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 3

Consider the system:

\frac{d x}{d t}=x-y^{2}, \quad \frac{d y}{d t}=y-x^{2}

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

Find the equilibrium points of the system.

[3]

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

[2]

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

[2]

Determine the stability of the equilibrium point at $(1,1)$ .

[4]

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

[3]

Question 8

HLPaper 3

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

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

[2]

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

[2]

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

[2]

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

[2]

Use Euler's method with step size 0.1 to approximate $y$ at $x=0.2$ for the solution through $(0,1)$ .

[3]

Determine the concavity of the solution curve at $(0,1)$ .

[4]

Question 9

HLPaper 1

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

Calculate the value of $\frac{d y}{d x}$ at the point $\left(0, \frac{\pi}{2}\right)$ .

[1]

Find the equation of the curve where the slope field has zero gradient.

[2]

Determine the nature of the points on the curve found in part (b) (e.g., maximum, minimum, or neither).

[3]

Question 10

HLPaper 3

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 $x \rightarrow 2$ .

[4]