I calculated \(\displaystyle{{\sin{{75}}}^{{\circ}}}\) as \(\displaystyle{\frac{{{1}}}{{{2}\sqrt{{{2}}}}}}+{\frac{{\sqrt{{{3}}}}}{{{2}\sqrt{{{2}}}}}}\), but

What is the average value of the function $f\left(x\right)=\cos \left(8x\right)$ on the interval ${\displaystyle [0,\frac{\pi }{2}]}$?

Find the values of b such that the function has the given maximum value.

Maximum value: 62

f(x) = −x2 + bx − 19 

b =            (smaller value)

b =           (larger value)

Solve algebraically:

${\displaystyle \underset{x\to 3}{lim}\frac{3-x}{5-\sqrt{x^2+16}}}$

• At the beginning of the year, there are 7650 individuals in a population of beavers whose per capita rate of for the year is 0.18. What is its population growth rate at the end of the year? 

How to solve $X_{xx}=(1+\delta \left\{\left(x\right)\right\})X$
I am trying to find the vibration modes of a string that has a uniform mass density, plus some point mass somewhere attached to it, modelled by an additional Dirac delta function in the mass density. The wave equation is of the form
$u_{xx}=(1+\delta \left(x\right))u_{tt}$
where u is the deformation, and ${\displaystyle (1+\delta \left(x\right))}$ the mass density. After separation of variables we find
$X_{xx}=(1+\delta \left(x\right))X$
where X is the spatial part of the solution. Is there any analytical solution for X?