A certain type of star emits energy according to the formula $E=150\cdot {10}^{0.5M}$, where $E$ is the energy in joules and $M$ is the magnitude of the star's brightness. Calculate the energy emitted by stars of magnitude $2$, $4$, $6$, and $8$. Then, plot the points on the graph and connect them through a smooth curve.

Determine the equation of the normal line to the following curve at the given point:

$x^3+y^3=8xy$; $(4,4)$

Find the value of $h^{\prime }\left(e\right)$ where $h\left(x\right)={\left(3x\right)}^{\frac{1}{3x}}$, and $e$ is the base of the natural logarithm.

A square's sides are shrinking at a constant rate of $2\text{ cm/s}$. What is the rate at which the length of the square's diagonal is changing?

The length of a rectangular prism is measured to be $120\pm 0.5\text{ cm}$, width $80\text{ cm}$, and height $60\text{ cm}$. Estimate the percentage error in the volume calculation based on the uncertainty in the length measurement.

Find the linear approximation of the function $g(x)$ for small values of $x$ using the formula $(1 + x)^k ≈ 1 + kx$.

$g(x)=(1+x)^4$

Consider a cuboidal 8 m³ container with a square base and no top: each side of the base measures a, and the height of the container is b. Sketch the graph of the function that depicts the surface area of the container S(a) for a > 0.

Using the graph, estimate the value of a that minimizes the surface area, and round your answer to 2 decimal places.

Find all critical points and domain endpoints for the function $y=x^3-192\sqrt{x}$.

Consider the quartic function $h\left(x\right)$ with its second derivative given by $h^{\prime \prime }\left(x\right)=(x+6)(x-5)$. Find the $x$-values where the graph of $h\left(x\right)$ has an inflection point.

Determine the points where the function $h$ has local maxima or minima given $h^{\prime }\left(x\right)=(x-15)(x+21)(x-13)$.

Calculate the critical points for the function $p\left(x\right)=2x^4-16x^2+32$ on the interval $[-2,3]$. Identify the absolute maximum and minimum values.

A small bakery makes a profit $P(c)$ in dollars from selling $c$ cakes per day according to the formula $P(c) = c(40 - c) - 80$. Considering that $P$ is a continuous function, how many cakes should the bakery sell per day to maximize its profit?