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AHL 5.15—Slope fields - IB Questionbank | RevisionDojo

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 1

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 2

HLPaper 1

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

Calculate the value of $\frac{d y}{d x}$ at the point $(1,2)$ .

[2]

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

[2]

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

[2]

Sketch the solution curve passing through $(1,2)$ .

[2]

Determine the nature (maximum, minimum, or neither) of the points on the curve from part (b).

[3]

Find the equation of the tangent line to the solution curve through $(1,2)$ at the point where it intersects the curve from part (b).

[4]

Question 3

HLPaper 1

Consider the differential equation $\frac{d y}{d x}=y-\cos (x)$ , for $-4 \leq x \leq 4,0 \leq y \leq 2$ .

Calculate $\frac{d y}{d x}$ at the point $(0,1)$ .

[1]

Find the equation of the curve where the slope is zero. [

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

[3]

Find the equation of the line tangent to the solution curve passing through $(0,1)$ at that point.

[3]

Question 4

HLPaper 1

The slope field for the differential equation $\frac{d y}{d x}=x^{2}+y$ , for $-2 \leq x \leq 2,-1 \leq y \leq 3$ , is shown below.

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

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

[2]

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

[2]

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

[3]

Question 5

HLPaper 1

System: $\frac{d x}{d t}=y-x^{3}, \frac{d y}{d t}=x-y^{3}$ , for $-2 \leq x \leq 2,-2 \leq y \leq 2$ . Slope field for $\frac{d y}{d x}=\frac{x-y^{3}}{y-x^{3}}$ :

Find the equilibrium points of the system.

[3]

Sketch the curve where $\frac{d y}{d x}=0$ .

[2]

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

[2]

Determine the stability of the equilibrium point at $(0,0)$ .

[4]

Use Euler's method with step size 0.1 to approximate $y$ at $x=0.2$ .

[3]

Analyze the asymptotic behavior of the solution curve through $(0,1)$ as $x \rightarrow 2$ .

Question 6

HLPaper 2

The slope field for $\frac{dy}{dx}=\frac{y^2-x^2}{y+x}$ , 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 $\frac{dy}{dx}=0$ .

[2]

Sketch the curve from Part 1 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 1.

[3]

Analyze the behavior of the solution curve 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 of the object. The following model was proposed, where the object’s displacement, $x$ , from the top of the tube, measured in metres, is given by the differential equation

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

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 occasions and its terminal velocity 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:

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

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 the object 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 object predicted by this model.

[2]

Write down the differential equation for the second model 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 of the 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 exact terminal velocity for the object predicted by the second model.

[2]

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

[2]

Question 8

HLPaper 1

The differential equation $\frac{dy}{dx} = x \ln(y) - y$ is defined for $y > 0$ in the region $-2 \leq x \leq 2$ and $1 \leq y \leq 3$ .

Calculate the value of $\frac{dy}{dx}$ at the point $(1, e)$ .

[2]

Find the equation of the curve (isocline) where $\frac{dy}{dx} = 0$ .

[2]

Determine the nature of the stationary points on the curve found in part (b) within the given region.

[3]

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

[3]

Determine the $x$ -coordinate where the solution curve through $(1, e)$ crosses the line $y = 1$ , if it exists within the domain $-2 \leq x \leq 2$ , using qualitative analysis.

[4]

Question 9

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 1 on the slope field.

[2]

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

[2]

Determine whether the solution curve from part 3 has any turning points. Justify your answer.

[3]

Question 10

HLPaper 2

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

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

[2]

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

[2]

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

[3]

Paper

Level

Question 1

HLPaper 1

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 2

HLPaper 1

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

Calculate the value of $\frac{d y}{d x}$ at the point $(1,2)$ .

[2]

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

[2]

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

[2]

Sketch the solution curve passing through $(1,2)$ .

[2]

Determine the nature (maximum, minimum, or neither) of the points on the curve from part (b).

[3]

Find the equation of the tangent line to the solution curve through $(1,2)$ at the point where it intersects the curve from part (b).

[4]

Question 3

HLPaper 1

Consider the differential equation $\frac{d y}{d x}=y-\cos (x)$ , for $-4 \leq x \leq 4,0 \leq y \leq 2$ .

Calculate $\frac{d y}{d x}$ at the point $(0,1)$ .

[1]

Find the equation of the curve where the slope is zero. [

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

[3]

Find the equation of the line tangent to the solution curve passing through $(0,1)$ at that point.

[3]

Question 4

HLPaper 1

The slope field for the differential equation $\frac{d y}{d x}=x^{2}+y$ , for $-2 \leq x \leq 2,-1 \leq y \leq 3$ , is shown below.

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

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

[2]

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

[2]

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

[3]

Question 5

HLPaper 1

System: $\frac{d x}{d t}=y-x^{3}, \frac{d y}{d t}=x-y^{3}$ , for $-2 \leq x \leq 2,-2 \leq y \leq 2$ . Slope field for $\frac{d y}{d x}=\frac{x-y^{3}}{y-x^{3}}$ :

Find the equilibrium points of the system.

[3]

Sketch the curve where $\frac{d y}{d x}=0$ .

[2]

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

[2]

Determine the stability of the equilibrium point at $(0,0)$ .

[4]

Use Euler's method with step size 0.1 to approximate $y$ at $x=0.2$ .

[3]

Analyze the asymptotic behavior of the solution curve through $(0,1)$ as $x \rightarrow 2$ .

Question 6

HLPaper 2

The slope field for $\frac{dy}{dx}=\frac{y^2-x^2}{y+x}$ , 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 $\frac{dy}{dx}=0$ .

[2]

Sketch the curve from Part 1 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 1.

[3]

Analyze the behavior of the solution curve 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.

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

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

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

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 the object 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 object predicted by this model.

[2]

Write down the differential equation for the second model 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 of the 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 exact terminal velocity for the object predicted by the second model.

[2]

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

[2]

Question 8

HLPaper 1

The differential equation $\frac{dy}{dx} = x \ln(y) - y$ is defined for $y > 0$ in the region $-2 \leq x \leq 2$ and $1 \leq y \leq 3$ .

Calculate the value of $\frac{dy}{dx}$ at the point $(1, e)$ .

[2]

Find the equation of the curve (isocline) where $\frac{dy}{dx} = 0$ .

[2]

Determine the nature of the stationary points on the curve found in part (b) within the given region.

[3]

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

[3]

Determine the $x$ -coordinate where the solution curve through $(1, e)$ crosses the line $y = 1$ , if it exists within the domain $-2 \leq x \leq 2$ , using qualitative analysis.

[4]

Question 9

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 1 on the slope field.

[2]

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

[2]

Determine whether the solution curve from part 3 has any turning points. Justify your answer.

[3]

Question 10

HLPaper 2

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

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

[2]

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

[2]

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

[3]

Paper

Level

Question 1

HLPaper 1

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 2

HLPaper 1

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

Calculate the value of $\frac{d y}{d x}$ at the point $(1,2)$ .

[2]

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

[2]

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

[2]

Sketch the solution curve passing through $(1,2)$ .

[2]

Determine the nature (maximum, minimum, or neither) of the points on the curve from part (b).

[3]

Find the equation of the tangent line to the solution curve through $(1,2)$ at the point where it intersects the curve from part (b).

[4]

Question 3

HLPaper 1

Consider the differential equation $\frac{d y}{d x}=y-\cos (x)$ , for $-4 \leq x \leq 4,0 \leq y \leq 2$ .

Calculate $\frac{d y}{d x}$ at the point $(0,1)$ .

[1]

Find the equation of the curve where the slope is zero. [

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

[3]

Find the equation of the line tangent to the solution curve passing through $(0,1)$ at that point.

[3]

Question 4

HLPaper 1

The slope field for the differential equation $\frac{d y}{d x}=x^{2}+y$ , for $-2 \leq x \leq 2,-1 \leq y \leq 3$ , is shown below.

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

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

[2]

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

[2]

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

[3]

Question 5

HLPaper 1

System: $\frac{d x}{d t}=y-x^{3}, \frac{d y}{d t}=x-y^{3}$ , for $-2 \leq x \leq 2,-2 \leq y \leq 2$ . Slope field for $\frac{d y}{d x}=\frac{x-y^{3}}{y-x^{3}}$ :

Find the equilibrium points of the system.

[3]

Sketch the curve where $\frac{d y}{d x}=0$ .

[2]

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

[2]

Determine the stability of the equilibrium point at $(0,0)$ .

[4]

Use Euler's method with step size 0.1 to approximate $y$ at $x=0.2$ .

[3]

Analyze the asymptotic behavior of the solution curve through $(0,1)$ as $x \rightarrow 2$ .

Question 6

HLPaper 2

The slope field for $\frac{dy}{dx}=\frac{y^2-x^2}{y+x}$ , 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 $\frac{dy}{dx}=0$ .

[2]

Sketch the curve from Part 1 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 1.

[3]

Analyze the behavior of the solution curve 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.

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

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

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

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 the object 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 object predicted by this model.

[2]

Write down the differential equation for the second model 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 of the 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 exact terminal velocity for the object predicted by the second model.

[2]

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

[2]

Question 8

HLPaper 1

The differential equation $\frac{dy}{dx} = x \ln(y) - y$ is defined for $y > 0$ in the region $-2 \leq x \leq 2$ and $1 \leq y \leq 3$ .

Calculate the value of $\frac{dy}{dx}$ at the point $(1, e)$ .

[2]

Find the equation of the curve (isocline) where $\frac{dy}{dx} = 0$ .

[2]

Determine the nature of the stationary points on the curve found in part (b) within the given region.

[3]

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

[3]

Determine the $x$ -coordinate where the solution curve through $(1, e)$ crosses the line $y = 1$ , if it exists within the domain $-2 \leq x \leq 2$ , using qualitative analysis.

[4]

Question 9

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 1 on the slope field.

[2]

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

[2]

Determine whether the solution curve from part 3 has any turning points. Justify your answer.

[3]

Question 10

HLPaper 2

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

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

[2]

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

[2]

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

[3]

AHL 5.15—Slope fields Questionbank