# Calculus

** Quick links**:

► downloadable teaching materials for calculus (calculus tests & review quizzes)

#### new __Quiz_HL_calculus_No_GDC_v1__ (and worked solutions) uploaded 4 November 2018

and __Quiz_SL_calculus_No_GDC_v1__ (with answers) uploaded 10 November 2018

#### Syllabus Content - SL & HL

The Calculus Topic is the largest syllabus topic in terms of the **recommended teaching hours** for both SL and HL. In SL, 40 teaching hours are recommended while it is 48 hours for HL. Although there are more hours recommended for teaching calculus in HL than in SL, it is worth noting that the Calculus Topic takes up a larger percentage (≈ 27%) of the total teaching hours for the **SL course** (40 out of a total of 150 hours) than the **HL course** where the Calculus Topic is 48 out of 240 hours (20%). Regardless of the details of recommended teaching hours, it is clear that the Calculus Topic is a very important one for Maths SL and Maths HL students.

The syllabus content that is **in HL but not in SL** are primarily the following items:

♦ informal idea of continuity

♦ derivatives of reciprocal trig functions

♦ derivatives of general exponential and logarithmic functions

♦ derivatives of inverse trig functions

♦ implicit differentiation

♦ related rates of change

♦ area of a region enclosed by a curve and the y-axis

♦ volume of revolution about the y-axis

♦ integration by substitution involving sophisiticated ('non-standard') substitutions

♦ integration by parts

Of these items, it could be argued that the most important in terms of teaching effort will be implicit differentiation, related rates, sophisticated integration by substitution, and integration by parts.

It is also worth noting how the content of the Calculus Topic changed from the previous syllabus (last exams 2013) to the current syllabus (first exams 2014).

► main changes for the **HL core syllabus**:

• differential equations (solution by separation of variables) removed and put in the Calculus Option Topic

• informal idea of continuity added

• oblique asymptotes no longer specifically mentioned

• although previously implied the following are now specifically stated:

- relationship between graphs of *f, f´ *and *f´´*

- calculating total distance travelled (linear motion)

► main changes for the **SL syllabus**:

• limit notation added

• relationship between graphs of *f, f´ *and *f´´* added

• calculating total distance travelled (linear motion) added

• integration by substitution of the form \(\int {f\left( {g\left( x \right)} \right)\,} g'\left( x \right)dx\) added

#### Teaching Approaches

Although the mathematical background of students entering Maths SL or HL can vary greatly, it is fair to say that calculus is one area of mathematics that few students have any significant experience with before entering either course. Many, if not most, students will have had exposure to some of the content in each of the other five syllabus topics (algebra, functions & equations, trigonometry, vectors, statistics & probability). It is also fair to say that gaining a sufficient fluency with many of the concepts and skills in calculus is often more difficult for students compared to concepts and skills in other syllabus topics. Because the content in the Calculus Topic is mostly, if not completely, new to students and because it often takes students a bit more time to adequately absorb the ideas and techniques in calculus it is strongly recommended that a teacher plan substantial **review of calculus material** during the 2nd year of the course.

Although there are several topics in HL that are not in SL (e.g. related rates, integration by parts), I would still recommend that the teaching of the calculus syllabus topic in both SL and HL is approached by dividing the content into three different **teaching units**: **Differential Calculus I - Fundamentals**, **Differential Calculus II - Further Techniques & Applications**, and **Integral Calculus**. It's possible to simply teach the Calculus Topic in two parts - differential and integral calculus, but I feel it's worth breaking up differential calculus into two parts for a few reasons. Perhaps the best reason is that since the topic is very new to students it's helpful to not rush too quickly through initial concepts such as limits of functions (and notation), the limit definition of the derivative (and finding slope from 'first principles'), and the rate of change of a function.

The support and teaching materials on this site are organized into these three teaching units and smaller teaching topics as folllows:

**Differential Calculus I - Fundamentals**

• differentiation basics

• maxima & minima

• tangents & normals

**Differential Calculus II - Further Techniques & Applications**

• further differentiation methods

• optimization

• implicit differentiation & related rates (HL only)

**Integral Calculus**

• integration basics

• integration by substitution

• further integration methods (HL only)

• areas & volumes

• modelling linear motion

go **here** for a set of unit tests on calculus - differential or integral calculus - for both HL and SL

## Selected Pages

### tangents & normals 8 March 2018

Quick links:► downloadable teaching materials for tangents & normals► syllabus content for the Calculus Topic: SL syllabus...

more

### Calculus Tests & Review 29 September 2019

here's an interesting problem (presented in GeoGebra applet below) to challenge your students ... solution will be appearing...

more

### implicit differentiation & related rates (HL) 10 September 2019

Quick links:► downloadable teaching materials for implicit differentiation and related rates► syllabus content for the...

more

### differentiation basics 1 September 2019

Quick links:► downloadable teaching materials for differentiation basics► syllabus content for the Calculus Topic: SL...

more

### optimization 22 April 2019

Quick links:► download teaching materials for optimization► syllabus content for the Calculus Topic: SL syllabus (see...

more

### further differentiation methods 3 April 2019

It seems appropriate to place Gottfried Wilhelm von Leibniz prominently at the top of this page. Along with major contributions...

more