Chapter 4

Find the given limit. $$ \lim _{x \rightarrow \infty} 2 /(x-3) $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ f(x)=-\frac{3}{2} x^{2}+x $$

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x^{2}-6 x ; a=0, b=4 $$

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(x)=x^{2}+6 x-11 $$

In an autocatalytic chemical reaction a substance \(A\) is converted into a substance \(B\) in such a manner that $$ \frac{d x}{d t}=k x(a-x) $$ where \(x\) is the concentration of substance \(B\) at time \(t, a\) is the initial concentration of substance \(A\), and \(k\) is a positive constant. Determine the value of \(x\) at which the rate \(d x / d t\) of the reaction is maximum.

Find all antiderivatives of the given function. 0

Find all critical numbers of the given function. $$ f(x)=x^{2}+4 x+6 $$

Find the given limit. $$ \lim _{x \rightarrow \infty} 4 /(2-x) $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ f(x)=x^{2}+2 x+4 $$

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x-3 x^{2} ; a=-1, b=3 $$

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(x)=x^{3}-x^{2}-x+2 $$

If \(C(x)\) is the cost of manufacturing an amount \(x\) of a given product and \(p\) is the price per unit amount, then the profit \(P(x)\) obtained by selling an amount \(x\) is $$P(x)=p x-C(x)$$ (Notice that there is a loss if \(P(x)\) is negative.) a. If \(C(x)=c x\) and \(c

Suppose that a colony of bacteria is growing exponentially. If 12 hours are required for the number of bacteria to grow from 4000 to 6000 , find the doubling time.

Find all antiderivatives of the given function. \(-2\)

Find all critical numbers of the given function. $$ f(x)=4 x^{3}-6 x^{2}-9 x $$

Find the given limit. $$ \lim _{x \rightarrow \infty} \frac{x}{3 x+2} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ f(x)=x^{3}-6 x^{2}+12 x-4 $$

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x^{3}-6 x ; a=-2, b=0 $$

Consider the circuit shown in Figure 4.52, consisting of a battery having a constant source voltage \(E\), constant internal resistance \(r\), and a variable external resistance \(R\). When current flows through the circuit, the power \(P\) dissipated in the external resistance is given by $$P=\frac{E^{2} R}{(R+r)^{2}}$$ Assume that \(E\) and \(r\) are positive constants. Show that the largest power dissipation occurs when \(R=r\).

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(x)=2 x^{4}-4 x^{2}+3 $$

Experiment has shown that under ideal conditions and a constant temperature of \(28.5^{\circ} \mathrm{C}\), the population of a certain type of flour beetle doubles in 6 days and 20 hours. Suppose that there are now 1500 such beetles. How long ago were there \(1200 ?\)

Find all antiderivatives of the given function. \(3 x\)

Find all critical numbers of the given function. $$ f(x)=3 x^{4}+4 x^{3}-12 x^{2}+1 $$

Find the given limit. $$ \lim _{x \rightarrow \infty} \frac{4 x^{2}}{\sqrt{2} x-3} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ f(x)=x^{4}-6 x^{2}+8 $$

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x^{3}-6 x ; a=-2, b=2 $$

The power output of an electric generator is \(V I\), where \(V\) is the constant terminal voltage and \(I\) is the variable current. The power loss is \(P+I^{2} R\), where \(P\) is constant and \(I^{2} R\) is the power lost to heat through the internal resistance \(R\). The efficiency \(E\) of the generator is given by $$E=\frac{\text { power output }}{\text { power input }}=\frac{V I}{V I+P+I^{2} R}$$ Assume that \(V, P\), and \(R\) are positive constants. Find the current for which the efficiency is maximum.

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(x)=x /\left(x^{3}-2\right) $$

If the population of the world is not unduly affected by war, famine, or new technology, then it is reasonable to assume that the population will grow (at least for a long period of time) at an exponential rate. Using this assumption and the census figures which show that the world population in 1962 was \(3,150,000,000\) the world population in 1978 was \(4,238,000,000\) determine what the world population should have been in \(2000 .\)

Find all antiderivatives of the given function. $$ -6 x+5 $$

Find all critical numbers of the given function. $$ f(x)=x^{5}-5 x^{3}+10 x-3 $$

Find the given limit. $$ \lim _{x \rightarrow \infty} \frac{2 x^{2}+x-1}{x^{2}-x+4} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ g(x)=\frac{x}{x^{2}+1} $$

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x^{3}+4 ; a=-2, b=1 $$

For an electron in the \(2 p\) state of an excited hydrogen atom, the probability function for the electron to be located at a distance \(r\) from the atom's center is given by $$P(r)=\frac{\pi r^{4}}{6 a^{5}} e^{-r / a} \quad \text { for } r>0$$ Find the most probable distance of the electron from the center of the atom.

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(t)=\frac{t^{2}-t+1}{t^{2}+t+1} $$

Suppose the populations of two countries are growing exponentially. Suppose also that one country has a population of \(50,000,000\) and a doubling time of 20 years, whereas the other has a population of \(20,000,000\) and a doubling time of 10 years. How long will it be until the two countries have the same population?

Find all antiderivatives of the given function. $$ -x^{2} $$

Find all critical numbers of the given function. $$ g(x)=x+1 / x $$

Find the given limit. $$ \lim _{t \rightarrow \infty} \frac{(t-1)(2 t+1)}{(3 t-2)(t+4)} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ g(x)=x \sqrt{x-1} $$

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x^{3}-2 ; a=-3, b=3 $$

According to one model, the time rate \(R\) at which a tumor grows is given by $$ R=A x \ln \frac{B}{x} \text { for } 0

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(t)=\frac{1}{\sqrt{t-t^{2}}} $$

Suppose the population \(f(t)\) of a given species grows exponentially, so that \(f(t)=f(0) e^{k t}\) for some positive constant \(k\) a. Show that the population doubles during any time interval of duration \((\ln 2) / k\). Thus \((\ln 2) / k\) is the doubling time \(d\). b. Show that \(f(t)=f(0) 2^{t / d}\).

Find all antiderivatives of the given function. $$ 4 x^{2}+6 x-1 $$

Find all critical numbers of the given function. $$ g(x)=x-4 / x^{2} $$

Find the given limit. $$ \lim _{t \rightarrow \infty} \frac{t}{t^{1 / 2}+2 t^{-1 / 2}} $$

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. $$ g(x)=x e^{x} $$

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x^{3}-3 x^{2}+3 x+1 ; a=-2, b=2 $$