Chapter 5

The number of dollars in the total cost of manufacturing \(x\) watches in a certain plant is given by \(C(x)=1500+30 x\) \(+20 / x\). Find (a) the marginal cost function, (b) the marginal cost when \(x=40\), and (c) the cost of manufacturing the forty-first watch.

If \(C(x)\) dollars is the total cost of manufacturing \(x\) toys and \(C(x)=110+4 x+0.02 x^{2}\), find (a) the marginal cost function, (b) the marginal cost when \(x=10\), and (c) the cost of manufacturing the eleventh toy.

F(x)=\frac{x+2}{x-2} ;[-4,4]

Suppose a liquid is produced by a certain chemical process and the total cost function \(C\) is given by \(C(x)=6+4 \sqrt{x}\), where \(C(x)\) dollars is the total cost of producing \(x\) gallons of the liquid. Find (a) the marginal cost when 16 gal are produced and (b) the number of gallons produced when the marginal cost is 40 cents per gal.

The number of dollars in the total cost of producing \(x\) units of a certain commodity is \(C(x)=40+3 x+9 \sqrt{2 x}\). Find (a) the marginal cost when 50 units are produced and (b) the number of units produced when the marginal cost is \(\$ 4.50\).

\(f(x)=(x-4)^{2}\)

f(x)=2 x+\frac{1}{2 x}

\(G(x)=(x-5)^{2 / 3} ;(-\infty,+\infty)\)

\(G(x)=\frac{2 x}{\left(x^{2}+4\right)^{3 / 2}}\)

If \(C(x)\) dollars is the total cost of producing \(x\) units of a commodity and \(C(x)=3 x^{2}-6 x+4\), find (a) the average cost function, (b) the marginal cost function, and (c) the marginal average cost function. (d) What is the range of \(C\) ? (e) Find the absolute minimum average unit cost. (f) Draw sketches of the total cost, average cost, and marginal cost curves on the same set of axes. Verify that the average cost and marginal cost are equal when the average cost has its least value.

\(f(x)=\frac{x}{\left(x^{2}+4\right)^{3 / 2}} ;[0,+\infty)\)

\(f(x)=(x-2)^{1 / 5}\)

\(f(x)=x^{4}-3 x^{3}+3 x^{2}+1\)

The total cost function \(C\) is given by \(C(x)=\frac{1}{3} x^{3}-2 x^{2}+5 x+2\). (a) Determine the range of \(C\). (b) Find the marginal cost function. (c) Find the interval on which the marginal cost function is decreasing and the interval on which it is increasing. (d) Draw a sketch of the graph of the total cost function; determine where the graph is concave upward and where it is concave downward, and find the points of inflection and an equation of any inflectional tangent.

\(F(x)=(2 x-6)^{3 / 2}+1\)

If \(C(x)\) dollars is the total cost of producing \(x\) units of a commodity and \(C(x)=2 x^{2}-8 x+18\), find (a) the domain and range of \(C\), (b) the average cost function, (c) the absolute minimum average unit cost, and (d) the marginal cost function. (e) Draw sketches of the total cost, average cost, and marginal cost curves on the same set of axes.

A rectangular field, having an area of \(2700 \mathrm{yd}^{2}\), is to be enclosed by a fence, and an additional fence is to be used to divide the field down the middle. If the cost of the fence down the middle is \(\$ 2\) per running yard, and the fence along the sides costs \(\$ 3\) per running yard, find the dimensions of the field so that the cost of the fencing will be the least.

\(f(x)= \begin{cases}x^{2} & \text { if } x<0 \\ -x^{2} & \text { if } x \geq 0\end{cases}\)

The fixed overhead expense of a manufacturer of children's toys is \(\$ 400\) per week, and other costs amount to \(\$ 3\) for each toy produced. Find (a) the total cost function, (b) the average cost function, and (c) the marginal cost function. (d) Show that there is no absolute minimum average unit cost. (e) What is the smallest number of toys that must be produced so that the average cost per toy is less than \(\$ 3.42 ?\) (f) Draw sketches of the graphs of the functions in (a), (b), and (c) on the same set of axes.

A rectangular open tank is to have a square base, and its volume is to be \(125 \mathrm{yd}^{3}\). The cost per square yard for the bottom is \(\$ 8\) and for the sides is \(\$ 4\). Find the dimensions of the tank in order for the cost of the material to be the least.

\(f(x)= \begin{cases}x^{2} & \text { if } x<1 \\ x^{3}-4 x^{2}+7 x-3 & \text { if } x \geq 1\end{cases}\)

The number of hundreds of dollars in the total cost of producing \(100 x\) radios per day in a certain factory is \(C(x)=4 x+5\). Find (a) the average cost function, (b) the marginal cost function, and (c) the marginal average cost function. (d) Show that there is no absolute minimum average unit cost. (e) What is the smallest number of radios that the factory must produce in a day so that the average cost per radio is less than \(\$ 7 ?\) (f) Draw sketches of the total cost, average cost, and marginal cost curves on the same set of axes.

If \(f(x)=a x^{3}+b x^{2}\), determine \(a\) and \(b\) so that the graph of \(f\) will have a point of inflection at \((1,2)\).

\(f(x)= \begin{cases}x^{2} & \text { if } x<0 \\ 2 x^{2} & \text { if } x \geq 0\end{cases}\)

If the demand equation for a particular commodity is \(3 x+4 p=12\), find (a) the price function, (b) the total revenue function, and (c) the marginal revenue function. Draw sketches of the demand, total revenue, and marginal revenue curves on the same set of axes. Verify that the marginal revenue curve intersects the \(x\) axis at the point whose abscissa is the value of \(x\) for which the total revenue is greatest and that the demand curve intersects the \(x\) axis at the point whose abscissa is twice that.

\(f(x)=(x+2)^{2}(x-1)^{2}\)

If \(f(x)=a x^{3}+b x^{2}+c x\), determine \(a, b\), and \(c\) so that the graph of \(f\) will have a point of inflection at \((1,2)\) and so that the slope of the inflectional tangent there will be \(-2\).

\(f(x)= \begin{cases}-x^{3} & \text { if } x<0 \\ x^{3} & \text { if } x \geq 0\end{cases}\)

The demand equation for a particular commodity is \(p x^{2}+9 p-18=0\) where \(p\) dollars is the price per unit when \(100 x\) units are demanded. Find (a) the price function, (b) the total revenue function, and (c) the marginal revenue function. (d) Find the absolute maximum total revenue.

\(f(x)= \begin{cases}-x^{4} & \text { if } x<0 \\ x^{4} & \text { if } x \geq 0\end{cases}\)

\(f(x)=2-(x-1)^{1 / 3}\)

If \(f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e\), determine the values of \(a, b, c, d\), and \(e\) so the graph of \(f\) will have a point of inflection at \((1,-1)\), have the origin on it, and be symmetric with respect to the \(y\) axis.

\(f(x)= \begin{cases}2(x-1)^{3} & \text { if } x<1 \\ (x-1)^{4} & \text { if } x \geq 1\end{cases}\)

Let \(R(x)\) dollars be the total revenue obtained when \(x\) units of a commodity are demanded and \(R(x)=2+3 \sqrt{x}-1\), where \(x\) is in the closed interval \([2,17] .\) Find (a) the demand equation, (b) the price function, and (c) the marginal revenue function. (d) Find the absolute maximum total revenue. (e) Draw sketches of the demand, total revenue, and marginal revenue curves on the same set of axes.

\(f(x)= \begin{cases}2 x+1 & \text { if } x \leq 4 \\ 13-x & \text { if } 4

Given \(f(x)=x^{3}+3 r x+5\), prove that (a) if \(r>0, f\) has no relative extrema; (b) if \(r<0, f\) has both a relative maximum value and a relative minimum value.

\(f(x)=(x+1)^{3}(x-2)^{2}\)

The demand equation for a certain commodity is \(x+p=14\), where \(x\) is the number of units produced daily and \(p\) is the number of hundreds of dollars in the price of each unit. The number of hundreds of dollars in the total cost of producing \(x\) units is given by \(C(x)=x^{2}-2 x+2\), and \(x\) is in the closed interval \([1,14] .\) (a) Find the profit function an draw a sketch of its graph. (b) On a set of axes different from that in (a), draw sketches of the total revenue and tot cost curves and show the geometrical interpretation of the profit function. (c) Find the maximum daily profit. (d) Fin the marginal revenue and marginal cost functions. (e) Draw sketches of the graphs of the marginal revenue and marg nal cost functions on the same set of axes and show that they intersect at the point for which the value of \(x\) makes th profit a maximum.

\(f(x)=\left\\{\begin{aligned} 5-2 x & \text { if } x<3 \\ 3 x-10 & \text { if } 3 \leq x \end{aligned}\right.\)

Given \(f(x)=x^{r}-r x+k\), where \(r>0\) and \(r \neq 1\), prove that (a) if \(01, f\) has a relative minimum value at \(1 .\)

\(f(x)= \begin{cases}2 x+9 & \text { if } x \leq-2 \\ x^{2}+1 & \text { if }-2

The demand equation for a certain commodity is \(p=(x-8)^{2}\), and the total cost function is given by \(C(x)=18 x-x^{2}\), where \(C(x)\) dollars is the total cost when \(x\) units are purchased. (a) Determine the permissible values of \(x .\) (b) Find the marginal revenue and marginal cost functions. (c) Find the value of \(x\) which yields the maximum profit. (d) Draw sketches of the marginal revenue and marginal cost functions on the same set of axes.

A page of print is to contain 24 in. \(^{2}\) of printed area, a margin of \(1 \frac{1}{2}\) in. at the top and bottom, and a margin of 1 in. at the sides. What are the dimensions of the smallest page that would fill these requirements?

A monopolist determines that if \(C(x)\) cents is the total cost of producing \(x\) units of a certain commodity, then \(C(x)=\) \(25 x+20,000\). The demand equation is \(x+50 p=5000\), where \(x\) units are demanded each week when the unit price is \(p\) cents. If the weekly profit is to be maximized, find (a) the number of units that should be produced each week, (b) the price of each unit, and (c) the weekly profit.

\(f(x)= \begin{cases}(x-2)^{2}-3 & \text { if } x \leq 5 \\ \frac{1}{2}(x+7) & \text { if } 5

A one-story building having a rectangular floor space of \(13,200 \mathrm{ft}^{2}\) is to be constructed where a 22 - \(\mathrm{ft}\) easement is required in the front and back and a \(15-\mathrm{ft}\) easement is required on each side. Find the dimensions of the lot having the least area on which this building can be located.

\(f^{\prime \prime}(c)=0 ; f^{\prime}(c)=\frac{1}{2} ; f^{\prime \prime}(x)>0\) if \(xc\)

\(f(x)= \begin{cases}3 x+5 & \text { if } x<-1 \\ x^{2}+1 & \text { if }-1 \leq x<2 \\ 7-x & \text { if } 2 \leq x\end{cases}\)

A direct current generator has an electromotive force of \(E\) volts and an internal resistance of \(r\) ohms. \(E\) and \(r\) are constants. If \(R\) ohms is the external resistance, the total resistance is \((r+R)\) ohms and if \(P\) watts is the power, then $$ P=\frac{E^{2} R}{(r+R)^{2}} $$ What external resistance will consume the most power?

Find the maximum tax revenue that can be received by the government if an additive tax for each unit produced is levied on a monopolist for which the demand equation is \(x+3 p=75\), where \(x\) units are demanded when \(p\) dollars is the price of one unit, and \(C(x)=3 x+100\), where \(C(x)\) dollars is the total cost of producing \(x\) units.