IB Mathematics Applications & Interpretation Topic AHL 5.15—slope Fields Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.15—slope Fields Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 5.15—slope Fields with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.15—slope Fields and mirrors Paper 1, 2, 3 style where relevant.

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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. The green curve should be ignored; the point

A (2, - 1)

is marked.

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

[2]

Sketch the lines $x = 0$ and $y = 0$ on the slope field.

[2]

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

[2]

Determine the concavity at $A (2, - 1)$ of the solution curve from Part 2.

[3]

Question 2

HLPaper 1

Consider the differential equation $\frac{d y}{d x} = e^{x} - y$ , for $- 2 \leq x \leq 2, 0 \leq y \leq 3$ .

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

[1]

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

[2]

Determine the coordinates of the point where the solution curve through $(0, 1)$ intersects the curve from Part 1.

[3]

Find the equation of the tangent line to the solution curve through $(0, 1)$ at the intersection point found in Part 2.

[3]

Question 3

HLPaper 1

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 0 on the slope field.

[2]

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

[2]

Determine whether the solution curve from part 2 has any turning points in $- 2 \leq x \leq 2$ . Justify your answer.

[3]

Question 4

HLPaper 1

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

Write down the value of $\frac{d y}{d x}$ at the point $(0, \frac{π}{2})$ .

[1]

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

[2]

Determine the nature of the points on the curve found in Part 1 (e.g., maximum, minimum, or neither).

[3]

Question 5

HLPaper 1

A slope field for

\frac{d y}{d x} = x^{2} - y^{2}

- \frac{1}{2} \leq x \leq \frac{1}{2}, - 2 \leq y \leq 2

is displayed here.

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

[2]

Sketch the curves obtained in part 0 on the slope field.

[2]

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

[2]

Determine whether the solution curve from part 2 has any turning points in $- \frac{1}{2} \leq x \leq \frac{1}{2}$ . Justify your answer.

[3]

IB Mathematics Applications & Interpretation AHL 5.15—slope Fields Questionbank

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IB Mathematics Applications & Interpretation AHL 5.15—slope Fields Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.15—slope Fields Questionbank

Unlock all 12 questions with a free account