Chapter 2

The left-hand and right-hand derivatives of \(f\) at a are defined by $$f_{-}^{\prime}(a)=\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h}$$ and $$\quad f_{+}^{\prime}(a)=\lim _{n \rightarrow 0^{+}} \frac{f(a+h)-f(a)}{h}$$ if these limits exist. Then \(\mathrm{f}^{\prime}(\mathrm{a})\) exists if and only if these one-sided derivatives exist and are equal. (a) Find \(\mathrm{f}_{-}^{\prime}(4)\) and \(\mathrm{f}_{+}^{\prime}(4)\) for the function $$f(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x \leqslant 0} \\ {5-x} & {\text { if } 0 < x < 4} \\ {\frac{1}{5-x}} & {\text { if } x \geq 4}\end{array}\right.$$ (b) Sketch the graph of f. (c) Where is f discontinuous? (d) Where is f not differentiable?

\(53-54\) (a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. \(\arctan x=1-x\)

Recall that a function \(f\) is called even if \(f(-x)=f(x)\) for all \(x\) in its domain and odd if \(f(-x)=-f(x)\) for all such \(x\) . Prove each of the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.

$$ \begin{array}{c}{\text { Let } P \text { and } Q \text { be polynomials. Find }} \\ {\lim _{x \rightarrow \pi} \frac{P(x)}{Q(x)}}\end{array} $$ $$ \begin{array}{l}{\text { If the degree of } P \text { is (a) less than the degree of } \mathrm{O} \text { and }} \\ {\text { (b) greater than the degree of } \mathrm{O} \text { . }}\end{array} $$

If $$\lim _{x \rightarrow 1} \frac{f(x)-8}{x-1}=10,$$ find $$\lim _{x \rightarrow 1} f(x).$$

Prove that \(f\) is continuous at a if and only if $$\lim _{h \rightarrow 0} f(a+h)=f(a)$$

When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. (a) Sketch a possible graph of T as a function of the time t that has elapsed since the faucet was turned on. (b) Describe how the rate of change of T with respect to t varies as t increases. (c) Sketch a graph of the derivative of T.

$$ \begin{array}{l}{\text { Make a rough sketch of the curve } y=x^{n}(n \text { an integer })} \\ {\text { for the following five cases: }}\end{array} $$ $$ \begin{array}{ll}{\text { (i) } \mathrm{n}=0} & {\text { (ii) } \mathrm{n}>0, \mathrm{n} \text { odd }} \\ {\text { (iii) } \mathrm{n}>0, \mathrm{n} \text { even }} & {\text { (iv) } \mathrm{n}<0, \mathrm{n} \text { odd }}\end{array} $$ $$ \begin{array}{l}{\text { (v) } \mathrm{n}<0, \text { n even }} \\ {\text { Then use these sketches to find the following limits. }}\end{array} $$ (a) $$ \lim _{x \rightarrow \mathbb{u}^{+}} x^{n} \quad \text { (b) } \lim _{x \rightarrow 0^{-}} x^{n} $$ (c) $$ \lim _{x \rightarrow \infty} x^{n} \quad \text { (d) } \lim _{x \rightarrow-\infty} x^{n} $$

Let \(\ell\) be the tangent line to the parabola \(y=x^{2}\) at the point \((1,1)\) . The angle of inclination of \(\ell\) is the angle \(\phi\) that \(\ell\) makes with the positive direction of the x-axis. Calculate \(\phi\) correct to the nearest degree.

$$ \lim _{x \rightarrow \infty} f(x) \text { if, for all } x>1 $$ $$ \frac{10 \mathrm{e}^{x}-21}{2 \mathrm{e}^{\mathrm{x}}}<\mathrm{f}(\mathrm{x})<\frac{5 \sqrt{\mathrm{x}}}{\sqrt{\mathrm{x}-1}} $$

If $$f(x)=\left\\{\begin{array}{ll}{x^{2}} & {\text { if } x \text { is rational }} \\ {0} & {\text { if } x \text { is irrational }}\end{array}\right.$$ prove that \(\lim _{x \rightarrow 0} f(x)=0.\)

Prove that cosine is a continuous function.

(a) \(A\) tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 \(\mathrm{L} / \mathrm{min}\) . Show that the concentration of salt after t minutes (in grams per liter) is $$ \begin{array}{c}{\mathrm{C}(\mathrm{t})=\frac{30 \mathrm{t}}{200+\mathrm{t}}} \\ {\text { (b) What happens to the concentration as } \mathrm{t} \rightarrow \infty ?}\end{array} $$

Show by means of an example that \(\lim _{x \rightarrow a}[f(x)+g(x)]\) may exist even though neither lim \(_{x \rightarrow a} f(x)\) nor \(\lim _{x \rightarrow a} g(x)\) exists.

Show by means of an example that \(\lim _{x \rightarrow a}[f(x) g(x)]\) may exist even though neither \(\lim _{x \rightarrow a} f(x)\) nor \(\lim _{x \rightarrow a} g(x)\) exists.

For what values of \(x\) is \(f\) continuous? $$f(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x \text { is rational }} \\\ {1} & {\text { if } x \text { is irrational }}\end{array}\right.$$

Evaluate $$\lim _{x \rightarrow 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}.$$

For what values of \(x\) is \(g\) continuous? $$g(\mathrm{x})=\left\\{\begin{array}{ll}{0} & {\text { if } x \text { is rational }} \\ {x} & {\text { if } x \text { is irrational }}\end{array}\right.$$

$$ \begin{array}{c}{\text { Use a graph to find a number } N \text { such that }} \\ {\text { If } \quad x>N \quad \text { then } \quad\left|\frac{3 x^{2}+1}{2 x^{2}+x+1}-1.5\right|<0.05}\end{array} $$

Is there a number a such that $$\lim _{x \rightarrow-2} \frac{3 x^{2}+a x+a+3}{x^{2}+x-2}$$ exists? If so, find the value of a and the value of the limit.

Is there a number that is exactly 1 more than its cube?

$$ \begin{array}{l}{\text { For the limit }} \\ {\quad \lim _{x \rightarrow \infty} \frac{\sqrt{4 x^{2}+1}}{x+1}=2} \\ {\text { illustrate Definition } 7 \text { by finding values of } N \text { that correspond }} \\ {\text { to } \varepsilon=0.5 \text { and } \varepsilon=0.1}\end{array} $$

If a and b are positive numbers, prove that the equation $$\frac{\mathrm{a}}{\mathrm{x}^{3}+2 \mathrm{x}^{2}-1}+\frac{\mathrm{b}}{\mathrm{x}^{3}+\mathrm{x}-2}=0$$ has at least one solution in the interval \((-1,1)\) .

$$ \begin{array}{l}{\text { For the limit }} \\ {\quad \lim _{x \rightarrow-\infty} \frac{\sqrt{4 x^{2}+1}}{x+1}=-2} \\ {\text { illustrate Dcfinition } 8 \text { by finding values of } N \text { that correspond }} \\\ {\text { to } \varepsilon=0.5 \text { and } \varepsilon=0.1}\end{array} $$

Show that the function $$f(x)=\left\\{\begin{array}{ll}{x^{4} \sin (1 / x)} & {\text { if } x \neq 0} \\\ {0} & {\text { if } x=0}\end{array}\right.$$ is continuous on \((-\infty, \infty)\)

(a) Show that the absolute value function \(F(x)=|x|\) is continuous everywhere. (b) Prove that if \(f\) is a continuous function on an interval, then so is \(|\mathrm{f}| .\) (c) Is the converse of the statement in part (b) also true? In other words, if \(|\mathrm{f}|\) is continuous, does it follow that \(\mathrm{f}\) is continuous? If so, prove it. If not, find a counterexample.

$$ \begin{array}{c}{\text { (a) How large do we have to take } x \text { so that } 1 / x^{2}<0.0001 ?} \\ {\text { (b) Taking } r=2 \text { in Theorem } 5, \text { we have the statement }} \\ {\quad \lim _{x \rightarrow \infty} \frac{1}{x^{2}}=0} \\ {\text { Prove this directly using Definition } 7}\end{array} $$

A Tibetan monk leaves the monastery at \(7 : 00 \mathrm{AM}\) and takes his usual path to the top of the mountain, arriving at \(7 : 00 \mathrm{PM}\) . The following morning, he starts at \(7 : 00\) AM at the top and takes the same path back, arriving at the monastery at \(7 : 00 \mathrm{PM} .\) Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.

$$ \lim _{x \rightarrow-\infty} \frac{1}{x}=0 $$

$$ \begin{array}{c}{\text { Formulate a precise definition of }} \\ {\quad \lim _{x \rightarrow-\infty} f(x)=-\infty} \\ {\text { Then use your definition to prove that }} \\ {\lim _{x \rightarrow-\infty}\left(1+x^{3}\right)=-\infty}\end{array} $$