Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 9; CCSS.HSF.IF, HSG.GPE, HSN.VM extension; CCSS.MP2, MP4, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-PPV; TEKS 19 TAC section 111.55

CAL2.96 Horizontal and Vertical Tangents in Parametric Form Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.96 Horizontal and Vertical Tangents in Parametric Form. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.96 Horizontal and Vertical Tangents in Parametric Form - Select where a parametric curve has horizontal or vertical tangents using dx/dt and dy/dt. Standards tags: AP Calculus BC Unit 9; CCSS.HSF.IF, HSG.GPE, HSN.VM extension; CCSS.MP2, MP4, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-PPV; 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

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Below are sample questions representative of the exercises included on this worksheet and related CAL2.96 Horizontal and Vertical Tangents in Parametric Form skill assessments:

Problem 1Difficulty: Mastery
For x=t4-t2 and y=t3-t, which statement correctly lists all basic horizontal and vertical tangent parameters?
Ahorizontal at t=0, -1/sqrt(2), and 1/sqrt(2); vertical at t=-1/sqrt(3) and 1/sqrt(3)
Bhorizontal at t=0 only; vertical at t=-1/sqrt(3) and 1/sqrt(3)
Cno horizontal or vertical tangents
Dhorizontal at t=-1/sqrt(3) and t=1/sqrt(3); vertical at t=0, -1/sqrt(2), and t=1/sqrt(2)
Solution & Explanation:dy/dt=3t2-1 gives horizontal candidates +/-1/sqrt(3), and dx/dt=2t(2t2-1) is nonzero there. dx/dt=0 at 0 and +/-1/sqrt(2), and dy/dt is nonzero at all three, so those are vertical.
Problem 2Difficulty: Mastery
For x=t3 and y=t5-t, which statement correctly lists horizontal and vertical tangent parameters?
Ahorizontal at t=0; vertical at t=-5^(-1/4) and t=5^(-1/4)
Bhorizontal at t=-1 and t=1; vertical at t=0
Chorizontal at t=-5^(-1/4) and t=5^(-1/4); vertical at t=0
Dno horizontal or vertical tangents
Solution & Explanation:dy/dt=5t4-1, so horizontal tangents occur at t=+/-5^(-1/4), where dx/dt=3t2 is nonzero. dx/dt=0 at t=0, and dy/dt=-1 there, so t=0 is vertical.
Problem 3Difficulty: Mastery
For x=(t2-1)3 and y=t2, which t-values give vertical tangents by the basic test?
At=0 only
Bt=-1 and t=1
Ct=-1, 0, and 1
Dno such t
Solution & Explanation:dx/dt=6t(t2-1)2, so dx/dt=0 at -1, 0, and 1. dy/dt=2t is nonzero at +/-1 but zero at 0, so only t=-1 and t=1 give vertical tangents.

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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 9; CCSS.HSF.IF, HSG.GPE, HSN.VM extension; CCSS.MP2, MP4, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-PPV; TEKS 19 TAC section 111.55 (Select where a parametric curve has horizontal or vertical tangents using dx/dt and dy/dt. Standards tags: AP Calculus BC Unit 9; CCSS.HSF.IF, HSG.GPE, HSN.VM extension; CCSS.MP2, MP4, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-PPV; TEKS 19 TAC section 111.55.).

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