Chapter 10

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ g(x)=\frac{2 x}{x^{2}+x-2} $$

Show that the function is concave upward wherever it is defined. $$ h(x)=\frac{1}{x^{2}} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=\frac{2}{3} x^{3}-2 x^{2}-6 x-2 $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ \begin{array}{l} f(x)=\frac{x+1}{x-1} \text { on }[2,4]\\\ \text { 24. } g(t)=\frac{t}{t-1} \text { on }[2,4] \end{array} $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ g(t)=2+\frac{5}{(t-2)^{2}} $$

Show that the function is concave upward wherever it is defined. $$ g(x)=-\sqrt{4-x^{2}} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ h(x)=x^{4}-4 x^{3}+10 $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(t)=\frac{t}{t-1} \text { on }[2,4] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=1+\frac{2}{x-3} $$

Show that the function is concave upward wherever it is defined. $$ h(x)=\sqrt{x^{2}+4} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ g(x)=x^{4}-2 x^{2}+4 $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=4 x+\frac{1}{x} \text { on }[1,3] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{x^{2}-2}{x^{2}-4} $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=2 x^{2}-3 x+4 $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=\frac{1}{x-2} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=9 x-\frac{1}{x} \text { on }[1,3] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ h(x)=\frac{2-x^{2}}{x^{2}+x} $$

Determine where the function is concave upward and where it is concave downward. $$ g(x)=-x^{2}+3 x+4 $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ h(x)=\frac{1}{2 x+3} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=\frac{1}{2} x^{2}-2 \sqrt{x} \text { on }[0,3] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ g(x)=\frac{x^{3}-x}{x(x+1)} $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=x^{3}-1 $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ h(t)=\frac{t}{t-1} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=\frac{1}{8} x^{2}-4 \sqrt{x} \text { on }[0,9] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{x^{4}-x^{2}}{x(x-1)(x+2)} $$

Determine where the function is concave upward and where it is concave downward. $$ g(x)=x^{3}-x $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ g(t)=\frac{2 t}{t^{2}+1} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=\frac{1}{x} \text { on }(0, \infty) $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=x^{4}-6 x^{3}+2 x+8 $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{3 / 5} $$

The demand for motorcycle tires imported by Dixie Import-Export is 40,000 /year and may be assumed to be uniform throughout the year. The cost of ordering a shipment of tires is $$\$ 400$$, and the cost of storing each tire for a year is $$\$ 2$$. Determine how many tires should be in each shipment if the ordering and storage costs are to be minimized. (Assume that each shipment arrives just as the previous one has been sold.)

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=\frac{1}{x+1} \text { on }(0, \infty) $$

Determine where the function is concave upward and where it is concave downward. $$ f(x)=3 x^{4}-6 x^{3}+x-8 $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{2 / 3}+5 $$

McDuff Preserves expects to bottle and sell \(2,000,000\) 32-oz jars of jam at a uniform rate throughout the year. The company orders its containers from Consolidated Bottle Company. The cost of ordering a shipment of bottles is $$\$ 200$$, and the cost of storing each empty bottle for a year is $$\$ .40$$. How many orders should McDuff place per year and how many bottles should be in each shipment if the ordering and storage costs are to be minimized? (Assume that each shipment of bottles is used up before the next shipment arrives.)

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=3 x^{2 / 3}-2 x \text { on }[0,3] $$

A skydiver leaps from the gondola of a hot-air balloon. As she free-falls, air resistance, which is proportional to her velocity, builds up to a point where it balances the force due to gravity. The resulting motion may be described in terms of her velocity as follows: Starting at rest (zero velocity), her velocity increases and approaches a constant velocity, called the terminal velocity. Sketch a graph of her velocity \(v\) versus time \(t\).

Determine where the function is concave upward and where it is concave downward. $$ f(x)=x^{4 / 7} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=\sqrt{x+1} $$

Neilsen Cookie Company sells its assorted butter cookies in containers that have a net content of \(1 \mathrm{lb}\). The estimated demand for the cookies is \(1,000,000\) 1-lb containers. The setup cost for each production run is $$\$ 500$$, and the manufacturing cost is $$\$ .50$$ for each container of cookies. The cost of storing each container of cookies over the year is $$\$ .40$$. Assuming uniformity of demand throughout the year and instantaneous production, how many containers of cookies should Neilsen produce per production run in order to minimize the production cost? Hint: Following the method of Example 5 , show that the total production cost is given by the function $$ C(x)=\frac{500,000,000}{x}+0.2 x+500,000 $$ Then minimize the function \(C\) on the interval \((0,1,000,000)\).

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=x^{2}+2 x^{2 / 3} \text { on }[-2,2] $$

Initially, 10 students at a junior high school contracted influenza. The flu spread over time, and the total number of students who eventually contracted the flu approached but never exceeded 200. Let \(P(t)\) denote the number of students who had contracted the flu after \(t\) days, where \(P\) is an appropriate function. a. Make a sketch of the graph of \(P\). (Your answer will not be unique.) b. Where is the function increasing? c. Does \(P\) have a horizontal asymptote? If so, what is it? d. Discuss the concavity of \(P\). Explain its significance. e. Is there an inflection point on the graph of \(P ?\) If so, explain its significance.

Determine where the function is concave upward and where it is concave downward. $$ f(x)=\sqrt[3]{x} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=(x-5)^{2 / 3} $$

A company expects to sell \(D\) units of a certain product per year. Sales are assumed to be at a steady rate with no shortages allowed. Each time an order for the product is placed, an ordering cost of \(K\) dollars is incurred. Each item costs \(p\) dollars, and the holding cost is \(h\) dollars per item per year. a. Show that the inventory cost (the combined ordering cost, purchasing cost, and holding cost) is $$ C(x)=\frac{K D}{x}+p D+\frac{h x}{2} \quad(x>0) $$ where \(x\) is the order quantity (the number of items in each order). b. Use the result of part (a) to show that the inventory cost is minimized if $$ x=\sqrt{\frac{2 K D}{h}} $$ This quantity is called the economic order quantity (EOQ).

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x^{2 / 3}\left(x^{2}-4\right) \text { on }[-1,2] $$

Use the information summarized in the table to sketch the graph of \(f\). \(f(x)=x^{3}-3 x^{2}+1\) Domain: \((-\infty, \infty)\) Intercept: \(y\) -intercept: 1 Asymptotes: None Intervals where \(f\) is \(\nearrow\) and \(\searrow: \nearrow\) on \((-\infty, 0) \cup(2, \infty)\); \(\searrow\) on \((0,2)\) extrema: Rel. max. at \((0,1) ;\) rel. min. at \((2,-3)\) Concavity: Downward on \((-\infty, 1) ;\) upward on \((1, \infty)\) Point of inflection: \((1,-1)\)

Determine where the function is concave upward and where it is concave downward. $$ f(x)=\sqrt{4-x} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=\sqrt{16-x^{2}} $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x^{2 / 3}\left(x^{2}-4\right) \text { on }[-1,3] $$