Chapter 2

Zoom in toward the points \((1,0),(0,1),\) and \((-1,0)\) on the graph of the function \(g(x)=\left(x^{2}-1\right)^{2 / 3} .\) What do you notice? Account for what you see in terms of the differentiability of \(g .\)

In the theory of relativity, the mass of a particle with velocity \(v\) is $$\mathrm{m}=\frac{\mathrm{m}_{0}}{\sqrt{1-v^{2} / \mathrm{c}^{2}}}$$ where \(m_{0}\) is the mass of the particle at rest and \(c\) is the speed of light. What happens as \(v \rightarrow c^{-}\) ?

\(39-44\) Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. $$ y=\frac{x^{2}+1}{2 x^{2}-3 x-2} $$

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-6} \frac{2 x+12}{|x+6|}$$

The gravitational force exerted by the earth on a unit mass at a distance \(r\) from the center of the planet is $$F(r)=\left\\{\begin{array}{ll}{\frac{G M r}{R^{3}}} & {\text { if } r

How close to \(-3\) do we have to take \(x\) so that $$\frac{1}{(\mathrm{x}+3)^{4}}>10,000$$

\(39-44\) Find the horlzontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. $$ y=\frac{2 x^{2}+x-1}{x^{2}+x-2} $$

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0.5^{-}} \frac{2 x-1}{\left|2 x^{3}-x^{2}\right|}$$

For what value of the constant \(c\) is the function f continuous on \((-\infty, \infty) ?\) $$f(x)=\left\\{\begin{array}{ll}{c x^{2}+2 x} & {\text { if } x<2} \\\ {x^{3}-c x} & {\text { if } x \geqslant 2}\end{array}\right.$$

(a) Use numerical and graphical evidence to guess the value of the limit $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{\sqrt{x}-1}$$ (b) How close to 1 does \(x\) have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?

Prove, using Definition \(6,\) that $$\lim _{x \rightarrow-3} \frac{1}{(x+3)^{4}}=\infty$$

\(39-44\) Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. $$ y=\frac{1+x^{4}}{x^{2}-x^{4}} $$

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-2} \frac{2-|x|}{2+x}$$

The cost (in dollars) of producing x units of a certain commodity is \(C(x)=5000+10 x+0.05 x^{2} .\) (a) Find the average rate of change of C with respect to \(x\) when the production level is changed (i) from \(x=100\) to \(x=105\) (ii) from \(x=100\) to \(x=101\) (b) Find the instantaneous rate of change of \(C\) with respect to \(x\) when \(x=100 .\) (This is called the marginal cost. Its significance will be explained in Section 3.7 )

Prove that $$\lim _{x \rightarrow 0^{+}} \ln x=-\infty$$

\(39-44\) Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. $$ y=\frac{x^{3}-x}{x^{2}-6 x+5} $$

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{|x|}\right)$$

Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function \(g\) that agrees with \(f\) for \(x \neq a\) and is continuous at a. (a) \(f(x)=\frac{x^{4}-1}{x-1}, \quad a=1\) (b) \(f(x)=\frac{x^{3}-x^{2}-2 x}{x-2}, \quad a=2\) (c) \(f(x)=[\sin x], \quad a=\pi\)

If a cylindrical tank holds \(100,000\) gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume \(V\) of water remaining in the tank after t minutes as $$\mathrm{V}(\mathrm{t})=100,000\left(1-\frac{\mathrm{t}}{60}\right)^{2} \quad 0 \leqslant \mathrm{t} \leqslant 60$$ Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t. What are its units? For times \(t=0,10,20,30,40,50,\) and 60 \(\mathrm{min}\) , find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two.At what time is the flow rate the greatest? The least?

Suppose that \(\lim _{x \rightarrow a} f(x)=\infty\) and \(\lim _{x \rightarrow a} g(x)=c,\) where \(c\) is a real number. Prove each statement. (a) $$\lim _{x \rightarrow a}[f(x)+g(x)]=\infty$$ (b) $$\lim _{x \rightarrow a}[f(x) g(x)]=\infty \quad\( if \)c>0$$ (c) $$\lim _{x \rightarrow a}[f(x) g(x)]=-\infty\( if \)c<0$$

\(39-44\) Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. $$ y=\frac{2 e^{x}}{e^{x}-5} $$

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\frac{1}{|x|}\right)$$

Suppose that a function \(f\) is continuous on \([0,1]\) except at 0.25 and that \(f(0)=1\) and \(f(1)=3 .\) Let \(N=2 .\) Sketch two possible graphs of \(f\) , one showing that \(f\) might not satisfy the conclusion of the Intermediate Value Theorem and one showing that \(f\) might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn't satisfy the hypothesis).

Use the definition of a derivative to find \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\) . Then graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) on a common screen and check to see if your answers are reasonable. \(f(x)=1+4 x-x^{2}\)

The cost of producing x ounces of gold from a new gold mine is \(C=f(x)\) dollars. (a) What is the meaning of the derivative \(f^{\prime}(x) ?\) What are its units? (b) What does the statement \(f^{\prime}(800)=17\) mean? (c) Do you think the values of \(f^{\prime}(x)\) will increase or decrease in the short term? What about the long term? Explain.

$$ \begin{array}{c}{\text { Estimate the horizontal asymptote of the function }} \\ {f(x)=\frac{3 x^{3}+500 x^{2}}{x^{3}+500 x^{2}+100 x+2000}}\end{array} $$ by graphing \(f\) for \(-10 \leqslant x \leqslant 10\) . Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

The signum (or sign) function, denoted by sgn, is defined by $$\operatorname{sgn} x=\left\\{\begin{aligned}-1 & \text { if } x<0 \\ 0 & \text { if } x=0 \\ 1 & \text { if } x>0 \end{aligned}\right.$$ (a) Sketch the graph of this function. (b) Find each of the following limits or explain why it does not exist. (i) $$\lim _{x \rightarrow 0^{+}} \operatorname{sgn} x$$ (ii) $$\lim _{x \rightarrow 0^{-}} \operatorname{sgn} x$$ (iii) $$\lim _{x \rightarrow 0} \operatorname{sgn} x$$ (iv) $$\lim _{x \rightarrow 0}|\operatorname{sgn} x|$$

If \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+10 \mathrm{sin} \mathrm{x}\) , show that there is a number c such that \(\mathrm{f}(\mathrm{c})=1000 .\)

Use the definition of a derivative to find \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\) . Then graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) on a common screen and check to see if your answers are reasonable. \(f(x)=1 / x\)

The number of bacteria after thours in a controlled laboratory experiment is \(\mathrm{n}=\mathrm{f}(\mathrm{t})\) . (a) What is the meaning of the derivative \(f^{\prime}(5) ?\) What are its units? (b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, \(f^{\prime}(5)\) or \(f^{\prime}(10) ?\) If the supply of nutrients is limited, would that affect your conclusion? Explain.

Let $$f(x)=\left\\{\begin{array}{ll}{4-x^{2}} & {\text { if } x \leqslant 2} \\\ {x-1} & {\text { if } x>2}\end{array}\right.$$ (a) Find \(\lim _{x \rightarrow 2-} f(x)\) and \(\lim _{x \rightarrow 2^{+}} f(x).\) (b) Does \(\lim _{x \rightarrow 2} f(x)\) exist? (c) Sketch the graph of \(f\) .

Suppose \(f\) is continuous on \([1,5]\) and the only solutions of the equation \(f(x)=6\) are \(x=1\) and \(x=4 .\) If \(f(2)=8\) , explain why \(f(3)>6\)

Let T(t) be the temperature (in \(^{\circ} \mathrm{F}\) ) in Dallas t hours after mid- night on June \(2,2001 .\) The table shows values of this function recorded every two hours. What is the meaning of \(\mathrm{T}^{\prime}(10) ?\) Estimate its value.

Let \(F(x)=\frac{x^{2}-1}{|x-1|}.\) (a) Find (i) $$\lim _{x \rightarrow 1^{+}} F(x)$$ (ii) $$\lim _{x \rightarrow 1^{-}} F(x)$$ (b) Does lim \(_{x \rightarrow 1}\) F(x) exist? (c) Sketch the graph of F.

\(47-50\) Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. \(x^{4}+x-3=0\), \((1,2)\)

Find a formula for a function that has vertical asymptotes \(x=1\) and \(x=3\) and horizontal asymptote \(y=1\)

Let $$g(x)=\left\\{\begin{array}{ll}{x} & {\text { if } x < 1} \\ {3} & {\text { if } x=1} \\ {2-x^{2}} & {\text { if } 1 < x \leqslant 2} \\ {x-3} & {\text { if } x > 2}\end{array}\right.$$ (a) Evaluate each of the following limits, if it exists. (i) $$\lim _{x \rightarrow 1^{-}} g(x)$$ (ii) $$\lim _{x \rightarrow 1} g(x)$$ (iii) \(g(1)\) (iv) $$\lim _{x \rightarrow 2^{-}} g(x)$$ (v) $$\lim _{x \rightarrow 2^{+}} g(x)$$ (vi) $$\lim _{x \rightarrow 2} g(x)$$ (b) Sketch the graph of \(g .\)

\(47-50\) Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. \(\sqrt[3]{\mathrm{x}}=1-\mathrm{x}_{,} \quad(0,1)\)

(a) If \(g(x)=x^{2 / 3}\) , show that \(g^{\prime}(0)\) does not exist. (b) If \(a \neq 0,\) find \(g^{\prime}(a).\) (c) Show that \(y=x^{2 / 3}\) has a vertical tangent line at \((0,0).\) (d) Illustrate part (c) by graphing y \(=x^{2 / 3}\).

\(47-50\) Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. \(\ln x=e^{-x},(1,2)\)

Show that the function \(f(x)=|x-6|\) is not differentiable at \(6 .\) Find a formula for \(f^{\prime}\) and sketch its graph.

\(51-52\) Determine whether \(\mathrm{f}^{\prime}(0)\) exists. \(f(x)=\left\\{\begin{array}{ll}{x \sin \frac{1}{x}} & {\text { if } x \neq 0} \\\ {0} & {\text { if } x=0}\end{array}\right.\)

\(51-52\) (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. \(\cos x=x^{3}\)

\(f(x)=\left\\{\begin{array}{ll}{x^{2} \sin \frac{1}{x}} & {\text { if } x \neq 0} \\ {0} & {\text { if } x=0}\end{array}\right.\)

\(49-52\) Find the limits as \(x \rightarrow \infty\) and as \(x \rightarrow-\infty\) . Use this infor- mation, together with intercepts, to give a rough sketch of the graph as in Example \(11 .\) \(y=x^{2}\left(x^{2}-1\right)^{2}(x+2)\)

In the theory of relativity, the Lorentz contraction formula $$\mathrm{L}=\mathrm{L}_{0} \sqrt{1-v^{2} / \mathrm{c}^{2}}$$ expresses the length \(L\) of an object as a function of its velocity \(v\) with respect to an observer, where \(L_{0}\) is the length of the object at rest and \(c\) is the speed of light. Find lim \(_{v \rightarrow c-} L\) and interpret the result. Why is a left-hand limit necessary?

\(51-52\) (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. \(\ln x=3-2 x\)

(a) Sketch the graph of the function \(f(x)=x|x|.\) (b) For what values of \(x\) is f differentiable? (c) Find a formula for \(f^{\prime}\) .'

If p is a polynomial, show that lim \(_{x \rightarrow a} p(x)=p(a).\)

\(53-54\) (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. \(100 e^{-x / 100}=0.01 x^{2}\)