# 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.

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: 1st 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 & 2nd 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

free

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

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

### Integration basics7 June 2021

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

### Mclaurin polynomials applet13 May 2021

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

### 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(...
more

### Differentiation basics8 June 2020

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

### 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...
more

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