Chapter 5

Use l'Hospital's rule to find the limits in Problems 1-50. $$ \lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5} $$

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. $$ y=(2-x)^{2},-2 \leq x \leq 3 $$

Assume a discrete-time population whose size at generation \(t+1\) is related to the size of the population at generation \(t\) by $$ N_{t+1}=(1.03) N_{t}, \quad t=0,1,2, \ldots $$ (a) If \(N_{0}=10\), how large will the population be at generation \(t=5 ?\) (b) How many generations will it take for the population size to reach double the size at generation \(0 ?\)

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. $$ y=3 x-x^{2}, x \in \mathbf{R} $$

Use the Newton-Raphson method to find a numerical approximation to the solution of $$ x^{2}-7=0 $$ that is correct to six decimal places.

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

In Problems \(1-8\), 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.) \(f(x)=2 x-1,0 \leq x \leq 1\)

$$ \begin{array}{l} \text { Find the smallest perimeter possible for a rectangle whose area }\\\ \text { is } 25 \mathrm{in} .^{2} \end{array} $$

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} $$

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. $$ y=\sqrt{x-1}, 1 \leq x \leq 2 $$

Suppose a discrete-time population evolves according to $$ N_{t+1}=(0.9) N_{t}, \quad t=0,1,2 \ldots $$ (a) If \(N_{0}=50\), how large will the population be at generation \(t=6 ?\) (b) After how many generations will the size of the population be one-quarter of its original size? (c) What will happen to the population in the long run - that is, as \(t \rightarrow \infty\) ?

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. $$ y=x^{2}+5 x, x \in \mathbf{R} $$

Use the Newton-Raphson method to find a numerical approximation to the solution of $$ e^{-x}=x $$ that is correct to six decimal places.

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=2-5 x^{2} $$

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

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow-2} \frac{3 x^{2}+5 x-2}{x+2} $$

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. $$ y=\ln (2 x-1), 1 \leq x \leq 2 $$

Assume the discrete-time population model $$ N_{t+1}=b N_{t}, \quad t=0,1,2 \ldots $$ Assume also that the population increases by \(2 \%\) each generation. (a) Determine \(b\). (b) Find the size of the population at generation 10 when \(N_{0}=\) 20 . (c) After how many generations will the population size have doubled?

Use the Newton-Raphson method to find a numerical approximation to the solution of $$ x^{2}+\ln x=0 $$ that is correct to six decimal places.

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

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.) \(f(x)=\sin (2 x), 0 \leq x \leq \pi\)

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

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. $$ y=\ln \frac{x}{x+1}, x>0 $$

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. $$ y=x^{2}-x+3, x \in \mathbf{R} $$

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

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.) \(f(x)=\cos \frac{x}{2}, 0 \leq x \leq 2 \pi\)

A rectangular study area is to be enclosed by a fence and divided into two equal parts, with the fence running along the division parallel to one of the sides. If the total area is \(384 \mathrm{ft}^{2}\), find the dimensions of the study area that will minimize the total length of the fence. How much fencing will be required?

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0} \frac{\sqrt{2 x+4}-2}{x} $$

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. $$ y=x e^{-x}, 0 \leq x \leq 1 $$

Assume the discrete-time population model $$ N_{t+1}=b N_{t}, \quad t=0,1,2, \ldots $$ Assume that the population increases by \(x \%\) each generation. (a) Determine \(b\). (b) After how many generations will the population size have doubled? Compute the doubling time for \(x=0.1,0.5,1,2,5\), and 10 .

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. $$ y=-\frac{2}{3} x^{3}+\frac{7}{2} x^{2}-3 x+4, x \in \mathbf{R} $$

Use the Newton-Raphson method to solve the equation $$ \sin x=\frac{1}{2} x $$ in the interval \((0, \pi)\).

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

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.) \(f(x)=|x|,-1 \leq x \leq 1\)

. A rectangular field is bounded on one side by a river and or the other three sides by a fence. Find the dimensions of the fiels that will maximize the enclosed area if the fence has a total lengtl of \(320 \mathrm{ft}\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0} \frac{3-\sqrt{2 x+9}}{2 x} $$

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. $$ y=\left|16-x^{2}\right|,-5 \leq x \leq 8 $$

(a) Find all equilibria of $$ N_{t+1}=1.3 N_{t}, \quad t=0,1,2, \ldots $$ (b) Use cobwebbing to determine the stability of the equilibria you found in (a).

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. $$ y=(x-2)^{3}+3, x \in \mathbf{R} $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=2 x^{3}+x^{2}-5 x $$

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

Find the largest possible area of a right triangle whose hypotenuse is \(4 \mathrm{~cm}\) long.

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. $$ y=(x-1)^{3}+1, x \in \mathbf{R} $$

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. $$ y=\sqrt{x+1}, x \geq-1 $$

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

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.) \(f(x)=e^{-|x|},-1 \leq x \leq 1\)

Suppose that \(a\) and \(b\) are the side lengths in a right triangle whose hypotenuse is \(5 \mathrm{~cm}\) long. What is the largest perimeter possible?

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

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. $$ y=x^{3}-3 x+1, x \in \mathbf{R} $$

(a) Find all equilibria of $$ N_{t+1}=N_{t}, \quad t=0,1,2, \ldots $$ (b) How will the population size \(N_{t}\) change over time, starting at time 0 with \(N_{0}\) ?