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Paper

Level

Difficulty

Question 1

HLPaper 2

View answer

A physicist models the velocity $v$ (in $\mathrm{m}\,\mathrm{s}^{-1}$ ) of a falling object under air resistance by

$\frac{d v}{d t}=9.8-0.2 v^{2}$ ,

where $t$ is measured in seconds and $\frac{dv}{dt}$ is in $\mathrm{m}\,\mathrm{s}^{-2}$ .

At $t=0$ , $v=0$ .

(a)

why the velocity is bounded and the terminal velocity.

[3]

(b)

Use Euler’s method with step size $h=0.1$ to $v(0.2)$ .

[3]

(c)

Find the equilibrium points and classify their stability.

[2]

(d)

the solution curve for $0\le t\le 2$ , indicating the Euler method points and equilibrium.

[3]

(e)

The analytical solution is $v=7\tanh(1.4t)$ . Use it to find the actual velocity at $t=0.2$ .

[2]

Question 2

HLPaper 2

View answer

A storage tank is being filled, and the water depth $d(t)$ metres after $t$ minutes is modelled by $\frac{dd}{dt}=1.2-0.08d^2$ , with $d(0)=2$ .

(a)

Using Euler's method with a step size $h=0.5$ , approximate $d(1)$ .

[6]

(b)

on whether using a smaller step size, such as $h=0.25$ , would lead to a more accurate solution.

[3]

(c)

one disadvantage of using a very small step size in Euler's method.

[2]

Question 3

HLPaper 2

View answer

An object is released from rest at the top of a long vertical tube filled with a thick viscous fluid at time $t = 0$ seconds. Take $x(0)=0$ and $v(0)=0$ , where $x$ is the object’s displacement from the top of the tube measured in metres, and $v=\frac{\text{d}x}{\text{d}t}$ is its velocity in $\text{m s}^{-1}$ .

Initially, the resistance is modelled as proportional to the velocity, giving the differential equation

$\frac{\text{d}^2x}{\text{d}t^2} = 9.81 - 0.9\left( \frac{\text{d}x}{\text{d}t} \right)$

where $9.81$ is in $\text{m s}^{-2}$ and the constant $0.9$ has units $\text{s}^{-1}$ .

The maximum velocity approached by the object as it falls is known as the terminal velocity.

An experiment is performed in which the object is placed in the fluid on a number of occasions and its terminal velocity is recorded. It is found that the terminal velocity was consistently smaller than that predicted by the model used. It was suggested that the resistance to motion is actually proportional to the velocity squared and so the following model was set up:

$\frac{\text{d}^2x}{\text{d}t^2} = 9.81 - 0.9\left( \frac{\text{d}x}{\text{d}t} \right)^2$

In this second model, the constant $0.9$ has units $\text{m}^{-1}$ .

At terminal velocity, the acceleration of an object is equal to zero.

(a)

By substituting $v = \frac{\text{d}x}{\text{d}t}$ into the equation, find an expression for the velocity of the object at time $t$ . Give your answer in the form $v = f(t)$ .

[7]

(b)

From your solution to part 1, or otherwise, find the terminal velocity of the object predicted by this model.

[2]

(c)

Write down the differential equation for the second model as a system of first-order differential equations.

[2]

(d)

Use Euler’s method, with a step length of $0.2\,\text{s}$ , to find the displacement and velocity of the object when $t = 0.6$ .

[4]

(e)

By repeated application of Euler’s method, an approximation for the terminal velocity, to five significant figures.

[1]

(f)

Use the differential equation to find the exact terminal velocity for the object predicted by the second model.

[2]

(g)

Use your answers to parts 4, 5 and 6 to on the accuracy of the Euler approximation to this model.

[2]

Question 4

HLPaper 2

View answer

A predator-prey system models the populations of deer ( $D$ , in thousands) and wolves ( $W$ , in hundreds) in a national park, given by:

\frac{\mathrm{d}D}{\mathrm{d}t} = 0.45D - 0.03DW, \quad \frac{\mathrm{d}W}{\mathrm{d}t} = 0.02DW - 0.4W

where $t$ is in years. Initially, $D = 22, W = 10$ .

(a)

Find the non-zero equilibrium point for the system.

[3]

(b)

the equilibrium point in the context of the model.

[1]

(c)

Use Euler's method with a step size of $0.2$ years to $D$ and $W$ after $0.2$ years.

[2]

(d)

On a phase plane with $D$ on the horizontal axis and $W$ on the vertical axis, for $18 \leq D \leq 24$ and $8 \leq W \leq 12$ , the initial direction of the trajectory from $(22,10)$ .

[2]

Question 5

HLPaper 1

View answer

Apply explicit Euler to $x'=x(2-y)$ , $y'=y(x-1)$ , starting from $(t,x,y)=(0,1.5,1)$ with $h=0.2$ for 4 steps.

(a)

Generate four predator-prey rows and verify positivity of both populations after every update.

[3]

(b)

Find the first row at which the sign of each population's Euler rate changes, and what the next update would do.

[3]

(c)

whether a negative population produced by a larger step would represent the model or a numerical failure.

[3]

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Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 5.16—eulers Method for 1st Order Des with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.16—eulers Method for 1st Order Des and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Difficulty

Question 1

HLPaper 2

View answer

A physicist models the velocity $v$ (in $\mathrm{m}\,\mathrm{s}^{-1}$ ) of a falling object under air resistance by

$\frac{d v}{d t}=9.8-0.2 v^{2}$ ,

where $t$ is measured in seconds and $\frac{dv}{dt}$ is in $\mathrm{m}\,\mathrm{s}^{-2}$ .

At $t=0$ , $v=0$ .

(a)

why the velocity is bounded and the terminal velocity.

[3]

(b)

Use Euler’s method with step size $h=0.1$ to $v(0.2)$ .

[3]

(c)

Find the equilibrium points and classify their stability.

[2]

(d)

the solution curve for $0\le t\le 2$ , indicating the Euler method points and equilibrium.

[3]

(e)

The analytical solution is $v=7\tanh(1.4t)$ . Use it to find the actual velocity at $t=0.2$ .

[2]

Question 2

HLPaper 2

View answer

A storage tank is being filled, and the water depth $d(t)$ metres after $t$ minutes is modelled by $\frac{dd}{dt}=1.2-0.08d^2$ , with $d(0)=2$ .

(a)

Using Euler's method with a step size $h=0.5$ , approximate $d(1)$ .

[6]

(b)

on whether using a smaller step size, such as $h=0.25$ , would lead to a more accurate solution.

[3]

(c)

one disadvantage of using a very small step size in Euler's method.

[2]

Question 3

HLPaper 2

View answer

Initially, the resistance is modelled as proportional to the velocity, giving the differential equation

$\frac{\text{d}^2x}{\text{d}t^2} = 9.81 - 0.9\left( \frac{\text{d}x}{\text{d}t} \right)$

where $9.81$ is in $\text{m s}^{-2}$ and the constant $0.9$ has units $\text{s}^{-1}$ .

The maximum velocity approached by the object as it falls is known as the terminal velocity.

$\frac{\text{d}^2x}{\text{d}t^2} = 9.81 - 0.9\left( \frac{\text{d}x}{\text{d}t} \right)^2$

In this second model, the constant $0.9$ has units $\text{m}^{-1}$ .

At terminal velocity, the acceleration of an object is equal to zero.

(a)

By substituting $v = \frac{\text{d}x}{\text{d}t}$ into the equation, find an expression for the velocity of the object at time $t$ . Give your answer in the form $v = f(t)$ .

[7]

(b)

From your solution to part 1, or otherwise, find the terminal velocity of the object predicted by this model.

[2]

(c)

Write down the differential equation for the second model as a system of first-order differential equations.

[2]

(d)

Use Euler’s method, with a step length of $0.2\,\text{s}$ , to find the displacement and velocity of the object when $t = 0.6$ .

[4]

(e)

By repeated application of Euler’s method, an approximation for the terminal velocity, to five significant figures.

[1]

(f)

Use the differential equation to find the exact terminal velocity for the object predicted by the second model.

[2]

(g)

Use your answers to parts 4, 5 and 6 to on the accuracy of the Euler approximation to this model.

[2]

Question 4

HLPaper 2

View answer

A predator-prey system models the populations of deer ( $D$ , in thousands) and wolves ( $W$ , in hundreds) in a national park, given by:

\frac{\mathrm{d}D}{\mathrm{d}t} = 0.45D - 0.03DW, \quad \frac{\mathrm{d}W}{\mathrm{d}t} = 0.02DW - 0.4W

where $t$ is in years. Initially, $D = 22, W = 10$ .

(a)

Find the non-zero equilibrium point for the system.

[3]

(b)

the equilibrium point in the context of the model.

[1]

(c)

Use Euler's method with a step size of $0.2$ years to $D$ and $W$ after $0.2$ years.

[2]

(d)

On a phase plane with $D$ on the horizontal axis and $W$ on the vertical axis, for $18 \leq D \leq 24$ and $8 \leq W \leq 12$ , the initial direction of the trajectory from $(22,10)$ .

[2]

Question 5

HLPaper 1

View answer

Apply explicit Euler to $x'=x(2-y)$ , $y'=y(x-1)$ , starting from $(t,x,y)=(0,1.5,1)$ with $h=0.2$ for 4 steps.

(a)

Generate four predator-prey rows and verify positivity of both populations after every update.

[3]

(b)

Find the first row at which the sign of each population's Euler rate changes, and what the next update would do.

[3]

(c)

whether a negative population produced by a larger step would represent the model or a numerical failure.

[3]

Unlock all 32 questions with a free account

Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.

IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Questionbank

IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Questionbank

Unlock all 32 questions with a free account

Unlock all 32 questions with a free account

IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Questionbank

IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Questionbank

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IB Mathematics Applications & Interpretation (AI) Topic AHL 5.16—eulers Method for 1st Order Des Questionbank

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IB Mathematics Applications & Interpretation (AI) Topic AHL 5.16—eulers Method for 1st Order Des Questionbank