Chapter 13

Heart Rate \(A\) cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after \(t\) min. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. $$\begin{array}{|c|c|}\hline t \text { (min) } & \text { Heartbeats } \\\\\hline 36 & 2530 \\\38 & 2661 \\\40 & 2806 \\\42 & 2948 \\\44 & 3080 \\\\\hline\end{array}$$ (a) Find the average heart rates (slopes of the secant lines) over the time intervals \([40,42]\) and \([42,44]\) (b) Estimate the patient's heart rate after 42 min by averaging the slopes of these two secant lines.

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

Water Flow \(A\) tank holds 1000 gal of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume \(V\) of water remaining in the tank (in gal) after \(t\) minutes. $$\begin{array}{c|c}\hline t \text { (min) } & V \text { (gal) } \\\\\hline 5 & 694 \\\10 & 444 \\ 15 & 250 \\\20 & 111 \\\25 & 28 \\\30 & 0 \\\\\hline\end{array}$$ (a) Find the average rates at which water flows from the tank (slopes of secant lines) for the time intervals \([10,15]\) and \([15,20]\) (b) The slope of the tangent line at the point \((15,250)\) represents the rate at which water is flowing from the tank after 15 min. Estimate this rate by averaging the slopes of the secant lines in part (a).

Does the Limit Exist? Let $$f(x)=\left\\{\begin{array}{ll} x-1 & \text { if } x<2 \\ x^{2}-4 x+6 & \text { if } x \geq 2 \end{array}\right.$$ (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 \(f\)

Does the Limit Exist? Let $$h(x)=\left\\{\begin{array}{ll} x & \text { if } x<0 \\ x^{2} & \text { if } 02 \end{array}\right.$$ (a) Evaluate each limit if it exists. (i) \(\lim _{x \rightarrow 0^{+}} h(x)\) (iv) \(\lim _{x \rightarrow 2^{-}} h(x)\) (ii) \(\lim _{x \rightarrow 0} h(x)\) (v) \(\lim _{x \rightarrow 2^{+}} h(x)\) (iii) \(\lim _{x \rightarrow 1} h(x)\) (vi) \(\lim _{x \rightarrow 2} h(x)\) (b) Sketch the graph of \(h.\)

DISCUSS: Cancellation and Limits (a) What is wrong with the following equation? $$\frac{x^{2}+x-6}{x-2}=x+3$$ (b) In view of part (a), explain why the equation $$\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}=\lim _{x \rightarrow 2}(x+3)$$ is correct.

DISCUSS: The Lorentz Contraction In the theory of relativity the Lorentz contraction formula $$L=L_{0} \sqrt{1-v^{2} / c^{2}}$$ 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?

DISCUSS PROVE: Limits of Sums and Products (a) 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. (b) 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.