Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.BF; CCSS.MP2, MP5, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-PS; TEKS 19 TAC section 111.55

CAL2.152 Taylor Polynomials and Graph Behavior Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.152 Taylor Polynomials and Graph Behavior. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.152 Taylor Polynomials and Graph Behavior - Select the graph or local behavior that best matches a Taylor polynomial near its center. Standards: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.BF; CCSS.MP2, MP5, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-PS; 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 CAL2.152 Taylor Polynomials and Graph Behavior skill assessments:

Problem 1Difficulty: Mastery

Let h=x-a and P_7(x)=4+h²-h³+h⁷. Which local graph behavior best matches P_7 near x=a?

Alocal maximum at (a,4), with the left branch slightly lower

Bincreasing horizontal inflection through (a,4)

Clocal minimum at (a,4), with the right branch slightly lower than the left at equal small distances

Dline through (a,4) with slope 1

Solution & Explanation:The +h² term dominates all higher powers near the center, so the graph has a local minimum. The -h³ term is negative on the right and positive on the left, making the right branch slightly lower at equal small distances.

Problem 2Difficulty: Mastery

For P_6(x)=1-x²/2+x⁴/24-x⁶/720, the Maclaurin polynomial for cos x through degree 6, which local graph behavior is best?

Aincreasing horizontal inflection through (0,1)

Beven-symmetric local maximum at (0,1), concave down at the center

Codd-symmetric crossing through the origin

Dlocal minimum at (0,1), concave up at the center

Solution & Explanation:Only even powers appear, so the graph is symmetric about the y-axis. The leading nonconstant term is -x²/2, so the graph has a local maximum at (0,1) and is concave down at the center.

Problem 3Difficulty: Mastery

For P_5(x)=x-x³/6+x⁵/120, the Maclaurin polynomial for sin x through degree 5, how does P_5 compare with its tangent line y=x near x=0?

Aabove y=x on both sides

Bbelow y=x on both sides

Cbelow y=x for x<0 and above y=x for x>0

Dabove y=x for x<0 and below y=x for x>0

Solution & Explanation:Subtract the tangent line: P_5-x=-x³/6+x⁵/120. The leading difference is -x³/6, which is positive for x<0 and negative for x>0.

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Frequently Asked Questions

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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 10; CCSS.HSA.SSE, HSF.BF; CCSS.MP2, MP5, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-PS; TEKS 19 TAC section 111.55 (Select the graph or local behavior that best matches a Taylor polynomial near its center. Standards: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.BF; CCSS.MP2, MP5, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-PS; TEKS 19 TAC section 111.55.).