91Ó°ÊÓ

The demand function for a product is modeled by \(p=75-0.25 x\) (a) If \(x\) changes from 7 to 8 , what is the corresponding change in \(p\) ? Compare the values of \(\Delta p\) and \(d p\). (b) Repeat part (a) when \(x\) changes from 70 to 71 units.

Find \(\lim _{x \rightarrow \infty} h(x)\), if possible. \(f(x)=5 x^{3}-3\) (a) \(h(x)=\frac{f(x)}{x^{2}}\) (b) \(h(x)=\frac{f(x)}{x^{3}}\) (c) \(h(x)=\frac{f(x)}{x^{4}}\)

You are designing a soft drink container that has the shape of a right circular cylinder. The container is supposed to hold 12 fluid ounces (1 fluid ounce is approximately \(1.80469\) cubic inches). Find the dimensions that will use a minimum amount of construction material.

Find the price elasticity of demand for the demand function at the indicated \(x\) -value. Is the demand elastic, inelastic, or of unit elasticity at the indicated \(x\) -value? Use a graphing utility to graph the revenue function, and identify the intervals of elasticity and inelasticity. \(p=600-5 x \quad x=30\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=1-x^{2 / 3}\)

A state game commission introduces 50 deer into newly acquired state game lands. The population \(N\) of the herd can be modeled by \(N=\frac{10(5+3 t)}{1+0.04 t}\) where \(t\) is the time in years. Use differentials to approximate the change in the herd size from \(t=5\) to \(t=6\).

Find the price elasticity of demand for the demand function at the indicated \(x\) -value. Is the demand elastic, inelastic, or of unit elasticity at the indicated \(x\) -value? Use a graphing utility to graph the revenue function, and identify the intervals of elasticity and inelasticity. \(p=400-3 x \quad x=20\)

Find \(\lim _{x \rightarrow \infty} h(x)\), if possible. \(f(x)=3 x^{2}+7\) (a) \(h(x)=\frac{f(x)}{x}\) (b) \(h(x)=\frac{f(x)}{x^{2}}\) (c) \(h(x)=\frac{f(x)}{x^{3}}\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=(1-x)^{2 / 3}\)

Find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{3}-1}\) (b) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}-1}\) (c) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x-1}\)