u/Fourierseriesagain
A difficult example involving the reverse process of differentiation
reddit.com
u/Fourierseriesagain — 6 days ago
A Special Integrand and Trigonometric Substitution
u/Fourierseriesagain — 9 days ago
▲ 5 r/askmath
The Newton Raphson Method Revisited
Hi,
The classical Newton-Raphson Method is a standard numerical method for solving certain equations. I have tried to reformulate the Newton-Raphson Method so that it is a direct corollary of the Fixed Point Theorem:
Is the above formulation a special case of some general result?
u/Fourierseriesagain — 11 days ago
▲ 1 r/SGExams
Complex Numbers for Further Maths and Engineering Maths
Hi,
Please find some facts on complex numbers related to H2 Further Maths and University Engineering Maths.
- Recall that arg(z) denotes the principal argument of a given nonzero complex number z (i.e. -pi < arg(z) <= pi).
(a) It is not correct to claim that arg(z_1 z_2) is identically equal to arg(z_1)+arg(z_2).
Example. arg(1) is not equal to arg(-1)+arg(-1).
Similarly, arg(z_1 / z_2) is not identically equal to arg(z_1) - arg(z_2).
(b) Let alpha and beta be real numbers. We cannot apply a law of indices to derive the result e^(i alpha) * e^(i beta) = e^(i*(alpha + beta)). In fact,
e^(i alpha) * e^(i beta)
=(cos alpha + i sin alpha)*(cos beta + i sin beta)
=(cos alpha cos beta-sin alpha sin beta)+i(sin alpha cos beta + cos alpha sin beta)
=cos(alpha+beta) + i sin (alpha+beta)
=e^(i*(alpha+beta)).
u/Fourierseriesagain — 12 days ago
An Example on Functions and Calculus
u/Fourierseriesagain — 11 days ago
▲ 0 r/SGExams
A question on functions and calculus
Hi,
Please find the following exercise on functions and calculus.
Exercise. A curve C has equation x^2-xy+y^2=3.
(i) Use an algebraic method to determine the range of values of x.
(ii) Determine the range of values of y.
(iii) Sketch the curve C.
(iv) Determine the area of the region enclosed by C.
u/Fourierseriesagain — 13 days ago
▲ 1 r/SGExams
Some short questions on vectors
Hi,
Please find some short questions on vectors.
- Let A, B and C be three non-collinear points. For example, you may begin with A=(1,1,1), B=(1,2,3) and C=(0,1,0).
(a) Let s=(1/2)*perimeter of triangle ABC. Use the formula sqrt{s(s-a)(s-b)(s-c)} to calculate the area of triangle ABC.
(b) Use vector product to compute the area of triangle ABC.
(c) Use scalar product to compute the area of triangle ABC.
(d) Determine a cartesian equation of the plane ABC.
(e) Determine a parametric form of the equation of the plane ABC.
u/Fourierseriesagain — 13 days ago
▲ 2 r/SGExams
A summary of trigonometry for Add Maths students
Hi,
Please find a summary of trigonometry for Add Maths students.
- Definition of 6 trigonometric functions (i.e. sin theta = y/r, cos theta = x/r, tan theta = sin theta / cos theta and so on). Periodic properties of such functions.
Remarks. (a) It would be good not to introduce identities like sin(180 deg - A)=sin A, cos(180 deg-A)=-cos A etc.
(b) Basic angle is meaningless if theta is a multiple of 90 degrees (i.e. Pi/2 radians).
- Graphs of y=a sin bx+c, y=a cos bx+c and y=a tan bx+c.
3(a) sin^2 A + cos^2 A = 1.
(b) 1+tan^2 A = sec^2 A.
(c) cot^2 A + 1 = cosec^2 A.
4(a) sin(A+B) = sin A cos B+cos A sin B.
(b) sin(A-B) = sin A cos B-cos A sin B.
(c) cos(A+B) = cos A cos B-sin A sin B.
(d) cos(A-B)=cos A cos B+sin A sin B.
(e) tan(A+B)=(tan A + tan B)/(1-tan A tan B ).
(f) tan(A-B)=(tan A - tan B)/(1+tan A tan B ).
Remark. Formulas (e) and (f) fail if cos A cos B is zero.
Trigonometric equations.
Three inverse trigonometric functions.
R-formulas.
u/Fourierseriesagain — 14 days ago
Going beyond Pythagoras' Theorem and its application to arc length
Hi,
I have been trying to generate parametric curve x=f(t) and y=g(t) (a <= t <= b) so that the arc length of this curve can be calculated using school calculus.
Although I have solved the above problem for certain polynomial functions f(t) and g(t), my method is not elegant (see attached document). I am keen to know more about such identities. Thank you.
u/Fourierseriesagain — 14 days ago
▲ 0 r/SGExams
Hi H2 Maths students,
Please find some questions on Maclaurin series:
Use MF27 to obtain the Maclaurin expansion of (ln(1-x))/(1-x) up to and including the term in x^4. Of course, no calculus is required here.
Use calculus to obtain the Maclaurin expansion of (ln(1-x))/(1-x) up to and including the term in x^4.
Determine the Maclaurin series of sin(5*sin^{-1} x) up to and including the term in x^5.
u/Fourierseriesagain — 15 days ago
u/Fourierseriesagain — 15 days ago
u/Fourierseriesagain — 16 days ago
▲ 2 r/SGExams
Hi H2 Maths students,
The question is from 2023 H2 Maths paper 2. Students are required to determine the exact area between the parametric curve C, the x-axis and the line x=21.
Although the above question has been highlighted as "not in the syllabus", it is still possible to compute the required area via the cartesian equation of C.
u/Fourierseriesagain — 17 days ago
u/Fourierseriesagain — 17 days ago
▲ 26 r/SGExams
Hi H2 Maths students,
There are some hidden facts in the formula list MF27.
- Using the following identities cos 2A=2cos^2 A-1 = 1-2 sin^2 A, we get cos^2 A + sin^2 A = 1.
- Using the Binomial Series with x=-r and n=-1, we get
1+r+r^2+r^3+... = 1/(1-r) for -1 < r < 1.
- Using the above infinite geometric series, we obtain
1+r+r^2+r^3+...+r^{n-1} = (1+r+r^2+...) - (r^n+r^{n+1}+...) = (1-r^n)/(1-r). ---(*)
Remark. The identity (*) is valid for all r not equal to 1.
u/Fourierseriesagain — 17 days ago