Finding the intersection of two functions: logarithmic functions
Question: There are two functions: \(y = 2+\log_2(23-x) \) \(y = \log_{\sqrt{2}}(x-8)\) Find the \(x\)-coordinate of the intersection of these […]
Category: Mathematics
Solving for the coefficients by using the factor theorem
Question: If the polynomial, \(P(x) = ax^4 + (b-a)x^3 + (1-2ab)x^2 + (ab-10)x + 2ab\), has a factor \(x-2\), then […]
Quadratic equation 2: Solving it with the discriminant
Question: The equation, \(kx^2 -(k+3)x – 1 =0\), which has real coefficients having complex roots, \(a+ib\) and \(a-ib\). In order […]
A problem for the quadratic equation, using the relationship between solutions and coefficients 1
Question: The roots of \(x^2 – 5x + 3 = 0\) are \(\alpha\) and \(\beta\). Another quadratic equation \(x^2 + […]
The bounded solution of a first order ordinary differential equation
Question: Find the bounded solution of the following ordinary differential equation: \[ \frac{dx}{dt} = 2x + \sin t \] for \(-\infty […]
The derivative of parametric equations
Question: \(x\) and \(y\) are given as follows: \[ x = e^t \cos t \] \[ y = e^t \sin […]
What should we do when complex variables involve trigonometric functions?
Question: Define \(z=x+iy\). Prove the following equation \[ \frac{\sin 2x + i\sinh 2y}{\cos 2x + \cosh 2y} = \tan z \] Answer: […]
Equation related to limit: Find the parameters
Question: If the following equation holds, \[ \lim_{x \rightarrow 2}\frac{\sqrt{9+ax}+b}{x-2}=-2 \] Find parameters, \(a\) and \(b\). Answer: Since the extreme […]
How do we find the figure of some of big number?
Question: Knowing that \(\log_{10}2=0.3010\) and \(\log_{10}3=0.4771\), (1) find the number of figures of \(6^{50}\). (2) find at which decimal place […]
Find the extrema: an optimization problem with Lagrange multipliers
Question: By using Lagrange multipliers, find the extrema of \(x^2+2y^2+3z^2\) when \(3x+2y+z=-1\). Answer: This is the exactly the same question […]