Chapter 5

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty}\left(1+\frac{3}{x}\right) $$

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty}\left(1+\frac{5}{x}\right)^{x} $$

In Problems \(41-46\), assume that \(a\) is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{e^{-a x}+e^{a x}}{2 a} $$

Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Show that if \(f(a)

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{2}{x}-x, x>0 $$

Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Assume further that \(f(a)=f(b)=0\) but \(f\) is not constant on \([a, b] .\) Explain why there must be a point \(c_{1} \in(a, b)\) with \(f^{\prime}\left(c_{1}\right)>0\) and a point \(c_{2} \in(a, b)\) with \(f^{\prime}\left(c_{2}\right)<0 .\)

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty}\left(1+\frac{3}{x^{2}}\right)^{x} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{2}{x^{3}}-x^{3}, x>0 $$

A car moves in a straight line. At time \(t\) (measured in seconds), its position (measured in meters) is $$ s(t)=\frac{1}{10} t^{2}, 0 \leq t \leq 10 $$ (a) Find its average velocity between \(t=0\) and \(t=10\). (b) Find its instantaneous velocity for \(t \in(0,10)\). (c) At what time is the instantaneous velocity of the car equal to its average velocity?

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

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=x(1+x), x>0 $$

A car moves in a straight line. At time \(t\) (measured in seconds), its position (measured in meters) is $$ s(t)=\frac{1}{100} t^{3}, 0 \leq t \leq 5 $$ (a) Find its average velocity between \(t=0\) and \(t=5\). (b) Find its instantaneous velocity for \(t \in(0,5)\). (c) At what time is the instantaneous velocity of the car equal to its average velocity?

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0^{+}}(\cos (2 x))^{3 / x} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=e^{-4 x}, x>0 $$

Denote the population size at time \(t\) by \(N(t)\), and assume that \(N(0)=50\) and \(|d N / d t| \leq 2\) for all \(t \in[0,5] .\) What can you say about \(N(5)\) ?

Find the limits in Problems \(51-60 .\) Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 0} x e^{x} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d t}=t(1-t), t \geq 0 $$

Denote the biomass at time \(t\) by \(B(t)\), and assume that \(B(0)=\) 3 and \(|d B / d t| \leq 1\) for all \(t \in[0,3]\). What can you say about \(B(3)\) ?

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 0^{+}} \frac{e^{x}}{x} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d t}=t^{2}\left(1-t^{2}\right), t \geq 0 $$

Suppose that \(f\) is differentiable for all \(x \in \mathbf{R}\) and, furthermore, that \(f\) satisfies \(f(0)=0\) and \(1 \leq f^{\prime}(x) \leq 2\) for all \(x>0\). (a) Use Corollary 1 of the MVT to show that $$ x \leq f(x) \leq 2 x $$ for all \(x \geq 0\). (b) Use your result in (a) to explain why \(f(1)\) cannot be equal to \(3 .\) (c) Find an upper and a lower bound for the value of \(f(1)\).

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow(\pi / 2)^{-}}(\tan x+\sec x) $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d t}=e^{-t / 2}, t \geq 0 $$

Suppose that \(f\) is differentiable for all \(x \in \mathbf{R}\) with \(f(2)=3\) and \(f^{\prime}(x)=0\) for all \(x \in \mathbf{R}\). Find \(f(x)\).

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow(\pi / 2)^{-}} \frac{\tan x}{1+\sec x} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d t}=1-e^{-3 t}, t \geq 0 $$

Suppose that \(f(x)=e^{-|x|}, x \in[-2,2]\). (a) Show that \(f(-2)=f(2)\). (b) Compute \(f^{\prime}(x)\), where defined. (c) Show that there is no number \(c \in(-2,2)\) such that \(f^{\prime}(c)=0\). (d) Explain why your results in (a) and (c) do not contradict Rolle's theorem. (e) Use a graphing calculator to sketch the graph of \(f(x)\).

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 1} \frac{x^{2}-1}{x+1} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d s}=\sin (\pi s), 0 \leq s \leq 1 $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d s}=\cos (2 \pi s), 0 \leq s \leq 1 $$

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow-\infty} x e^{x} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=\sec ^{2}\left(\frac{x}{2}\right),-1

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\frac{1}{\sqrt{x}}\right) $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=1+\sec ^{2}\left(\frac{x}{4}\right),-1

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 0^{+}} x^{3 x} $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d y}{d x}=3 x^{2}, \text { for } x \geq 0 \text { with } y=1 \text { when } x=0 $$

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow \infty}\left(\frac{x+1}{x+2}\right) $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d y}{d x}=\frac{x^{2}}{3}, \text { for } x \geq 0 \text { with } y=2 \text { when } x=0 $$

use l'Hospital's rule to find $$ \lim _{x \rightarrow 0} \frac{a^{x}-1}{b^{x}-1} $$ where \(a, b>0\).

In Problems 59-72, solve the initial-value problem. $$ \frac{d y}{d x}=2 \sqrt{x}, \text { for } x \geq 0 \text { with } y=2 \text { when } x=1 $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d y}{d x}=\frac{1}{2 \sqrt{x}}, \text { for } x \geq 1 \text { with } y=3 \text { when } x=4 $$

For \(p>0\), determine the values of \(p\) for which the following limit is either 1 or \(\infty\) or a constant that is neither 1 nor \(\infty\) : $$ \lim _{x \rightarrow \infty}\left(1+\frac{c}{x^{p}}\right)^{x} $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d N}{d t}=\frac{1}{t}, \text { for } t \geq 1 \text { with } N(1)=10 $$

Show that $$ \lim _{x \rightarrow \infty} x^{p} e^{-x}=0 $$ for any positive number \(p .\) Graph \(f(x)=x^{p} e^{-x}, x>0\), for \(p=1 / 2,1\), and \(2 .\) Since \(f(x)=x^{p} e^{-x}=x^{p} / e^{x}\), the limiting behavior \(\left(\lim _{x \rightarrow \infty} \frac{x^{p}}{e^{x}}=0\right)\) shows that the exponential function grows faster than any power of \(x\) as \(x \rightarrow \infty\).

In Problems 59-72, solve the initial-value problem. $$ \frac{d N}{d t}=\frac{t}{t+2}, \text { for } t \geq 0 \text { with } N(0)=2 $$

Show that $$ \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}}=0 $$ for any number \(p>0 .\) This shows that the logarithmic function grows more slowly than any positive power of \(x\) as \(x \rightarrow \infty\).

In Problems 59-72, solve the initial-value problem. $$ \frac{d W}{d t}=e^{t}, \text { for } t \geq 0 \text { with } W(0)=1 $$

When l'Hospital introduced indeterminate limits in his textbook, his first example was $$ \lim _{x \rightarrow a} \frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}} $$ where \(a\) is a positive constant. (This example was communicated to him by Bernoulli.) Show that this limit is equal to \((16 / 9) a\).

In Problems 59-72, solve the initial-value problem. $$ \frac{d W}{d t}=e^{-3 t}, \text { for } t \geq 0 \text { with } W(0)=2 $$

The height \(y\) in feet of a tree as a function of the tree's age \(x\) in years is given by $$ y=121 e^{-17 / x} \quad \text { for } x>0 $$ (a) Determine (1) the rate of growth when \(x \rightarrow 0^{+}\) and (2) the limit of the height as \(x \rightarrow \infty\). (b) Find the age at which the growth rate is maximal. (c) Show that the height of the tree is an increasing function of age. At what age is the height increasing at an accelerating rate and at what age at a decelerating rate? (d) Sketch the graph of both the height and the rate of growth of the tree as functions of age.