# 5. Calculus

The **Calculus topic** has the largest number of *suggested teaching hours* of the five syllabus topics: **28 hours for SL** (just one hour more than the Statistics & Probability topic at SL) and **55 hours for HL** (4 hours more than the Geometry & Trigonometry topic at HL). For many students, calculus is often the most challenging part of the syllabus. I think this is mostly due to the fact that the majority of students will not have previously studied any calculus and that calculus requires a great deal of conceptual understanding, not just learning technical processes.

**∼ quick link ∼**__downloadable exercise sets, quizzes & tests__ for teaching units in **Calculus**

**Changes compared to previous Maths HL-SL syllabusses**:

∗ No formal definition of derivative from first principles in SL

∗ No volumes of revolution in SL

∗ The following ‘new’ content was in previous HL Calculus option topic:** HL 5.18:** 1^{st} order differential equations; numerical solution using Euler’s method; separation of variables method; homogeneous differential equations; linear differential equations and integrating factor method** HL 5.19:** Maclaurin series; use of simple substitution, products, integration and differentiation to obtain other series; Maclaurin series developed from differential equations

**Calculus - syllabus overview**

Syllabus item numbers are in brackets.

__SL - Calculus__

**SL core (AA & AI)**

differentiation basics (5.1, 5.2, 5.3)

tangents & normals (5.4)

integration basics (5.5)

**SL (AA)**

differentiation rules & 2^{nd} derivative (5.6, 5.7)

maxima, minima & optimization (5.8)

kinematics (5.9)

integration & areas (5.10, 5.11)

__HL - Calculus__

**HL (AA)**

further calculus (5.12)

evaluating limits (5.13)

implicit differentiation & related rates (5.14)

optimization (5.14)

further differentiation & integration (5.15, 5.16)

areas & volumes (5.17)

differential equations (5.18)

Maclaurin series (5.19)

### Exercise Sets, Quizzes & Tests

AA_SL_5.4(8)_diff_calc4_v1

Exercise set with 4 questions. GDC allowed on all questions. **Worked solutions** included. __Syllabus content__: equations of tangents & normals; differentiation of \({x^n}\), chain rule, product rule; maximum and minimum points, points of inflexion.

AA_HL_5.4(14)_diff_calc6_v1

Exercise set with 11 questions (1-6 no GDC, 7-11 GDC allowed). **Worked solutions** available below. __Syllabus content__: chain rule; product rule; quotient rule; tangent lines; implicit differentiation; points of inflexion; related rates; optimization

AA_HL_5.4(14)_diff_calc6_SOL_KEY_v1 **Worked solutions** for HL_diff_calc6 exercise set above.

AA_SL_Quiz_2_diff_calc_v1

SL differential calculus quiz with 5 questions. GDC allowed on all questions. Content: equation of normal, optimization, points of inflexion, finding maxima & minima, applying 1st and 2nd derivative tests. **Worked solutions** available below.

AA_SL_Quiz_2_diff_calc_SOL_KEY_v1 **Worked solutions** for SL differential calculus quiz above

AA_HL_Quiz_2_diff_calc_v1

HL differential calculus quiz with 6 questions. GDC allowed on all questions. Content: equation of normal, derivative from first principles, optimization, points of inflexion, finding maxima & minima, applying 1st and 2nd derivative tests. **Worked solutions** available below.

AA_HL_Quiz_2_diff_calc_SOL_KEY_v1 **Worked solutions** for HL differential calculus quiz above

AA_HL_Test1_diff_calc_v1

First HL test on differential calculus; syllabus content covered: chain rule; product rule; quotient rule; finding maxima & minima; points of inflexion; derivative from first principles; optimization; **worked solutions** available below

AA_HL_Test1_diff_calc_SOL_KEY_v1 **worked solutions** for first HL differential calculus test above

AA_HL_Test2_diff_calc_v1

2nd HL test on differential calculus; syllabus content covered: chain rule; product rule; quotient rule; finding maxima & minima; points of inflexion; implicit differentiation; related rates; optimization; **worked solutions** available below

AA_HL_Test2_diff_calc_SOL_KEY_v1 **worked solutions** for 2nd HL differential calculus test above

AA_HL_Test1_integral_calculus_v1

First HL test on integral calculus; syllabus content covered: definite integrals using technology; areas between a curve and *x*-axis; kinematic problems; integration by inspection; areas between curves; integration by substitution; integration by parts; volumes of revolution about the *x*-axis; **worked solutions** available below

AA_HL_Test1_integral_calc_SOL_KEY_v1 **worked solutions** for first HL integral calculus test above

## Selected Pages

### Maclaurin series HL (TN) 31 May 2021

HL syllabus content: Maclaurin series; use of simple substitution, products, integration and differentiation to obtain...

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### Integration basics 7 June 2021

When first teaching integration, I think it is helpful to look back at differentiation and try to impress upon students...

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### Mclaurin polynomials applet 13 May 2021

Here is a Geogebra applet which produces and graphs the \(n\)th degree Maclaurin polynomial (up to \(n = 10\)) for an entered...

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### Differential equations HL (TN) 28 February 2021

HL syllabus content: first order differential equations; numerical solution of \(\frac{{{\rm{d}}y}}{{{\rm{d}}x}} = f\left(...

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### Differentiation basics 8 June 2020

The introductory content for differential calculus is not too different between SL and HL; however, SL students do not need...

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### Integration by parts (tutorial) 29 April 2020

In developing a technique for integrating functions it is important to remember the fact that differentiation and integration...

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