Find the length of the loop of the curve given by $x=6t-2t^3$, $y=6t^2$.

Find the exact length of the curve $y=\frac{x^3}{3}+\frac{1}{4x}$ for $1\le x\le 3$.

Let $f(x,y)=x^2+3y^2$. Find the maximum rate of change of $f$ at the point $(1,2)$ and the direction in which it occurs.

Find the exact length of the curve $y=2\surd 3x^{\frac{3}{2}}$ for $0\le x\le 6$.

Find the global maximum and minimum values of the function $f(x,y)=e^{-xy}$ on the region defined by $x^2+4y^2\le 1$.

Use symmetry to evaluate the double integral of $f(x,y)=xy$ over the region $R$, where $R$ is the disk $x^2+y^2\le 4$.

Find the exact length of the curve $y=\ln (\sec (x\left)\right)$ for $0\le x\le \frac{\pi }{6}$.

Which of the following points is a location where the function $f\left(x\right)=\frac{1}{x-2}$ is discontinuous?

Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is

f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0

where ln x has zero mean and standard deviation σ > 0.

e. For what value of σ > 0 in part (d) does ƒ(x*) have a minimum?