Chapter 3

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ f(x)=-2 x^{3}+3 x^{2}+12 x-5 $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ f(x)=x^{2}-6 x+1 $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ f(x)=3 x^{3}-4 x^{2}-12 x+17 $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ f(x)=x^{3}-3 x^{2}+2 $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ f(t)=3 t^{5}-20 t^{3} $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ f(x)=\frac{x^{2}+3}{x-1} $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ g(t)=\frac{t^{2}}{t+1} $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ G(x)=(2 x-1)^{2}(x-3)^{3} $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ F(x)=2 x+\frac{8}{x}+2 $$

In Exercises 1 through 10 , determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical tangents. $$ f(x)=\frac{1}{x^{3}}+\frac{2}{x^{2}}+\frac{1}{x} $$

In Exercises 17 through 20, sketch the graph of a function \(f\) that has all the given properties. a. \(f^{\prime}(x)>0\) when \(x<0\) and when \(x>5\) b. \(f^{\prime}(x)<0\) when \(00\) when \(-62\) d. \(f^{\prime \prime}(x)<0\) when \(x<-6\) and when \(-3

In Exercises 17 through 20, sketch the graph of a function \(f\) that has all the given properties. a. \(f^{\prime}(x)>0\) when \(x<-2\) and when \(-23\) c. \(f^{\prime}(-2)=0\) and \(f^{\prime}(3)=0\)

In Exercises 17 through 20, sketch the graph of a function \(f\) that has all the given properties. a. \(f^{\prime}(x)>0\) when \(12\) c. \(f^{\prime \prime}(x)>0\) for \(x<2\) and for \(x>2\) d. \(f^{\prime}(1)=0\) and \(f^{\prime}(2)\) is undefined.

In Exercises 17 through 20, sketch the graph of a function \(f\) that has all the given properties. a. \(f^{\prime}(x)>0\) when \(x<1\) b. \(f^{\prime}(x)<0\) when \(x>1\) c. \(f^{\prime \prime}(x)>0\) when \(x<1\) and when \(x>1\) d. \(f^{\prime}(1)\) is undefined.

In Exercises 21 through 24 , find all critical numbers for the given function \(f(x)\) and use the second derivative test to determine which (if any) critical points are relative maxima or relative minima. $$ f(x)=-2 x^{3}+3 x^{2}+12 x-5 $$

In Exercises 21 through 24 , find all critical numbers for the given function \(f(x)\) and use the second derivative test to determine which (if any) critical points are relative maxima or relative minima. $$ f(x)=x(2 x-3)^{2} $$

In Exercises 21 through 24 , find all critical numbers for the given function \(f(x)\) and use the second derivative test to determine which (if any) critical points are relative maxima or relative minima. $$ f(x)=\frac{x^{2}}{x+1} $$

In Exercises 21 through 24 , find all critical numbers for the given function \(f(x)\) and use the second derivative test to determine which (if any) critical points are relative maxima or relative minima. $$ f(x)=\frac{1}{x}-\frac{1}{x+3} $$

In Exercises 25 through 28, find the absolute maximum and the absolute minimum values (if any) of the given function on the specified interval. $$ f(x)=-2 x^{3}+3 x^{2}+12 x-5 ;-3 \leq x \leq 3 $$

In Exercises 25 through 28, find the absolute maximum and the absolute minimum values (if any) of the given function on the specified interval. $$ f(t)=-3 t^{4}+8 t^{3}-10 ; 0 \leq t \leq 3 $$

In Exercises 25 through 28, find the absolute maximum and the absolute minimum values (if any) of the given function on the specified interval. $$ g(s)=\frac{s^{2}}{s+1} ;-\frac{1}{2} \leq s \leq 1 $$

The first derivative of a certain function is \(f^{\prime}(x)=x(x-1)^{2} .\) a. On what intervals is \(f\) increasing? Decreasing? b. On what intervals is the graph of \(f\) concave up? Concave down? c. Find the \(x\) coordinates of the relative extrema and inflection points of \(f\). d. Sketch a possible graph of \(f(x)\).

The first derivative of a certain function is \(f^{\prime}(x)=x^{2}(5-x)\). a. On what intervals is \(f\) increasing? Decreasing? b. On what intervals is the graph of \(f\) concave up? Concave down? c. Find the \(x\) coordinates of the relative extrema and inflection points of \(f\). d. Sketch a possible graph of \(f(x)\).

A manufacturer can produce sunglasses at a cost of \(\$ 5\) apiece and estimates that if they are sold for \(x\) dollars apiece, consumers will buy \(100(20-x)\) sunglasses a day. At what price should the manufacturer sell the sunglasses to maximize profit?

A box with a rectangular base is to be constructed of material costing \(\$ 2 / \mathrm{in} .{ }^{2}\) for the sides and bottom and \(\$ 3 / \mathrm{in} .{ }^{2}\) for the top. If the box is to have volume 1,215 in. \({ }^{3}\) and the length of its base is to be twice its width, what dimensions of the box will minimize its cost of construction? What is the minimal cost?

A cylindrical container with no top is to be constructed for a fixed amount of money. The cost of the material used for the bottom is 3 cents per square inch, and the cost of the material used for the curved side is 2 cents per square inch. Use calculus to derive a simple relationship between the radius and height of the container having the greatest volume.

Bernardo is a real estate developer. He estimates that if 60 luxury houses are built in a certain area, the average profit will be \(\$ 47,500\) per house. The average profit will decrease by \(\$ 500\) per house for each additional house built in the area. How many houses should Bernardo build to maximize the total profit? (Remember, the answer must be an integer.)

OPTIMAL DESIGN A farmer wishes to enclose a rectangular pasture with 320 feet of fence. What dimensions give the maximum area if a. the fence is on all four sides of the pasture? b. the fence is on three sides of the pasture and the fourth side is bounded by a wall?

It is estimated that between the hours of noon and 7:00 P.M., the speed of highway traffic flowing past a certain downtown exit is approximately $$ S(t)=t^{3}-9 t^{2}+15 t+45 $$ miles per hour, where \(t\) is the number of hours past noon. At what time between noon and 7:00 P.M. is the traffic moving the fastest, and at what time between noon and 7:00 P.M. is it moving the slowest?

Loni is standing on the bank of a river that is 1 mile wide and wants to get to a town on the opposite bank, 1 mile upstream. She plans to row on a straight line to some point \(P\) on the opposite bank and then walk the remaining distance along the bank. To what point \(P\) should Loni row to reach the town in the shortest possible time if she can row at 4 miles per hour and walk at 5 miles per hour?

A manufacturing firm has received an order to make 400,000 souvenir medals. The firm owns 20 machines, each of which can produce 200 medals per hour. The cost of setting up the machines to produce the medals is \(\$ 80\) per machine, and the total operating cost is \(\$ 5.76\) per hour. How many machines should be used to minimize the cost of producing the 400,000 medals? (Remember, the answer must be an integer.)

Suppose that the demand equation for a certain commodity is \(q=60-0.1 p\) (for \(0 \leq p \leq 600\) ). a. Express the elasticity of demand as a function of \(p\). b. Calculate the elasticity of demand when the price is \(p=200\). Interpret your answer. c. At what price is the elasticity of demand equal to 1 ?

Suppose that the demand equation for a certain commodity is \(q=200-2 p^{2}(\) for \(0 \leq p \leq 10)\). a. Express the elasticity of demand as a function of \(p\). b. Calculate the elasticity of demand when the price is \(p=6\). Interpret your answer. c. At what price is the elasticity of demand equal to 1 ?

Suppose that \(q=500-2 p\) units of a certain commodity are demanded when \(p\) dollars per unit are charged, for \(0 \leq p \leq 250\). a. Determine where the demand is elastic, inelastic, and of unit elasticity with respect to price. b. Use the results of part (a) to determine the intervals of increase and decrease of the revenue function and the price at which revenue is maximized. c. Find the total revenue function explicitly, and use its first derivative to determine its intervals of increase and decrease and the price at which revenue is maximized.

A cruise line estimates that when each deluxe balcony stateroom on a particular cruise is priced at \(p\) thousand dollars, then \(q\) tickets for staterooms will be demanded by travelers, where \(q=300-0.7 p^{2}\). a. Find the elasticity of demand for the stateroom tickets. b. When the price is \(\$ 8,000(p=8)\) per stateroom, should the cruise line raise or lower the price to increase total revenue?

Lamar is an artist who has been commissioned to construct an ornate window. The window is to be in the form of an equilateral triangle surmounted on a rectangle, and the entire window is to have a perimeter of 20 feet. The rectangular part will be made of clear glass and will admit twice as much light as the stained glass triangular part. What dimensions should Lamar choose for the window if he wants it to admit the maximum amount of light?

Oil from an offshore rig located 3 miles from the shore is to be pumped to a location on the edge of the shore that is 8 miles east of the rig. The cost of constructing a pipe in the ocean from the rig to the shore is \(1.5\) times more expensive than the cost of construction on land. How should the pipe be laid to minimize cost?

Through its franchised stations, an oil company gives out 16,000 road maps per year. The cost of setting up a press to print the maps is \(\$ 100\) for each production run. In addition, production costs are 6 cents per map and storage costs are 20 cents per map per year. The maps are distributed at a uniform rate throughout the year and are printed in equal batches timed so that each arrives just as the preceding batch has been used up. How many maps should the oil company print in each batch to minimize cost?

An electronics firm uses 600 cases of components each year. Each case costs $$\$ 1,000$$. The cost of storing one case for a year is 90 cents, and the ordering fee is $$\$ 30$$ per shipment. How many cases should the firm order each time to keep total cost at a minimum? (Assume that the components are used at a constant rate throughout the year and that each shipment arrives just as the preceding shipment is being used up.)

A manufacturing firm receives raw materials in equal shipments arriving at regular intervals throughout the year. The cost of storing the raw materials is directly proportional to the size of each shipment, while the total yearly ordering cost is inversely proportional to the shipment size. Show that the total cost is lowest when the total storage cost and total ordering cost are equal.

Austin needs $$\$ 10,000$$ spending money each year, which he takes from his savings account by making \(N\) equal withdrawals. Each withdrawal incurs a transaction fee of $$\$ 8$$, and money in his account earns interest at the simple interest rate of \(4 \%\). a. The total cost \(C\) of managing the account is the transaction cost plus the loss of interest due to withdrawn funds. Express \(C\) as a function of \(N\). [Hint: You may need the fact that \(\left.1+2+\cdots+N=\frac{N(N+1)}{2} .\right]\) b. How many withdrawals should Austin make each year to minimize the total transaction \(\operatorname{cost} C(N)\) ?

A manufacturing firm receives an order for \(q\) units of a certain commodity. Each of the firm's machines can produce \(n\) units per hour. The setup cost is \(s\) dollars per machine, and the operating cost is \(p\) dollars per hour. a. Derive a formula for the number of machines that should be used to keep total cost as low as possible. b. Prove that when the total cost is minimal, the cost of setting up the machines is equal to the cost of operating the machines.

A manufacturer finds that in producing \(x\) units per day (for \(0

A fundamental problem in crystallography is the determination of the packing fraction of a crystal lattice, which is the fraction of space occupied by the atoms in the lattice, assuming that the atoms are hard spheres. When the lattice contains exactly two different kinds of atoms, it can be shown that the packing fraction is given by the formula* $$f(x)=\frac{K\left(1+c^{2} x^{3}\right)}{(1+x)^{3}}$$ where \(x=\frac{r}{R}\) is the ratio of the radii, \(r\) and \(R\), of the two kinds of atoms in the lattice, and \(c\) and \(K\) are positive constants. a. The function \(f(x)\) has exactly one critical number. Find it, and use the second derivative test to classify it as a relative maximum or a relative minimum. b. The numbers \(c\) and \(K\) and the domain of \(f(x)\) depend on the cell structure in the lattice. For ordinary rock salt: \(c=1, K=\frac{2 \pi}{3}\), and the domain is the interval \((\sqrt{2}-1) \leq x \leq 1\). Find the largest and smallest values of \(f(x)\). c. Repeat part (b) for \(\beta\)-cristobalite, for which \(c=\sqrt{2}, K=\frac{\sqrt{3} \pi}{16}\), and the domain is \(0 \leq x \leq 1\) d. What can be said about the packing fraction \(f(x)\) if \(r\) is much larger than \(R ?\) Answer this question by computing \(\lim _{x \rightarrow \infty} f(x)\). e. Read the article on which this problem is based, and write a paragraph on how packing factors are computed and used in crystallography.

If air resistance is neglected, it can be shown that the stream of water emitted by a fire hose will have height $$ y=-16\left(1+m^{2}\right)\left(\frac{x}{v}\right)^{2}+m x $$ feet above a point located \(x\) feet from the nozzle, where \(m\) is the slope of the nozzle and \(v\) is the velocity of the stream of water as it leaves the nozzle. Assume \(v\) is constant. a. Suppose \(m\) is also constant. What is the maximum height reached by the stream of water? How far away from the nozzle does the stream reach (that is, what is \(x\) when \(y=0\) )? b. If \(m\) is allowed to vary, find the slope that allows a firefighter to spray water on a fire from the greatest distance. c. Suppose the firefighter is \(x=x_{0}\) feet from the base of a building. If \(m\) is allowed to vary, what is the highest point on the building that the firefighter can reach with the water from her hose?