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Question 1

HLPaper 2

A tank contains a saline solution with concentration $S$ (in $\mathrm{g} / \mathrm{L}$ ) at time $t$ (in minutes). Water with salt at $2 \mathrm{~g} / \mathrm{L}$ flows in at $3 \mathrm{~L} / \mathrm{min}$ , and the solution flows out at $3 \mathrm{~L} / \min$ . The tank volume is 100 L . The differential equation is:

\frac{d S}{d t}=0.06-0.03 S

Initial concentration:

S=1

g/L at $t=0$ .

1.

Solve the differential equation to find $S(t)$ .

[4]

2.

Use Euler's method with $h=1$ minute to estimate $S$ after 2 minutes.

[3]

3.

Sketch the solution curve and Euler's approximations for $0 \leq t \leq 5$ .

[3]

4.

Determine the time when $S=1.5$ $\mathrm{g} / \mathrm{L}$ using the analytical solution.

[3]

Question 2

HLPaper 1

A researcher models the concentration of a chemical pollutant $C$ (in $\mathrm{mg} / \mathrm{L}$ ) in a lake over time $t$ (in months). The rate of change of the concentration is influenced by natural degradation and an external source, modeled by the differential equation:

\frac{d C}{d t}=-0.2 C+5

Initially, at $t=0$ , the concentration is $C=10 \mathrm{mg} / \mathrm{L}$ .

1.

Solve the differential equation to find the general solution for $C$ in terms of $t$ .

[4]

2.

Use the initial condition to find the particular solution and determine the concentration after 3 months.

[3]

3.

Use Euler's method with a step size of 0.5 months to estimate the concentration after 1 month.

[3]

4.

Sketch the solution curve for $0 \leq t \leq 5$ and the Euler's method approximations for the first two steps on the same graph.

[3]

5.

The lake is considered safe for aquatic life when $C<15 \mathrm{mg} / \mathrm{L}$ . Using your particular solution, determine if the lake is safe after 10 months.

[2]

Question 3

HLPaper 1

A predator-prey system models rabbits $R$ and foxes $F$ (in hundreds) over time $t$ (in years):

\frac{d R}{d t}=2 R-0.02 R F, \quad \frac{d F}{d t}=0.01 R F-0.5 F

Initial populations:

\begin{aligned} & R=10 \\ & F=2 \end{aligned}

at $t=0$ .

1.

Interpret the terms in both differential equations.

[4]

2.

Use Euler's method with

h=0.1

years to estimate

R

and

F

after 0.2 years.

[4]

3.

Find the non-zero equilibrium point.

[3]

4.

Sketch the phase portrait trajectory for

\begin{aligned} & 0 \leq t \leq 0.2 \\ & 5 \leq R \leq 15 \\ & 1 \leq F \leq 3 \end{aligned}

[2]

Question 4

HLPaper 1

Consider the curve defined by $x^{2}+y^{2}=e^{x y}$ , where $y \neq 0$ . The slope of the curve is given by the differential equation derived from implicit differentiation. (d) Use implicit differentiation to show that $\frac{d y}{d x}=\frac{y e^{x y}-2 x}{2 y - x e^{x y}}$ . (b) At the point $(0,1)$ , use Euler's method with a step size of 0.1 to estimate $y$ when $x=0.2$ . [3] (c) Verify that $(0,1)$ lies on the curve. [1] (d) Sketch the curve and the Euler's method approximation near $(0,1)$ for $0 \leq x \leq 0.2$ . [3]

1.

Use implicit differentiation to show that $\frac{d y}{d x}=\frac{y e^{x y}-2 x}{2 y-x e^{x y}}$

[4]

2.

At the point $(0,1)$ , use Euler's method with a step size of 0.1 to estimate $y$ when $x=0.2$ .

[3]

3.

Verify that $(0,1)$ lies on the curve.

[1]

4.

Sketch the curve and the Euler's method approximation near $(0,1)$ for $0 \leq x \leq 0.2$ .

[3]

Question 5

HLPaper 2

A model describes the growth of a coral reef's surface area $S$ (in $\mathrm{m}^{2}$ ) over time $t$ (in years), given by:

\frac{d S}{d t}=0.1 S\left(1-\frac{S}{200}\right)

At $t=0, S=10$ .

1.

Explain why the reef's surface area is limited to $200 \mathrm{~m}^{2}$ .

[2]

2.

Use Euler's method with a step size of 0.2 years to estimate $S$ after 0.4 years.

[3]

3.

Solve the differential equation to find $S(t)$ .

[4]

4.

Sketch the solution curve for $0 \leq t \leq 5$ , indicating the Euler's method points and the limiting value.

[3]

5.

Find the actual value of $S$ at $t=0.4$ and compare it with the Euler's method estimate.

[3]

Question 6

HLPaper 1

A model describes the decay of a radioactive isotope with activity $A$ (in becquerels) over time $t$ (in days), given by:

\frac{d A}{d t}=-0.08 A+4

At $t=0, A=100$ .

1.

Solve the differential equation to find the general solution for $A$ .

[4]

2.

Use the initial condition to find the particular solution and calculate $A$ after 5 days.

[3]

3.

Use Euler's method with a step size of 0.5 days to estimate $A$ after 1 day.

[3]

4.

Sketch the solution curve for $0 \leq t \leq 10$ and the Euler's method approximations for the first two steps.

[3]

5.

Explain, with a reason, whether the Euler's method estimate in part 3 is an over or under-approximation compared to the actual value.

[2]

Question 7

HLPaper 1

A model describes the spread of a new technology adoption $A$ (in thousands of users) over time $t$ (in months):

\frac{d A}{d t}=0.3 A\left(1-\frac{A}{50}\right)

At $t=0, A=2$ .

1.

Explain why the maximum number of adopters is 50,000.

[2]

2.

Use Euler's method with a step size of 0.2 months to estimate $A$ after 0.4 months.

[3]

3.

Solve the differential equation to find $A(t)$ .

[4]

Question 8

HLPaper 2

A scientist models the temperature $T$ (in ${ }^{\circ} \mathrm{C}$ ) of a cooling reactor core over time $t$ (in hours), governed by:

\frac{d T}{d t}=-0.15(T-20)

At $t=0, T=80$ .

1.

Solve the differential equation to find the general solution for $T$ .

[4]

2.

Use the initial condition to find the particular solution and calculate $T$ after 4 hours.

[3]

3.

Use Euler's method with a step size of 1 hour to estimate $T$ after 2 hours.

[3]

4.

Sketch the solution curve for $0 \leq t \leq 10$ and the Euler's method approximations for the first two steps on the same graph.

[3]

5.

Determine the time when $T=30^{\circ} \mathrm{C}$ using the particular solution.

[3]

Question 9

HLPaper 2

A researcher models the concentration of a chemical pollutant $C$ (in $\mathrm{mg} / \mathrm{L}$ ) in a lake over time $t$ (in months). The rate of change of the concentration is influenced by natural degradation and an external source, modeled by the differential equation:

\frac{d C}{d t}=-0.2 C+5

Initially, at $t=0$ , the concentration is $C=10 \mathrm{mg} / \mathrm{L}$ .

1.

Solve the differential equation to find the general solution for $C$ in terms of $t$ .

[4]

2.

Use the initial condition to find the particular solution and determine the concentration after 3 months.

[3]

3.

Use Euler's method with a step size of 0.5 months to estimate the concentration after 1 month.

[3]

4.

Sketch the solution curve for $0 \leq t \leq 5$ and the Euler's method approximations for the first two steps on the same graph.

[3]

5.

The lake is considered safe for aquatic life when $C<15 \mathrm{mg} / \mathrm{L}$ . Using your particular solution, determine if the lake is safe after 10 months.

[2]

Question 10

HLPaper 1

A biologist studies the growth of a bacterial colony, where $P$ (in thousands) represents the population at time $t$ (in hours). The growth is modeled by:

$\frac{d P}{d t}=0.3 P\left(1-\frac{P}{50}\right)$

At $t=0, P=5$ .

1.

Explain why the population has a carrying capacity of 50,000 bacteria.

[2]

2.

Use Euler's method with a step size of 0.2 hours to estimate $P$ after 1 hour.

[4]

3.

Find the equilibrium points by setting $\frac{d P}{d t}=0$ .

[2]

4.

Sketch the direction field for the differential equation for $0 \leq t \leq 2,0 \leq P \leq 60$ , and indicate the behavior near the equilibrium points.

[3]

AHL 5.16—Eulers method for 1st order DEs Questionbank