91Ó°ÊÓ

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 \(f\) .

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 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.

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

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.

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. \(\ln x=e^{-x},(1,2)\)