Arc Length (1 Viewer)

[Sober]

Does anybody know why arc length is not dealt with in 4 unit maths? It is not very difficult at all to deal with and is included in most non-school calculus books.

 

[buchanan]

It used to be in the course, but was removed.

The current 1-year 4 unit course used to be a 2-year course (Level 1). Arc length was in Level 1 and used to be taught in high schools in NSW. Other countries' syllabuses for high school maths also have arc length.

Level 1 was watered down to the 1-year course in 1981.

Many people would like the 2-year course to return.

The only excuse for why the harder Level 1 course can't be taught in high schools again, is teacher incompetence. And that's a very poor excuse.<!-- google_ad_section_end -->

Here are the topics previously in the course which have been removed, but many people want back:

Leaving Certificate (1916-1966)

• 3rd derivative test for inflections
• Substitution x=acos<SUP>2</SUP>θ+bsin<SUP>2</SUP>θ for ∫√((x-a)(x-b))dx, ∫(1/√((x-a)(x-b)))dx, ∫√((x-a)/(b-x))dx
• Euler's Formula
• Integration as a summation
• Determinants and solutions of equations
• Convergence and divergence of infinite series
• Logarithmic and Exponential series and Euler's constant
• Binomial series for fractional or negative index.

Level 1 (1966-1980)

• Euclidean algorithm
• Proof of the fundamental theorem of arithmetic
• Determinants and the solution of equations and area of triangle
• Geometry of matrices
• Geometrical transformations using matrices
• Algebra of matrices
• Rolle's theorem and mean value theorem
• Integration as summation
• Euler's formula
• Length of arc
• Group theory, isomorphism
• Applications of matrices to geometry and probability
• Work, Kinetic Energy, Potential energy
• Convergence and divergence of infinite series
• Riemann Zeta function
• Logarithmic and exponential series
• Series for sinx, cosx, tan<SUP>-1</SUP>x
• Taylor's series.

Level 2F (1966-1982)

• Mid-ordinate rule
• Change of coordinate systems - transformations
• Analytical geometry in three dimensions

Quite a lot of this stuff which used to be taught in school, is now postponed till university in Australia. However, other countries are still teaching it in school. So Australia is seen as a bit behind the rest of the world, particularly in regard to school maths and 1st year uni maths.

If we bring these topics back and make 4 unit a 2-year course again, we won't have to postpone them till university, and perhaps the uni's can then focus on what they should be doing and not have to teach what should have been done in school, as they currently do.

I reckon you could also add surface area of revolution to the high school maths syllabuses as well as arc length.

Here are some past HSC questions on arc length:

• 1967 Q8(iv)
• 1970 Q3(i)
• 1972 Q7(iii)
• 1974 Q5(i)
• 1977 Q8(ii)
• 1979 Q7(iii)
• 1984 Q2(a) (note in 1984 the formula was given)

available in the files

http://www.boredofstudies.org/mirror/4u/hsc1967-1974.pdf
http://www.boredofstudies.org/mirror/4u/hsc1975-1980.pdf
http://www.boredofstudies.org/mirror/4u/hsc1981-1989.pdf

 

[Affinity]

Here are the topics previously in the course which have been removed, but many people want back:

It will be unpopular though.

 

[SeDaTeD]

And a lot of functions you end up with by using the formula can't really be integrated using 4unit techniques.

 

[buchanan]

Trying to find the perimeter of an ellipse will lead to non-elementary functions.

However, my favourite question on arc-length is Coroneos 3F Q11:

Prove that d((1/2)t(1+t<SUP>2</SUP>)<SUP>1/2</SUP>+(1/2)ln(t+(1+t<SUP>2</SUP>)<SUP>1/2</SUP>))/dt=(1+t<SUP>2</SUP>)<SUP>1/2</SUP>. Find the arc length of the parabola x=2at, y=at<SUP>2</SUP> between the origin and the point (2a,a).

(Ans: a(√2+ln(1+√2)))

BTW, my favourite question on surface area of revolution it to prove the surface area of a sphere of radius r is 4πr<SUP>2</SUP>.

 

[XcarvengerX]

Convergence and divergence of infinite series

Love to learn that topic. It is very interesting.

 

[SeDaTeD]

You probably won't get to that until first or second year uni, when you study analysis. Pretty much the only series studied in detail for the HSC are geometric and arithmetic series.

 

[buchanan]

Delaying stuff which you should be doing at school until uni is part of the problem.

Quite a lot of this stuff which used to be taught in school, is now postponed till university in Australia. However, other countries are still teaching it in school. So Australia is seen as a bit behind the rest of the world, particularly in regard to school maths and 1st year uni maths.

If we bring these topics back and make 4 unit a 2-year course again, we won't have to postpone them till university, and perhaps the uni's can then focus on what they should be doing and not have to teach what should have been done in school, as they currently do.

 

[locked.on]

It used to be in the course, but was removed.

available in the files

http://www.boredofstudies.org/mirror/4u/hsc1967-1974.pdf
http://www.boredofstudies.org/mirror/4u/hsc1975-1980.pdf
http://www.boredofstudies.org/mirror/4u/hsc1981-1989.pdf

Do you happen to have worked solutions to these papers as well?