Chapter 5

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=x-2 x^{2}-3 x^{3}-4 x^{4} $$

Each function is continuous and defined on a closed interval. It therefore satisfies the assumptions of the extremevalue theorem. With the help of a graphing calculator, graph each function and locate its global extrema. (Note that a function may assume a global extremum at more than one point.) \(f(x)=\ln (x+1), 0 \leq x \leq 2\)

Suppose that \(a\) and \(b\) are the side lengths in a right triangle whose hypotenuse is \(10 \mathrm{~cm}\) long. Show that the area of the triangle is largest when \(a=b\).

Determine whether each function has absolute maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. $$ y=\cos \left(\pi x^{2}\right),-1 \leq x \leq 1 $$

Use the stability criterion to characterize the stability of the equilibria of $$ x_{t+1}=\frac{2}{3}-\frac{2}{3} x_{t}^{2}, \quad t=0,1,2, \ldots $$

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\frac{1}{x}, x \neq 0 $$

Use the Newton-Raphson method to find a numerical approximation to the solution of $$ x^{2}-16=0 $$ when your initial guess is (a) \(x_{0}=3\) and (b) \(x_{0}=4\).

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=1+\frac{1}{x}+\frac{1}{x^{2}} $$

Sketch the graph of a function that is continuous on the closed interval \([0,3]\) and has a global maximum at the left endpoint and a global minimum at the right endpoint.

A rectangle has its base on the \(x\) -axis, its lower left corner at \((0,0)\), and its upper right corner on the curve \(y=1 / x .\) What is the smallest perimeter the rectangle can have?

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \pi / 2} \frac{\sin \left(\frac{\pi}{2}-x\right)}{\cos x} $$

Determine whether each function has absolute maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. $$ y=\sin [2 \pi(x-3)], 2 \leq x \leq 3 $$

Use the stability criterion to characterize the stability of the equilibria of $$ x_{t+1}=\frac{3}{5} x_{t}^{2}-\frac{2}{5}, \quad t=0,1,2, \ldots $$

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\frac{-2}{x^{2}+3} $$

Suppose that you wish to use the Newton-Raphson method to solve $$ f(x)=0 $$ numerically. It just so happens that your initial guess \(x_{0}\) satisfies \(f\left(x_{0}\right)=0 .\) What happens to subsequent iterations? Give a graphical illustration of your results. [Assume that \(f^{\prime}\left(x_{0}\right) \neq 0\).]

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=x^{2}-\frac{2}{x^{2}}+\frac{3}{x^{3}} $$

Sketch the graph of a function that is continuous on the closed interval \([-2,1]\) and has a global maximum and a global minimum in the interior of the domain of the function.

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0^{+}} \frac{\sqrt{x}}{\ln (x+1)} $$

Determine whether each function has absolute maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. $$ y=e^{-|x|}, x \in \mathbf{R} $$

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ \left(x^{2}+1\right)^{1 / 3}, x \in \mathbf{R} $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=1-\frac{1}{x^{2}} $$

Sketch the graph of a function that is continuous on the open interval \((0,2)\) and has neither a global maximum nor a global minimum in its domain.

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}} $$

Determine whether each function has absolute maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. $$ y=e^{-x^{2} / 4}, x \in \mathbf{R} $$

Use the stability criterion to characterize the stability of the equilibria of $$ x_{t+1}=\frac{x_{t}}{0.3+x_{t}}, \quad t=0,1,2, \ldots $$

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\frac{5}{x-2}, x \neq 2 $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=x^{3}-\frac{1}{x^{3}} $$

Sketch the graph of a function that is continuous on the closed interval \([1,4]\), except at \(x=2\), and has neither a global maximum nor a global minimum in its domain.

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} \frac{\ln (\ln x)}{x} $$

Determine whether each function has absolute maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. $$ y=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}-6 x+2, x \in \mathbf{R} $$

(a) Use the stability criterion to characterize the stability of the equilibria of $$ x_{t+1}=\frac{5 x_{t}^{2}}{4+x_{t}^{2}}, \quad t=0,1,2, \ldots $$ (b) Use cobwebbing to decide to which value \(x_{t}\) converges as \(t \rightarrow \infty\) if (i) \(x_{0}=0.5\) and (ii) \(x_{0}=2\).

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\frac{1}{(1+x)^{2}}, x \neq-1 $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\frac{1}{1+x} $$

In Problems 13-18, use a graphing calculator to determine all local and global extrema of the functions on their respective domains. \(f(x)=3-x, x \in[-1,3)\)

How close does the curve \(y=1 / x\) come to the origin? (Hint: Find the point on the curve that minimizes the square of the distance between the origin and the point on the curve. If you use the square of the distance instead of the distance, you avoid dealing with square roots.)

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} \frac{\ln (\ln x)}{\ln x} $$

Determine whether each function has absolute maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. $$ y=x^{2}(1-x), x \in \mathbf{R} $$

(a) Use the stability criterion to characterize the stability of the equilibria of $$ x_{t+1}=\frac{10 x_{t}^{2}}{9+x_{t}^{2}}, \quad t=0,1,2, \ldots $$ (b) Use cobwebbing to decide to which value \(x_{t}\) converges as \(t \rightarrow \infty\) if (i) \(x_{0}=0.5\) and (ii) \(x_{0}=3\).

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\frac{x^{2}}{x^{2}+1}, x \geq 0 $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\frac{x}{1+x} $$

Use a graphing calculator to determine all local and global extrema of the functions on their respective domains. \(f(x)=5+2 x, x \in(-2,1)\)

How close does the circle with radius \(\sqrt{2}\) and center \((2,2)\) come to the origin.

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0} \frac{2^{x}-1}{3^{x}-1} $$

Determine whether each function has absolute maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. $$ y=(x-1)^{1 / 3}, x \in \mathbf{R} $$

Ricker's curve is given by $$ R(P)=\alpha P e^{-\beta P} $$ for \(P \geq 0\), where \(P\) denotes the size of the parental stock and \(R(P)\) the number of recruits. The parameters \(\alpha\) and \(\beta\) are positive constants. (a) Show that \(R(0)=0\) and \(R(P)>0\) for \(P>0\). (b) Find $$ \lim _{P \rightarrow \infty} R(P) $$ (c) For what size of the parental stock is the number of recruits maximal? (d) Does \(R(P)\) have inflection points? If so, find them. (e) Sketch the graph of \(f(x)\) when \(\alpha=2\) and \(\beta=1 / 2\).

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\sin x, 0 \leq x \leq 2 \pi $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=5 x^{4}+\frac{5}{x^{4}} $$

Use a graphing calculator to determine all local and global extrema of the functions on their respective domains. \(f(x)=x^{2}-2, x \in[-1,1]\)

. Show that if \(f(x)\) is a positive twice-differentiable function that has a local minimum at \(x=c\), then \(g(x)=[f(x)]^{2}\) has a local minimum at \(x=c\) as well.

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0} \frac{5^{x}-1}{7^{x}-1} $$