MA-C4 Integral calculus-y12€¦ · Web viewThe reverse chain rule cannot be applied to any other...

Year 12 Mathematics Advanced

MA-C4 Integral calculus Unit duration

The topic Calculus involves the study of how things change and provides a framework for developing quantitative models of change and deducing their consequences. It involves the development of two key aspects of calculus, namely differentiation and integration. The study of calculus is important in developing students’ capacity to operate with and model situations involving change, using algebraic and graphical techniques to describe and solve problems and to predict outcomes in fields such as biomathematics, economics, engineering and the construction industry.

5 weeks

Subtopic focus Outcomes

The principal focus of this subtopic is to introduce the anti-derivative or indefinite integral and to develop and apply methods for finding the area under a curve, including the Trapezoidal rule and the definite integral, for a range of functions in a variety of contexts. Students develop their understanding of how integral calculus relates to area under curves and a further understanding of the interconnectedness of topics from across the syllabus. Geometrical representation assists in understanding the development of this topic, but careful sequencing of the ideas is required so that students can see that integration has many applications, not only in mathematics but also in other fields such as the sciences and engineering.

A student: applies calculus techniques to model and solve problems MA12-3 applies the concepts and techniques of indefinite and definite integrals in the

solution of problems MA12-7 chooses and uses appropriate technology effectively in a range of contexts,

models and applies critical thinking to recognise appropriate times for such use MA12-9

constructs arguments to prove and justify results and provides reasoning to support conclusions which are appropriate to the context MA12-10

Prerequisite knowledge Assessment strategies

The material in this topic builds on content from MA-C1 Introduction to calculus and MA-C2 Differential calculus.

Formative assessment: Students will investigate anti-derivatives and integration by considering the reverse of the differentiation process to establish the formal process or rule for each type of question. Challenge students to verbalise what they are doing, using generalisations, at each stage of the process. They should draw their responses as a chain of events. Students should be given the opportunity to demonstrate their understanding of the connections between the approximation methods and the calculus methods of integration.

All outcomes referred to in this unit come from Mathematics Advanced Syllabus© NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2017

© NSW Department of Education, 2019 MA-C4 Integral calculus 1

http://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/stage-6-learning-areas/stage-6-mathematics/mathematics-advanced-2017

Glossary of terms

Term Description

anti-derivative An anti-derivative, primitive or indefinite integral of a function f ( x ) is a function F ( x ) whose derivative is f ( x ), ie

F ' ( x )=f ( x ) .The process of finding the anti-derivative is called integration.

even function Algebraically, a function is even if f (−x )=f ( x ), for all values of x in the domain.An even function has line symmetry about the y-axis.

odd function Algebraically, a function is odd if f (−x)=−f (x), for all values of x in the domain.An odd function has point symmetry about the origin.

Lesson sequence

ContentStudents learn to:

Suggested teaching strategies and resources Date and initial

Comments, feedback, additional resources used

Introducing the anti-derivative

(1 lesson)

C4.1: The anti-derivative define anti-differentiation as the

reverse of differentiation and use the notation ∫ f (x )dx for anti-derivatives or indefinite integrals (ACMMM114, ACMMM115)

recognise that any two anti-derivatives of f (x) differ by a constant

Introducing the anti-derivative Introduce the anti-derivative as the result of reversing

the process of differentiation. It is an expression of x, generally, and is shown as F (x), whereddx (F ( x ) )=f (x ).

The anti-derivation is also known as the indefinite integral or a primitive, and the process of reversing differentiation is known as Integration. The process of integration of a function f (x) is notated by

∫ f ( x ) . dx and is read as “the integration of f (x) with respect to x”.Resource: antiderivative-matching-activity.DOCX


Lesson sequence




Determining simple indefinite integrals

(1 lesson)

establish and use the formula

∫ xndx= 1n+1

xn+1+c , for

n≠−1 (ACMMM116)

recognise and use linearity of anti-differentiation (ACMMM119)

Determining simple indefinite integrals Lead students to the rule for integrating polynomial

terms, by facilitating examples of the form ∫ x3 . dx.

A key question may be “Does ∫ x3 . dx= x4

4?”

Discuss that x4

4 may be a solution but we do not

have enough information to tell, as the constant term would have been eliminated during differentiation. To acknowledge the constant in the integral, usually a c or d is added to the expression,

i.e. ∫ x3 . dx= x4

4+c

Students need to cement their understanding of this concept by being challenged with questions of the form

∫ x2 . dx, ∫ x5 . dx, ∫ 1x2. dx, ∫ 1

x4. dx and

∫√x .dxLinearity of integrals

Establish the following scaling and distributive properties of integrals

∫ kf (x ) . dx=k∫ f ( x ) . dx

and

∫ f ( x )+g ( x ) .dx=∫ f ( x ) . dx+∫ g(x ). dx Students need to apply these properties to solve

integrals of the form

∫5 x3 .dx


Lesson sequence




∫ (x¿¿3+3 x¿¿2+9x ). dx ¿¿

Establishing the reverse chain rule

(2 lessons)

establish and use the formula

∫ f '(x )[ f (x)]ndx= 1n+1 [ f (x) ]n+1+c

where n≠−1 (the reverse chain rule)

determine indefinite integrals of the form ∫ f (ax+b)dx (ACMMM120)

Establishing the reverse chain rule Students need to review the chain rule using

polynomial expressions, i.e. ddx

( [3 x−1 ]4).

Challenge students to verbalise what they are doing, using generalisations, at each stage of the process. Draw their responses as a chain of events.

Highlight the existence of the derivative f ' (x) as a factor of the result of the chain rule for [ f (x )]n.

Challenge students to reverse the process by providing them with results from the chain rule and ask them to complete original derivative statement,

e.g. ddx

(? )=10 x (x2−1)4

Consider the result for ddx

(? )=2 x(x2−1)4. How

does the result above help us identify the answer? And how does this result help us answer

∫2 x (x2−1)4 .dx By using this result, lead students to the generalised

solution to ∫ f '(x )[ f (x)]ndx= 1n+1 [ f (x) ]n+1+c

Resource: reverse-chain-rule-matching-activity.docx

Applying the reverse chain rule to functions with a linear expression of x

Students need to build on the concept of the reverse chain rule for integrals not in the form


Lesson sequence




∫ f '(x )[ f (x)]ndx The reverse chain rule can also be applied to

integrals in the form ∫ f (ax+b ) . dx, where the expression of x is linear. The reverse chain rule cannot be applied to any other expressions of x, i.e. the reverse chain rule cannot be applied to

∫ f (a x2+b ) . dx as the expression of x is not linear.

Challenge students to develop a solution to an integral of the form ∫(3 x−2)

5. dx

Lead students to develop the relationship

∫ f (ax+b ) . dx=F (ax+b )a

+C,

where F (x) is the primitive of f (x).


Lesson sequence




Applying standard integrals

(2 lessons)

establish and use the formulae for the anti-derivatives of sin(ax+b), cos (ax+b) and

sec2(ax+b) establish and use the formulae ∫ exdx=ex+c and

∫ eax+bdx=1aeax+b+c

Review standard derivatives Review the standard derivatives established in

MA-C2 Differential calculus Refer to the 2020 HSC reference sheet produced by

NESA, which can be found under the Assessment and examination materials on the NESA Mathematics Advanced syllabus page

Applying standard integrals Students need to build on their understanding of the

reverse chain rule and apply it to a variety of functions

Lead students to utilise the results on the reference sheet and their understanding of the reverse chain rule to establish

∫sin (ax+b ) .dx=¿ 1acos (ax+b )+c¿

∫cos (ax+b ) . dx=−1asin(ax+b)+c

∫ sec2(ax+b) . dx=1a tan (ax+b)+c

∫ exdx=ex+c

∫ eax+bdx=1aeax+b+c

Integrals resulting in natural logarithms

(1 lesson)

establish and use the formulae

∫ 1xdx=ln∨x∨+c and

∫ f'(x )f (x)

dx= ln∨ f (x)∨+c for

x≠0 , f (x)≠0, respectively

Integrals resulting in natural logarithms Students need to revisit the results for differentiating ln (x) and ln (f ( x )) in MA-C2 Differential calculus and use the results in reverse to establish the integral results

∫ 1xdx=ln|x|+c where x≠0




Lesson sequence




∫ f'(x )f (x)

dx= ln∨ f (x)∨+c wheref (x)≠0

as ln|0| is undefined

Students are able to solve questions of the form

∫ 3x2

x3+2. dx

∫ 2 x−1x2−x

.dx

∫ x3

x4+1. dx

Integrating exponentials of any base

(1 lesson)

establish and use the formulae

∫ ax dx= ax

ln a+c

Integrating exponentials of any base Lead students to establish the formula

∫ ax dx= ax

ln a+c by using the following techniques:

Establish a=e ln a and therefore

ax=(e¿¿ ln a)x=exlna¿

∴ ∫ axdx¿∫exlnadx

¿ exlna

ln a+c

¿ ax

ln a+c


Lesson sequence




Sketching anti-derivatives with slope fields

(1 lesson)

examine families of anti-derivatives of a given function graphically

Sketching anti-derivatives with slope fields Introduce students to slope fields using the sketching

anti-derivatives activity. A slope field shows the family of curves for the anti-derivative.

Key questions: “Why is there more than one possible curve?” and “Why are they vertical translations of each other?”

It is important to establish that integrating alone is not enough to identify the anti-derivative, as there is not enough information to identify the constant term. This can only be established if one point on the curve is known.

It is important to note that Mathematics Extension 1 students will experience slope fields for expressions that may involve x and y values. The curves produced in this instance are not vertical translations of each other. Resource: sketching-anti-derivative-activity.DOCX


Lesson sequence




Integrating to determine a function

(1 lesson)

determine f (x), given f ' (x) and an initial condition f (a)=b in a range of practical and abstract applications including coordinate geometry, business and science

Integrating to determine a function Staff need to build on the key idea that integrating

alone is not enough information to determine an anti-derivative as the constant term is unknown.

The constant term can be determined by substituting in an initial condition f (a)=b, which represents a point (a ,b) on the slope field.

For example,

If f ’ ( x )=3 x3−4 x+1 find f (x) given f (1)=2.

Step 1: Integrate,

f (x)=∫ f ' (x ) . dx

f (x)=∫ 3x3−4 x+1.dxf ( x )=3

4x4−2x2+x+c

Step 2: Substitute in the values of the initial condition to find the constant term

∴2=34×14−2×12+1+c

2=−14

+c

c=2 14

Step 3: Determine the anti-derivative function

f ( x )=34x4−2x2+x+2 1

4


Lesson sequence




The area under a curve and its approximations

(1 lesson)

C4.2: Areas and the definite integral

know that ‘the area under a curve’ refers to the area between a function and the x-axis, bounded by two values of the independent variable and interpret the area under a curve in a variety of contexts AAM

determine the approximate area under a curve using a variety of shapes including squares, rectangles (inner and outer rectangles), triangles or trapezia

o consider functions which cannot be integrated in the scope of this syllabus, for example f (x)=ln x, and explore the effect of increasing the number of shapes used

The area under a curve and its approximations Introduce area under a curve using a practical

context:o Finding the distance travelled from a velocity time

graph.

0 1 2 3 4 5 6 7 8 9 10 1102468

10121416

Velocity-time graph

Time (s)

Velo

city

(m/s

)

o Finding the amount of water wasted from a leaking tap. (Graph water mL/min)

Define the area under a curve as the area between a function and the x-axis, bounded by two values of the independent variable.

Explore various methods of approximating areas for curves that cannot be integrated in the scope of this course. Examine the effect of increasing the number of shapes used to approximate the area using the resources below.

Resources: approximating-areas.DOCX, approximating-areas.XLSX

Trapezoidal Rule use the notation of the definite Trapezoidal Rule


Lesson sequence




(1 lesson)integral ∫

a

b

f (x )dx for the area

under the curve y=f (x ) from x=a to x=b if f (x)≥0

use the Trapezoidal rule to estimate areas under curves AAM

o use geometric arguments (rather than substitution into a given formula) to approximate a definite integral of the form

∫a

b

f (x )dx , where f (x)≥0,

on the interval a≤ x≤b, by dividing the area into a given number of trapezia with equal widths

o demonstrate understanding of the formula:

∫a

b

f (x )dx ≈ b−a2n

[ f (a )+f (b )+2{f (x1 )+…+ f (xn−1 ) }]

where a=x0 and b=xn, and

the values of x0 , x1 , x2 ,…,xn are found by dividing the interval a≤ x≤b into n equal sub-intervals

Define the definite integral ∫a

b

f (x )dx for the area

under the curve y=f (x ) from x=a to x=b if f (x)≥0

Approximate a definite integral by dividing the area into a given number of trapezia with equal widths, calculating the areas of individual trapezia, then the total area.

Lead the discovery of the trapezoidal rule.Suggested method: Find an expression for the area of a number of trapezia, repeat for another number of trapezia, then generalise for n trapezia.

NESA exemplar questions The following table shows the velocity (in metres per

second) of a moving object evaluated at 10-second intervals. Use the trapezoidal rule to obtain an estimate of the distance travelled by the object over the time interval 30≤t ≤70.

Discuss other methods for obtaining the estimate. An object is moving on the x-axis. The graph shows

the velocity, dxdt , of the object as a function of time t .

The coordinates of the points shown on the graph are A (2,1 ) ,B (4,5 ) ,C (5,0 ) and D (6 ,−5 ). The velocity is constant for t ≥6.


Lesson sequence




o Use the trapezoidal rule to estimate the distance travelled between t=0 and t=4 (noting that distance is given on a velocity-time graph by the area under the graph).

o The object is initially at the origin. When is the displacement of the object decreasing?

o Estimate the time at which the object returns to the origin. Justify your answer.

o Sketch the displacement x as a function of time.

Integrals using the area of basic shapes (1 lesson)

use geometric ideas to find the

definite integral ∫a

b

f (x )dx

where f (x) is positive throughout an interval a≤ x≤b and the shape of f (x) allows such calculations, for example when f (x) is a straight line in the interval or f (x) is a semicircle in the interval AAM

Integrals using the area of basic shapes Students apply the area of basic shapes to find the

definite integral ∫a

b

f (x )dx where f (x) is positive

throughout an interval a≤ x≤b.

Examples in desmos:o Triangle

o Trapezium

o Circle

o Semi-circle

NESA exemplar questionsGraphs A and B shown below represent the functions y=f ( x ) and y=g ( x ) respectively.


https://www.desmos.com/calculator/80tr2pv5a1

https://www.desmos.com/calculator/9g7okfagav

https://www.desmos.com/calculator/xwabx3jxin

https://www.desmos.com/calculator/c4sukxq9lm

Lesson sequence




Graph A

Graph B

Evaluate the integral ∫0

10

f ( x )dx .

Use the formula for the area of a circle to find

∫0

3

g ( x )dx .


Lesson sequence




Signed areas

(1 lesson)

understand the relationship of position to signed areas, namely that the signed area above the horizontal axis is positive and the signed area below the horizontal axis is negative

using technology or otherwise, investigate the link between the anti-derivative and the area under a curve

o interpret ∫a

b

f (x)dx as a sum

of signed areas (ACMMM127)

o understand the concept of the signed area function

F ( x )=∫a

x

f ( t )dt

(ACMMM129)

Signed areas Students investigate definite integrals to develop an

understanding of signed areas.Students could

o Evaluate ∫1

7

(2x−6)dx using an integral calculator

such as wolfram alpha. Enter: integrate 2x-6 from 1 to 7

o Examine the individual areas using desmos. Enter the functions and bounds

o Interpret the result: ∫a

b

f (x)dx as the sum of signed

areas, area above the horizontal axis is positive and the area below the horizontal axis is negative.

o Note: The area between f (x)=2x−6 and the x-axis bounded by x=1 and 7 is 20units2, the corresponding definite integral equals 12.

Define the area under a curve as:

o ∫a

b

f ( x )dx where f(x) is positive for a≤ x≤b.

o |∫a

b

f (x )dx| where f(x) is negative for a≤ x≤b.


https://www.desmos.com/calculator/be5ne9vwi8

https://www.wolframalpha.com/calculators/integral-calculator/

Lesson sequence




Definite integrals

(1 lesson)

● use the formula

∫a

b

f (x)dx=F (b)−F (a) ,

where F (x) is the anti-derivative

of f (x) , to calculate definite integrals (ACMMM131) AAM o understand and use the

Fundamental Theorem of

Calculus, F ' ( x )= ddx

¿ and

illustrate its proof geometrically (ACMMM130)

o calculate total change by integrating instantaneous rate of change

Definite Integrals: Demonstration of increasing the number of shapes

as a link to integration (Geogebra)

Prove ∫a

b

f ( x )dx=F (b )−F (a ) using an

understanding of the fundamental theorem of calculus. NESA’s Mathematics Advanced Year 12 topic guidance for calculus contains two approaches.

Use the formula ∫a

b

f (x)dx=F (b)−F (a) , where

F (x) is the anti-derivative of f (x) , to calculate definite integrals.

Calculate total change by integrating instantaneous rate of change. Examples:

o Given a function of velocity, dxdt

=2x+1. Integrate

the instantaneous rate of change (velocity) to find total displacement for a given time period..

o Given a function of accelerationdvdt

. Integrate the

instantaneous rate of change to find total change in velocity.




https://www.geogebra.org/m/nmw6Dhdk

https://www.geogebra.org/m/nmw6Dhdk

Lesson sequence




Area under a curve

(1 lesson)

● calculate the area under a curve (ACMMM132) o use symmetry properties of

even and odd functions to simplify calculations of area

o recognise and use the additivity and linearity of definite integrals (ACMMM128)

Area under a curve: Apply definite integrals to calculate the area under

curves including:o Area bounded by the curve and the x-axis

o Area between set bounds (e.g. x=1 and x=4)

o Area between set bounds where f(x) changes sign.

Teacher to model dissecting the definite integral where f(x) changes sign.

The teacher can use a desmos resource to construct graphs for demonstrations of finding the area under a curve.

Properties useful for calculations of area:o Even and odd functions.

o Additivity of definite integrals:

∫a

b

f (x )dx=∫a

c

f (x )dx+∫c

b

f (x ) dx

Visual representation of the sum of additivity

∫a

b

f (x )dx=−∫b

a

f (x )dx

∫a

a

f (x )dx=0

o Linearity of definite integrals:

∫a

b

f (x )+g ( x )dx=∫a

b

f (x)dx+∫a

b

g (x)dx

∫a

b

f (x )−g ( x )dx=∫a

b

f ( x )dx−∫a

b

g (x)dx


https://www.geogebra.org/m/s2asvWnh

https://www.desmos.com/calculator/be5ne9vwi8

Lesson sequence




∫a

b

kf (x)=k∫a

b

f ( x)dx

NESA exemplar questions Find the area bounded by the graph of y=3 x2+6,

the x-axis, and the lines x=−2 and x=2.

Show that ∫−2

2

x3dx=0.

Explain why this is not representative of the area bounded by the graph of y=x3, thex-axis, and the lines x=−2 and x=2.

Area between two curves

(1 lesson)

● calculate areas between curves determined by any functions within the scope of this syllabus (ACMMM134) AAM

Area between two curves: Apply definite integrals to calculate the area between

two curves including:o Area between two curves over set bounds

Visual representation of area between 2 curveso Area between two curves (bounds defined by the

points of intersection)Visual representation of subtracting the areas.

o Area between two curves over set bounds where f(x) and g(x) intersect. Teacher to model dissecting the definite integral where f(x) and g(x) intersect. Visual representation of area between 2 curvesNote: equations need to be adjusted to examine when curves intersect.

The teacher can also use a desmos resource to construct graphs for demonstrations of finding the area between curves.

NESA exemplar questions


https://www.desmos.com/calculator/x3nts4fqb9

https://www.geogebra.org/m/MpF9wtfA

https://www.geogebra.org/m/YCt3bvBE

https://www.geogebra.org/m/MpF9wtfA

Lesson sequence




Find the area bounded by the line y=5 and the curve y=x2−4.

Sketch the region bounded by the curve y=x2 and the lines y=4 , y=9. Evaluate the area of this region.

Problem solving

(1 lesson)

integrate functions and find indefinite or definite integrals and apply this technique to solving practical problems AAM

Problem solving Trapezoidal rule: See NESA exemplar questions in

this unit. Economics: Consumer and producer surplus Average value of a function Work Braking Distance

Resource: integral-calculus-applications.DOCX

Reflection and evaluationPlease include feedback about the engagement of the students and the difficulty of the content included in this section. You may also refer to the sequencing of the lessons and the placement of the topic within the scope and sequence. All ICT, literacy, numeracy and group activities should be recorded in the ‘Comments, feedback, additional resources used’ section.


MA-C4 Integral calculus-y12€¦ · Web viewThe reverse chain rule cannot be applied to any other...

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