Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 7; FL MA.912.C.5.2-C.5.4; CCSS.HSM; CCSS.MP1, MP2, MP4, MP5, MP6; Integrated Calculus II Standards CAL2-ODE; TEKS 19 TAC section 111.55

CAL2.90 Qualitative Differential Equation Analysis Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.90 Qualitative Differential Equation Analysis. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.90 Qualitative Differential Equation Analysis - Select increasing/decreasing behavior, equilibria, and long-run behavior without solving explicitly. Standards: AP Calculus BC Unit 7; FL MA.912.C.5.2-C.5.4; CCSS.HSM; CCSS.MP1, MP2, MP4, MP5, MP6; Integrated Calculus II Standards CAL2-ODE; TEKS 19 TAC section 111.55

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Calculus II

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related CAL2.90 Qualitative Differential Equation Analysis skill assessments:

Problem 1Difficulty: Mastery

An autonomous differential equation has slopes positive for y < -1, negative for -1 < y < 4, and positive for y > 4. What happens to a solution with y(0) = 0?

AIt crosses y = -1 and continues downward

BIt decreases toward y = -1

CIt increases toward y = 4

DIt decreases without bound

Solution & Explanation:Starting at y = 0 places the solution in (-1, 4), where slopes are negative. The solution decreases, and arrows on both sides of y = -1 point toward that equilibrium, so it approaches -1.

Problem 2Difficulty: Mastery

For an autonomous equation dy/dt = f(y) with f continuously differentiable, which situation would violate uniqueness?

AA solution remains constant when it starts at an equilibrium

BA solution is increasing on an interval where f(y) > 0

CA solution starts between two equilibria and crosses one of them to enter another interval

DA solution approaches a stable equilibrium asymptotically

Solution & Explanation:If a solution crossed an equilibrium, then at the crossing time both the nonconstant solution and the constant equilibrium solution would pass through the same point, contradicting uniqueness. Approaching asymptotically is allowed.

Problem 3Difficulty: Mastery

For dy/dt = e^(-t²)(y - 1), t >= 0, and y(0) = 2, which conclusion follows from the qualitative sign information?

AThe solution decreases toward 1

BThe solution crosses 1 after a long time

CThe solution is constant because e^(-t²) becomes small

DThe solution stays above 1 and is increasing

Solution & Explanation:The coefficient e^(-t²) is positive for every t. With y > 1, the derivative is positive, so the solution remains above the equilibrium and increases; a small positive coefficient still gives positive slope.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Calculus II Math students. It aligns with educational standard AP Calculus BC Unit 7; FL MA.912.C.5.2-C.5.4; CCSS.HSM; CCSS.MP1, MP2, MP4, MP5, MP6; Integrated Calculus II Standards CAL2-ODE; TEKS 19 TAC section 111.55 (Select increasing/decreasing behavior, equilibria, and long-run behavior without solving explicitly. Standards: AP Calculus BC Unit 7; FL MA.912.C.5.2-C.5.4; CCSS.HSM; CCSS.MP1, MP2, MP4, MP5, MP6; Integrated Calculus II Standards CAL2-ODE; TEKS 19 TAC section 111.55).