91Ó°ÊÓ

Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. $$f(x)=\frac{x^{2}-4}{x+3}$$

An open-top cylindrical container is to have a volume of \(400 \mathrm{cm}^{2} .\) What dimensions (radius and height) will minimize the surface area?

The postal service places a limit of 84 in. on the combined length and girth of (distance around) a package to be sent parcel post. What dimensions of a rectangular box with square cross-section will contain the largest volume that can be mailed? (Hint: There are two different girths.)

Sketch a graph that possesses the characteristics listed. Answers may vary. \(f\) is concave up at \((1,-3),\) concave down at (8,7) and has an inflection point at (5,4)

Body surface area. Certain chemotherapy dosages depend on a patient's surface area. According to the Gehan and George model, $$S=0.02235 h^{0.42246} w^{0.51456} $$where \(h\) is the patient's height in centimeters, \(w\) is his or her weight in kilograms, and S is the approximation to his or her surlace area in square meters. (Source: www.halls.md.)Joanne is \(160 \mathrm{cm}\) tall and weighs \(60 \mathrm{kg}\). Use a differential to cstimate how much her surface area changes after her weight decreases by \(1 \mathrm{kg}\).

Healing wound. The circular area of a healing wound is given by \(A=\pi r^{2},\) where \(r\) is the radius, in centimeters. By approximately how much does the area decrease when the radius is decreased from \(2 \mathrm{cm}\) to \(1.9 \mathrm{cm} ?\) Use 3.14 for \(\pi\)

A power line is to be constructed from a power station at point \(A\) to an island at point \(C,\) which is 1 mi directly out in the water from a point \(B\) on the shore. Point \(B\) is 4 mi downshore from the power station at \(A\). It costs 5000 dollars per mile to lay the power line under water and 3000 dollars per mile to lay the line under ground. At what point \(S\) downshore from \(A\) should the line come to the shore in order to minimize cost? Note that \(S\) could very well be \(B\) or \(A\). (Hint: The length of \(C S \text { is } \sqrt{1+x^{2}} .)\)

It is known that homing pigeons tend to avoid flying over water in the daytime, perhaps because the downdrafts of air over water make flying difficult. Suppose that a homing pigeon is released on an island at point \(C,\) which is \(3 \mathrm{mi}\) directly out in the water from a point \(B\) on shore. Point \(B\) is 8 mi downshore from the pigeon's home loft at point \(A\). Assume that a pigeon flying over water uses energy at a rate 1.28 times the rate over land. Toward what point \(S\) downshore from \(A\) should the pigeon fly in order to minimize the total energy required to get to the home loft at \(A\) ? Assume that Total energy \(=\) (Energy rate over water) \(\cdot\) (Distance over water) \(+\) (Energy rate over land) \(\cdot\) (Distance over land).

Determine a rational function that meets the given conditions, and sketch its graph. The function \(f\) has a vertical asymptote at \(x=0, a\) horizontal asymptote at \(y=3,\) and \(f(1)=2\).

Marginal average cost. In Section \(1.6,\) we defined the average cost of producing \(x\) units of a product in terms of the total cost \(C(x)\) by \(A(x)=C(x) / x .\) Find a general expression for marginal average cost, \(A^{\prime}(x)\)