅)𝑒 𝜆𝑡
E. None of these options
̅, the time 𝑡 ∗ taken for the bacteria to vanish in the
ii) If the initial number of bacteria 𝑁0 is less than 𝑁
culture medium is given by:
A. None of these options
̅
𝑁
B. 𝑡 ∗ = 𝜆ln (𝑁 ̅
)
0 −𝑁
1 ̅
𝑁
C. 𝑡 ∗ = ln ( ̅
)
𝜆 𝑁0 −𝑁
̅
𝑁
D. 𝑡 ∗ = 𝜆ln ( ̅ )
𝑁−𝑁0
̅−𝑁0
𝑁
E. 𝑡 ∗ = 𝜆ln ( ̅
𝑁
)
QN 6. (Qn5 Look-alike) If the growth rate of bacteria in a culture medium is directly proportional to
the difference between the number of bacteria and a certain optimal number 𝑃̅ .
i)Derive an expression for the number of bacteria P as a function of the time 𝑡 given that the initial
number of bacteria placed in the culture medium is 𝑃0 and that the constant of proportionality is 𝜆
ii) How long does it take for the difference in the number of bacteria and the optimal number to reduce
to ¼ of the initial difference?
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