Chapter 4

Evaluate $$ \lim _{x \rightarrow \infty} \frac{x^{(x+1) / x}}{\sqrt{1+x^{2}}} $$ .

Discrete Dynamical Systems. Suppose that \(\Phi: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function. The collection of sequences \(\left\\{x_{n}\right\\}_{n} \geq 0\) that satisfies \(x_{n+1}=\Phi\left(x_{n}\right)\) for \(n \geq 0\) is called the discrete dynamical system associated with \(\Phi .\) Notice that each element \(\left\\{x_{n}\right\\}\) of a discrete dynamical system is determined by its first element \(x_{0}\). A number \(x_{*}\) such that \(\Phi\left(x_{*}\right)=x_{*}\) is called an equilibrium point of the dynamical system. We say that an equilibrium point \(x *\) of \(\Phi\) is stable if there is a \(\delta>0\) such that each element \(\left\\{x_{n}\right\\}\) of the dynamical system associated to \(\Phi\) converges to \(x_{*}\) provided that \(\left|x_{*}-x_{0}\right|<\delta .\) Exercises \(84-89\) are concerned with these ideas. Suppose that \(x\) * is an equilibrium point of the dynamical system. Show that if \(x_{N}=x_{*},\) then \(x_{n}=x_{*}\) for all \(n \geq N\). If \(x_{n}=x_{*},\) then \(x_{N+1}=\Phi\left(x_{N}\right)=\Phi\left(x_{*}\right)=x_{*}\). Continuing, \(x_{N+2}=\Phi\left(x_{N+1}\right)=\Phi\left(x_{*}\right)=x_{*},\) and so on. (In words, once a member of the sequence \(\left\\{x_{n}\right\\}\) equals \(x_{*},\) its successor equals \(x\), as well.) It follows that once a member of the sequence \(\left\\{x_{n}\right\\}\) equals \(x_{*},\) all subsequent members equal \(x_{*}\)

In each of Exercises 82-89, use the first derivative to determine the intervals on which the function is increasing and on which the function is decreasing. $$ f(x)=\frac{x+e^{x}}{x^{2}+e^{x}} $$

For any constants \(\alpha\) and \(\beta\) with \(\beta>0,\) show that $$ \lim _{x \rightarrow \infty} \frac{\ln (x)^{\alpha}}{x^{\beta}}=0 $$

Discrete Dynamical Systems. Suppose that \(\Phi: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function. The collection of sequences \(\left\\{x_{n}\right\\}_{n} \geq 0\) that satisfies \(x_{n+1}=\Phi\left(x_{n}\right)\) for \(n \geq 0\) is called the discrete dynamical system associated with \(\Phi .\) Notice that each element \(\left\\{x_{n}\right\\}\) of a discrete dynamical system is determined by its first element \(x_{0}\). A number \(x_{*}\) such that \(\Phi\left(x_{*}\right)=x_{*}\) is called an equilibrium point of the dynamical system. We say that an equilibrium point \(x *\) of \(\Phi\) is stable if there is a \(\delta>0\) such that each element \(\left\\{x_{n}\right\\}\) of the dynamical system associated to \(\Phi\) converges to \(x_{*}\) provided that \(\left|x_{*}-x_{0}\right|<\delta .\) Exercises \(84-89\) are concerned with these ideas. Suppose that \(\left\\{x_{n}\right\\}_{n} \geq 0\) is an element of the discrete dyna\(\begin{array}{lllll}\text { mical system associated with } & \Phi . \text { Suppose that }\end{array}\) \(\xi=\lim _{n \rightarrow \infty} x_{n} .\) Show that \(\xi\) is an equilibrium point of the discrete dynamical system associated with \(\Phi .\)

In each of Exercises 82-89, use the first derivative to determine the intervals on which the function is increasing and on which the function is decreasing. $$ f(x)=\left(x^{2}+e^{x}+1\right) /\left(x^{4}+1\right) $$

Cauchy's Mean Value Theorem states the following: If \(f(x)\) and \(g(x)\) are functions that are continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b),\) then there is a point \(\xi \in(a, b)\) such that \((g(b)-g(a)) f^{\prime}(\xi)=(f(b)-f(a)) g^{\prime}(\xi) .\) Notice that, if both \(g(a) \neq g(b)\) and \(g^{\prime}(\xi) \neq 0,\) then $$ \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)} $$ Complete the following outline to obtain a proof of Cauchy's Mean Value Theorem a. Let \(r(x)=\) $$ (f(b)-f(a))(g(x)-g(a))-(f(x)-f(a))(g(b)-f(a)) $$ Check that \(r(a)=r(b)\) b. Apply Rolle's Theorem to the function \(r\) on the interval \([a, b]\) to conclude that there is a point \(\xi \in(a, b)\) such $$ \text { that } r^{\prime}(\xi)=0 $$ c. Rewrite the conclusion of (b) to obtain the conclusion of Cauchy's Mean Value Theorem

Discrete Dynamical Systems. Suppose that \(\Phi: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function. The collection of sequences \(\left\\{x_{n}\right\\}_{n} \geq 0\) that satisfies \(x_{n+1}=\Phi\left(x_{n}\right)\) for \(n \geq 0\) is called the discrete dynamical system associated with \(\Phi .\) Notice that each element \(\left\\{x_{n}\right\\}\) of a discrete dynamical system is determined by its first element \(x_{0}\). A number \(x_{*}\) such that \(\Phi\left(x_{*}\right)=x_{*}\) is called an equilibrium point of the dynamical system. We say that an equilibrium point \(x *\) of \(\Phi\) is stable if there is a \(\delta>0\) such that each element \(\left\\{x_{n}\right\\}\) of the dynamical system associated to \(\Phi\) converges to \(x_{*}\) provided that \(\left|x_{*}-x_{0}\right|<\delta .\) Exercises \(84-89\) are concerned with these ideas. Suppose that \(x\) * is an equilibrium point of the dynamical system \Phi. Suppose that \(\Phi^{\prime}\) exists and is continuous on an open interval centered at \(x_{*}\). Suppose also that \(\left|\Phi^{\prime}\left(x_{*}\right)\right|<1\). Show that there are positive numbers \(\delta\) and \(K\) with \(K<1\) such that \(\left|\Phi^{\prime}(x)\right|

In each of Exercises 82-89, use the first derivative to determine the intervals on which the function is increasing and on which the function is decreasing. $$ f(x)=x^{3}-x^{2}+e^{-x} $$

In each of Exercises 82-89, use the first derivative to determine the intervals on which the function is increasing and on which the function is decreasing. $$ f(x)=\left(4 x^{3}-3\right) \exp \left(-x^{2}\right) $$

In each of Exercises 82-89, use the first derivative to determine the intervals on which the function is increasing and on which the function is decreasing. $$ f(x)=x^{1 / 3}-\ln \left(1+x^{2}\right) $$

In each of Exercises \(89-92,\) investigate the given limit numerically and graphically. \(\lim _{x \rightarrow 0}(1-\cos (x)) / \sin (x)^{2}\)

Discrete Dynamical Systems. Suppose that \(\Phi: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function. The collection of sequences \(\left\\{x_{n}\right\\}_{n} \geq 0\) that satisfies \(x_{n+1}=\Phi\left(x_{n}\right)\) for \(n \geq 0\) is called the discrete dynamical system associated with \(\Phi .\) Notice that each element \(\left\\{x_{n}\right\\}\) of a discrete dynamical system is determined by its first element \(x_{0}\). A number \(x_{*}\) such that \(\Phi\left(x_{*}\right)=x_{*}\) is called an equilibrium point of the dynamical system. We say that an equilibrium point \(x *\) of \(\Phi\) is stable if there is a \(\delta>0\) such that each element \(\left\\{x_{n}\right\\}\) of the dynamical system associated to \(\Phi\) converges to \(x_{*}\) provided that \(\left|x_{*}-x_{0}\right|<\delta .\) Exercises \(84-89\) are concerned with these ideas. Continuation: Prove the following theorem: If \(x_{*}\) is an equilibrium point of the dynamical system \(\Phi\) such that (1) \(\Phi^{\prime}\) exists and is continuous on an open interval centered at \(x *\) and \((2)\left|\Phi^{\prime}(x *)\right|<1,\) then \(x_{*}\) is a stable equilibrium of \(\Phi\)

In each of Exercises 90-93, a function \(f\) is given. Calculate \(f^{\prime}\) and plot \(y=f^{\prime}(x)\) in a suitable viewing window. Use this plot to identify the points at which \(f\) has local extrema. $$ f(x)=x^{3}-x+\sin ^{2}(\pi x / 2),-1

Investigate the given limit numerically and graphically. \(\lim _{x \rightarrow 0} \tan (2 \sin (x)) / \sin (2 \tan (x))\)

In each of Exercises 90-93, a function \(f\) is given. Calculate \(f^{\prime}\) and plot \(y=f^{\prime}(x)\) in a suitable viewing window. Use this plot to identify the points at which \(f\) has local extrema. $$ f(x)=2+x^{3} / 3-\sin ^{2}(x+1),-3

Investigate the given limit numerically and graphically. \(\lim _{x \rightarrow 0^{+}} x^{(1-\sin (x) / x)}\)

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=x^{3}-2 x+\cos (x) $$

Investigate the given limit numerically and graphically. \(\lim _{x \rightarrow 0}(1+\tan (2 x))^{1 / x}\)

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=x^{2}-2 x \ln \left(1+x^{2}\right)+x-4 $$

In each of Exercises 90-93, a function \(f\) is given. Calculate \(f^{\prime}\) and plot \(y=f^{\prime}(x)\) in a suitable viewing window. Use this plot to identify the points at which \(f\) has local extrema. $$ f(x)=\exp \left(-x^{2}\right)+\exp \left(-(x-1)^{2}\right)+\exp \left(-(x-3)^{2}\right) $$

In each of Exercises \(93-95,\) illustrate the assertion of l'Hôpital's Rule by plotting \(f / g\) and \(f^{\prime} / g^{\prime}\) in an interval centered at \(c\) \(f(x)=(1-\cos (x)), g(x)=x \tan (x), c=0\)

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=x-2 \exp \left(-x^{2}\right) $$

For each given function \(f,\) graph the function \(S(x)=\) signum \(\left(f^{\prime}(x)\right)\) for \(x\) in the given interval \(I .\) Use the graph of \(S\) to determine and classify the local extrema of \(f\). a. \(f(x)=x^{5}-12 x^{4}+55 x^{3}-120 x^{2}+124 x-48\) \(I=[0.8,4.1]\) b. \(f(x)=4 x^{4}-24 x^{3}+51 x^{2}-44 x+12, \quad I=[0.3,2.6]\) c. \(f(x)=x^{4}-x^{3}-7 x^{2}+x+6, \quad I=[-2.4,3.2]\)

Illustrate the assertion of l'Hôpital's Rule by plotting \(f / g\) and \(f^{\prime} / g^{\prime}\) in an interval centered at \(c\) \(f(x)=\left(1+x-e^{x}\right), g(x)=x \sin (x), c=0\)

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=\sin ^{2}(x)-x^{3}+5 x+20 $$

Illustrate the assertion of l'Hôpital's Rule by plotting \(f / g\) and \(f^{\prime} / g^{\prime}\) in an interval centered at \(c\) \(f(x)=(1+\cos (x)), g(x)=(x-\pi) \sin (x), c=\pi\)

In Exercises \(95-98,\) approximate the value \(c\) guaranteed by the application of the Mean Value Theorem to the given function \(f\) on the given interval \(I=[a, b]\). Graph the function, the tangent line at \((c, f(c)),\) and the line segment between \((a, f(a))\) and \((b, f(b))\) $$ f(x)=x^{5}+x^{3}+1, \quad I=[0,2] $$

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=x e^{x}, \quad I=[-3 / 2,1] $$

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=x^{4}-2 x^{3}+x^{2}-2 x+13, \quad I=[-1,3] $$

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=x \sin (1 / x), \quad I=[1 /(4 \pi), \pi / 24] $$