AHL 5.15—Slope fields Questionbank

Practice AHL 5.15—Slope fields with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

The slope field for $\frac{d y}{d x}=\frac{y^{2}-x^{2}}{y+x}$ , for $-2 \leq x \leq 2,-2 \leq y \leq 2$ , excluding $x+y=0$ , is shown below.

Find the equation of the curve where $\frac{d y}{d x}=0$ .

[2]

Sketch the curve from part (a) on the slope field.

[2]

Sketch the solution curve through $(1,1)$ .

[2]

Use Euler's method with step size 0.1 to approximate $y$ at $x=1.2$ .

[3]

Determine the coordinates of the intersection of the solution curve through $(1,1)$ with the curve from part (a).

[3]

Analyze the behavior of the solution curve as $x \rightarrow 2$ .

[4]

Question 2

HLPaper 1

The differential equation $\frac{d y}{d x}=x \ln (y)-y$ , for $-2 \leq x \leq 2,1 \leq y \leq 3$ , is defined for $y>0$ .

Calculate $\frac{d y}{d x}$ at $(1, e)$ .

[2]

Find the equation of the curve where $\frac{d y}{d x}=0$ .

[2]

Determine the nature of the points on the curve from part (b).

[3]

Find the equation of the tangent line to the solution curve through $(1, e)$ at $(1, e)$ .

[3]

Determine the $x$ -coordinate where the solution curve through $(1, e)$ crosses the $y=1$ line, if it exists, using qualitative analysis.

[4]

Question 3

HLPaper 2

The slope field for $\frac{d y}{d x}=x^{2}-y^{2}$ , for $-2 \leq x \leq 2,-2 \leq y \leq 2$ , is shown below.

Find the equations of the curves where $\frac{d y}{d x}=0$ .

[2]

Sketch the curves from part (a) on the slope field.

[2]

Sketch the solution curve passing through $(0,1)$ .

[2]

Determine whether the solution curve from part (c) has any turning points. Justify your answer.

[3]

Question 4

HLPaper 2

The slope field for $\frac{d y}{d x}=x y$ , for $-2 \leq x \leq 2,-2 \leq y \leq 2$ , is shown below.

Write down the equation of the curves where $\frac{d y}{d x}=0$ .

[2]

Sketch the curves from part (a) on the slope field.

[2]

Sketch the solution curve passing through $(1,1)$ .

[2]

Determine the concavity of the solution curve from part (c) at (1, 1).

[3]

Question 5

HLPaper 2

The slope field for the differential equation $\frac{d y}{d x}=x-y^{2}$ , for $-3 \leq x \leq 3,0 \leq y \leq 3$ , is shown below.

Find the equation of the curve on which the points with zero gradient lie.

[2]

Sketch the curve from part (a) on the slope field above.

[2]

Sketch the solution curve that passes through the point $(1,1)$ .

[2]

Determine the coordinates of the point where the solution curve from part (c) intersects the curve found in part (a).

[3]

Question 6

HLPaper 1

The differential equation $\frac{d y}{d x}=e^{x+y}-x$ , for $-2 \leq x \leq 2,-1 \leq y \leq 2$ .

Calculate $\frac{d y}{d x}$ at $(0,0)$ .

[2]

Find the equation of the curve where $\frac{d y}{d x}=0$ .

[2]

Determine the nature of the points on the curve from part (b).

[3]

Find the equation of the tangent line to the solution curve through $(0,0)$ .

[3]

Analyze the asymptotic behavior of the solution curve through $(0,0)$ as $x \rightarrow 2$ .

[4]

Question 7

HLPaper 2

An object is placed into the top of a long vertical tube, filled with a thick viscous fluid,at time $t = 0$ seconds.

Initially it is thought that the resistance of the fluid would be proportional to the velocity ofthe object. The following model was proposed, where the object’s displacement, $x$ , fromthe top of the tube, measured in metres, is given by the differential equation

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

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 occasionsand its terminal velocity recorded. It is found that the terminal velocity was consistently smallerthan that predicted by the model used. It was suggested that the resistance to motion is actuallyproportional to the velocity squared and so the following model was set up.

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

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

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

[7]

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

[2]

Write down the differential equation as a system of first order differential equations.

[2]

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

[4]

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

[1]

Use the differential equation to find the terminal velocity for the object.

[2]

Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Eulerapproximation to this model.

[2]

Question 8

HLPaper 3

This question will investigate the solution to a coupled system of differential equations when there is only one eigenvalue.

It is desired to solve the coupled system of differential equations

\dot{x} = 3x + y

\dot{y} = -x + y

The general solution to the coupled system of differential equations is hence given by

\begin{pmatrix} x \\ y \end{pmatrix} = A\begin{pmatrix} 1 \\ -1 \end{pmatrix}e^{2t} + B\begin{pmatrix} t \\ -t + 1 \end{pmatrix}e^{2t}

As $t \to \infty$ the trajectory approaches an asymptote.

State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.

[1]

Show that the matrix

\left( \begin{array}{cc} 3 & 1 \\ -1 & 1 \end{array} \right)

has (sadly) only one eigenvalue. Find this eigenvalue and an associated eigenvector.

[7]

Verify that

\left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} t \\ -t + 1 \end{array} \right)e^{2t}

is also a solution.

[5]

Find the values of $x$ and $y$ when $t = 2$ .

[2]

Find the equation of this asymptote.

[3]

If initially at $t = 0$ , $x = 20$ , $y = 10$ , find the particular solution.

[3]

Hence, verify that

\left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} 1 \\ -1 \end{array} \right)e^{2t}

is a solution to the above system.

[5]

Question 9

HLPaper 1

The diagram shows the slope field for the differential equation $\frac{dy}{dx} = \sin(x+y), -4 \leq x \leq 5, 0 \leq y \leq 5$ The graphs of the two solutions to the differential equation that pass through points (0, 1) and (0, 3) are shown. For the two solutions given, the local minimum points lie on the straight line L₁.

Find the equation of L₁, giving your answer in the form y = mx + c.

[1]

For the two solutions given, the local maximum points lie on the straight line L₂. Find the equation of L₂.

[1]

Question 10

HLPaper 3

This question will investigate the solution to a coupled system of differential equations and how to transform it to a system that can be solved by the eigenvector method.

It is desired to solve the coupled system of differential equations

$\dot x = x + 2y - 50$

$\dot y = 2x + y - 40$

where $x$ and $y$ represent the population of two types of symbiotic coral and $t$ is time measured in decades.

Find the equilibrium point for this system.

[2]

If initially $x = 100$ and $y = 50$ use Euler’s method with an time increment of 0.1 to find an approximation for the values of $x$ and $y$ when $t = 1$ .

[3]

Extend this method to conjecture the limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

Show how using the substitution $X = x - 10{\text{,}}\,\,Y = y - 20$ transforms the system of differential equations into $\begin{array}{*{20}{c}} {\dot X = X + 2Y} \\ {\dot Y = 2X + Y} \end{array}$ .

[3]

Solve this system of equations by the eigenvalue method and hence find the general solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system.

[8]

Find the particular solution to the original system, given the initial conditions of part (b).

[2]

Hence find the exact values of $x$ and $y$ when $t = 1$ , giving the answers to 4 significant figures.

[2]

Use part (f) to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

With the initial conditions as given in part (b) state if the equilibrium point is stable or unstable.

[1]

10.

If instead the initial conditions were given as $x = 20$ and $y = 10$ , find the particular solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system, in this case.

[2]

11.

With the initial conditions as given in part (j), determine if the equilibrium point is stable or unstable.

[2]

An object is placed into the top of a long vertical tube, filled with a thick viscous fluid,at time $t = 0$ seconds.

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

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

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

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