Finding the extrema; an optimization problem you can see math class
Question: Find the extrema of \(x^2+2y^2+3z^2\) under the condition that \(3x+2y+z=-1\). Answer: From the condition, solve for \(z\). \[ z […]
Category: Mathematics
Residue theorem and integral of a complex-valued function: The climax of complex variables
Question: Find the value of \[ \int_C \frac{dz}{z^2+2iz-4} \] The contour, \(C\), is a circle that has center \(z=1\) and […]
The basic of simultaneous ordinary differential equation: Easy one
Question: Solve for the following simultaneous differential equations: \[ (1): \frac{dx}{dt}+ay = e^{bt} \] \[ (2): \frac{dy}{dt}-ax = 0 […]
The integral of \(\frac{1}{1+x^6}\) using Beta and Gamma functions
Question: Find the value of integral \[ \int^{\infty}_{-\infty}\frac{dx}{x^6+1} \] Answer: The integrand is even function, so we can reduce the […]
Differentiation with trigonometric and logarithmic functions
Question: Here is a function of \(x\). \[ y=\alpha \cos x + 2 – \cos x \log\frac{1+\cos x}{1-\cos x} \] \(\alpha\) […]
What we can derive from the trigonometric addition theorem
Question: Show that \(\frac{\sin(\alpha + \beta)\sin(\alpha – \beta)}{\sin\alpha – \sin\beta}=\sin\alpha + \sin\beta\). Answer: Use \(\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm […]
Can you relate \(\tan \theta \) to \(\sin 2\theta\) and \(\cos 2\theta\)?
Question: When \(\tan\theta = -2\), find \(\cos 2\theta\) and \(\sin 2\theta\). Answer: Maybe, you don’t believe that you can derive […]
You will be cool if you handle with tangent!!
Question: Find \((\tan 15^o + \sqrt{3})^2\) without a calculator. Answer: It is not easy to obtain \(\tan 15^o\) without a […]
Only this makes you understand calculus!
Speaking of calculus, most of math and science students learn it at high school, I guess, but how many people, […]
The previous question and its answer: What is the area covered with squares?
Question: What is the area covered by three squares? Answer: You can use simultaneous equations to solve it, but there […]