Chapter 5

Compute \(P^{\prime}(t)\) for: a. \(P(t)=e^{5 t}\) b. \(P(t)=\ln 5 t\) c. \(P(t)=e^{t \sqrt{t}}\) d. \(\quad P(t)=e^{\sqrt{2 t}}\) e. \(P(t)=\ln (\ln t)\) f. \(P(t)=e^{\ln t}\) g. \(P(t)=1 /\left(1+e^{t}\right)\) h. \(P(t)=1 / \ln t\) i. \(\quad P(t)=1 /\left(1+e^{-t}\right)\) j. \(\quad P(t)=\left(1+e^{t}\right)^{3}\) k. \(P(t)=\left(e^{\sqrt{t}}\right)^{3} \quad\) l. \(\quad P(t)=\ln \sqrt{t}\)

Use one rule for each step and identify the rule to differentiate a. \(P(t)=3 \ln t+e^{3 t}\) b. \(P(t)=t^{2}+\ln 2 t\) c. \(P(t)=\ln 5\) d. \(P(t)=\ln \left(e^{2 t}\right)\) e. \(P(t)=\ln \left(t^{2}+t\right)\) f. \(P(t)=e^{t^{2}-t}\) g. \(P(t)=e^{1 / x}\) h. \(P(t)=e^{\sqrt{x}}\) i. \(P(t)=\ln \left((t+1)^{2}\right)\) j. \(P(t)=e^{-t^{2} / 2}\)

Write a solution for each of the following derivative equations. Sketch the graph of the solution. For each, find the doubling time, \(t_{d b l},\) or half life, \(t_{1 / 2},\) which ever is applicable. a. \(P(0)=5 \quad P^{\prime}(t)=2 P(t)\) b. \(P(0)=5 \quad P^{\prime}(t)=-2 P(t)\) c. \(P(0)=2 \quad P^{\prime}(t)=0.1 P(t)\) d. \(P(0)=2 \quad P^{\prime}(t)=-0.1 P(t)\) e. \(P(0)=10 \quad P^{\prime}(t)=P(t)\) f. \(P(0)=10 \quad P^{\prime}(t)=-P(t)\) g. \(P(0)=0 \quad P^{\prime}(t)=0.01 P(t)\) h. \(P(0)=0 \quad P^{\prime}(t)=-0.01 P(t)\)

Derivatives of functions are computed below. Identify the rule used in each step. In a few steps the rule is an algebraic rule of exponents and not a derivative rule. a. \(\left[5 t^{4}-7 e^{t}\right]^{\prime}\) \(\left[5 t^{4}\right]^{\prime}-\left[7 e^{t]^{\prime}}\right.\) \(5\left[t^{4}\right]^{\prime}-7\left[e^{t}\right]^{\prime}\) \(5 \times 4 t^{3}-7\left[e^{t}\right]^{\prime}\) \(5 \times 4 t^{3}-7 \times e^{t}\) \(\begin{array}{l}\text { b. } \quad\left[\left(1+e^{t}\right)^{8}\right]^{\prime} \\ & 8\left(1+e^{t}\right)^{7}\left[1+e^{t}\right]^{\prime} \\ & 8\left(1+e^{t}\right)^{7}\left([1]^{\prime}+\left[e^{t}\right]^{\prime}\right) \\\ & 8\left(1+e^{t}\right)^{7}\left(0+\left[e^{t}\right]^{\prime}\right) \\\ & 8\left(1+e^{t}\right)^{7}\left(0+e^{t}\right) \\ & 8 e^{t}\left(1+e^{t}\right)^{7}\end{array}\) c. \(\left[e^{3 t}\right]^{\prime}\) \(\left[\left(e^{t}\right)^{3}\right]^{\prime}\) \(3\left(e^{t}\right)^{2}\left[e^{t}\right]^{\prime}\) \(3\left(e^{t}\right)^{2} \times e^{t}\) \(3 e^{2 t} \times e^{t}\) \(3 e^{3 t}\)

(a) Compute the centered difference $$\frac{P(a+h)-P(a-h)}{2 h}$$ which is an approximation to \(P^{\prime}(a),\) for \(P(t)=t^{2}\) and compare your answer with \(P^{\prime}(a)\). (b) Compute the centered difference $$\frac{P(a+h)-P(a-h)}{2 h}$$ for \(P(t)=5 t^{2}-3 t+7\) and compare your answer with \(P^{\prime}(a)\).

Use the logarithmic differentiation to compute \(y^{\prime}(t)\) for a. \(y(t)=10^{t}\) b. \(y(t)=\frac{t-1}{t+1}\) c. \(y(t)=(t-1)^{3}\left(t^{3}-1\right)\) d. \(y(t)=(t-1)(t-2)(t-3)\) e. \(y(t)=u(t) v(t) w(t)\) f. \(y(t)=u(t) v(t)\)

Compute the derivatives of a. \(\quad P(t)=e^{\left(t^{2}\right)}\) b. \(P(t)=\ln \left(t^{2}\right)\) c. \(P(t)=\left(e^{t}\right)^{2}\) d. \(P(t)=e^{(2 \ln t)}\) e. \(P(t)=\ln \left(e^{3 t}\right)\) f. \(\quad P(t)=\sqrt{e^{t}}\) g. \(\quad P(t)=e^{5}\) h. \(P(t)=\ln (\sqrt{t})\) i. \(\quad P(t)=e^{t+1}\)

Use Equation 5.12, \(\log _{d} A=\ln A / \ln d\) to compute \(\log _{2} A\) for \(A=1,2,3, \cdots, 10\).

Differentiate (means compute the derivative of) \(P\). Use one rule for each step and identify the rule as, C (Constant Rule), \(t^{n}\left(t^{n}\right.\) Rule), S (Sum Rule), CF (Constant Factor Rule), PC (Power Chain Rule), or E (Exponential Rule). For example, $$ \begin{aligned} \left[\pi t^{-2}-5\left(e^{t}\right)^{7}\right]^{\prime} &=\left[\pi t^{-2}\right]^{\prime}-\left[5\left(e^{t}\right)^{7}\right]^{\prime} \\ &=\pi\left[t^{-2}\right]^{\prime}-5\left[\left(e^{t}\right)^{7}\right]^{\prime} \\\ &=\pi \times(-2) t^{-3}-5\left[\left(e^{t}\right)^{7}\right]^{\prime} \\ &=-2 \pi t^{-3}-5(7)\left(e^{t}\right)^{6}\left[e^{t}\right]^{\prime} \\ &=-2 \pi t^{-3}-35\left(e^{t}\right)^{6} \times e^{t} \\ &=-2 \pi t^{-3}-35\left(e^{t}\right)^{7} \end{aligned} $$ a. \(\quad P(t)=5 t^{2}+32 e^{t}\) b. \(\quad P(t)=3\left(e^{t}\right)^{5}-6 t^{5}\) c. \(\quad P(t)=7-8\left(e^{t}\right)^{-1}\) d. \(\quad P(t)=\frac{2}{5}+\frac{t}{3}\) e. \(\quad P(t)=t^{25}+3 e\) f. \(\quad P(t)=\frac{4}{e}+\frac{t^{5}}{7}\) g. \(\quad P(t)=1+t+\frac{t^{2}}{2}-e^{t}\) h. \(P(t)=\frac{\left(e^{t}\right)^{2}}{2}+\frac{t^{3}}{3}\)

Technology Sketch the graphs of \(y=2^{t}\) and \(y=4+2.7725887(t-2)\) a. Using a window of \(0 \leq x \leq 2.5,0 \leq y \leq 6\). b. Using a window of \(1.5 \leq x \leq 2.5,0 \leq y \leq 6\). c. Using a window of \(1.8 \leq x \leq 2.2,3.3 \leq y \leq 4.6\). Mark the point (2,4) on each graph.

Use a semilog graph to determine which of the following data sets are exponential. a.$$ \begin{array}{|c|c|} \hline \mathrm{t} & \mathrm{P}(\mathrm{t}) \\ \hline 0 & 5.00 \\ 1 & 3.53 \\ 2 & 2.50 \\ 3 & 1.77 \\ 4 & 1.25 \\ 5 & 0.88 \\ \hline \end{array} $$ b. $$ \begin{array}{|c|c|} \hline \mathrm{t} & \mathrm{P}(\mathrm{t}) \\ \hline 0 & 5.00 \\ 1 & 1.67 \\ 2 & 1.00 \\ 3 & 0.71 \\ 4 & 0.55 \\ 5 & 0.45 \\ \hline \end{array} $$ c. $$ \begin{array}{|c|c|} \hline \mathrm{t} & \mathrm{P}(\mathrm{t}) \\ \hline 0 & 5.00 \\ 1 & 3.63 \\ 2 & 2.50 \\ 3 & 1.63 \\ 4 & 1.00 \\ 5 & 0.63 \\ \hline \end{array} $$

Find values of \(C\) and \(k\) so that \(P(t)=C e^{k t}\) matches the data. a. \(P(0)=5 \quad P(2)=10\) b. \(P(0)=10 \quad P(2)=5\) c. \(P(0)=2 \quad P(5)=10\) d. \(P(0)=10 \quad P(5)=10\) e. \(P(0)=5 \quad P(2)=2\) f. \(P(0)=8 \quad P(10)=6\) g. \(P(1)=5 \quad P(2)=10\) h. \(P(2)=10 \quad P(10)=20\)

The function \(b^{t}\) for \(b=1\) is a special exponential function. Confirm that the derivative equation \(\left[b^{t}\right]^{\prime}=b^{t} \ln b\) is valid for \(b=1\). Draw some graphs of \(b^{t}\) for \(b=1\) and its derivative.

We found in Section 1.3 that light intensity, \(I_{d},\) as a function of depth, \(d\) was given by $$I_{d}=0.4 \times 0.82^{d}$$ Find \(k\) so that \(I_{d}=0.4 e^{k \times d}\).

Draw the graphs of $$y_{1}(t)=e^{t} \quad y_{2}(t)=t^{2} \quad y_{3}(t)=t^{3} \quad y_{4}(t)=t^{4} \quad-1 \leq t \leq 5$$ The graphs are close together near \(t=0\) and increase as \(t\) increases. Which one grows the most as \(t\) increases? Expand the domain and range to \(-1 \leq t \leq 10,0 \leq y \leq 25,000,\) and answer the same question.

Let \(E(t)=10^{t}\) a. Approximate \(E^{\prime}(0)\) using the centered difference quotient on [-0.0001,0.0001] . b. Use your value for \(E^{\prime}(0)\) and \(E^{\prime}(t)=E^{\prime}(0) E(t)\) to approximate \(E^{\prime}(-1), E^{\prime}(1),\) and \(E^{\prime}(2)\). c. Sketch the graphs of \(E(t)\) and \(E^{\prime}(t)\). d. Repeat a., b., and c. for \(E(t)=8^{t}\).

Use one rule for each step and identify the rule to differentiate $$ \begin{array}{ll} \text { a. } P(t)=3 e^{5 t}+\pi & \text { b. } P(t)=\frac{e^{2}}{2}+\frac{t^{3}}{3} \\ \text { c. } P(t)=5^{t} & \text { d. } P(t)=e^{2 t} e^{3 t} \end{array} $$

Write each of the following functions in the form \(f(t)=A e^{k t} .\) a. \(\quad f(t)=5 \cdot 10^{t}\) b. \(\quad f(t)=5 \cdot 10^{-t}\) c. \(\quad f(t)=7 \cdot 2^{t}\) d. \(\quad f(t)=5 \cdot 2^{-t}\) e. \(\quad f(t)=5\left(\frac{1}{2}\right)^{t}\) f. \(\quad f(t)=5\left(\frac{1}{2}\right)^{-t}\)

Draw the graphs of $$y(t)=e^{t} \quad \text { and } \quad p(t)=1+t+\frac{t^{2}}{2}+\frac{t^{3}}{6}+\frac{t^{4}}{24}$$ Set the domain and range to \(-1 \leq t \leq 2\). and \(0 \leq y \leq 8\).

Let \(E(t)=\left(\frac{1}{2}\right)^{t}\). a. Approximate \(E^{\prime}(0)\) using the centered difference quotient on [-0.0001,0.0001] . b. Use your value for \(E^{\prime}(0)\) and \(E^{\prime}(t)=E^{\prime}(0) E(t)\) to approximate \(E^{\prime}(-1), E^{\prime}(1),\) and \(E^{\prime}(2)\). c. Sketch the graphs of \(E(t)\) and \(E^{\prime}(t)\).

A pristine lake of volume \(1,000,000 \mathrm{~m}^{3}\) has a river flowing through it at a rate of \(10,000 \mathrm{~m}^{3}\) per day. A city built beside the river begins dumping \(1000 \mathrm{~kg}\) of solid waste into the river per day. 1\. Write a derivative equation that describes the amount of solid waste in the lake \(t\) days after dumping begins. 2\. What will be the concentration of solid waste in the lake after one year?

Suppose a barrel has 100 liters of water and 400 grams of salt and at time \(t=0\) minutes a stream of water flowing at 5 liters per minute and carrying \(3 \mathrm{~g} /\) liter of salt starts flowing into the barrel, the barrel is well mixed, and a stream of water and salt leaves the barrel at 5 liters per minute. What is the amount of salt in the barrel \(t\) minutes after the flow begins? Draw a candidate solution graph for this problem before computing the solution.

Compute \(y^{\prime}(x)\) or assert that you do not yet have forumlas to compute \(\mathbf{y}^{\prime}(\mathbf{x})\) for a. \(y(x)=e^{5 x}\) b. \(y(x)=e^{-3 x}\) c. \(y(x)=e^{\sqrt{x}}\) d. \(y(x)=\left(e^{x}\right)^{2}\) e. \(y(x)=\left(e^{\sqrt{x}}\right)^{2}\) f. \(y(x)=\left(e^{-x}\right)^{2}\) g. \(y(x)=\frac{e^{x}+e^{-x}}{2}\) h. \(y(x)=\frac{e^{x}-e^{-x}}{2}\) i. \(y(x)=5 e^{-0.06 x}+3 e^{-0.1 x} \quad\) j. \(y(x)=e^{\left(x^{2}\right)}\) k. \(y(x)=\sqrt{e^{x}}\) l. \(y(x)=8 e^{-0.0001 x}-16 e^{-0.001 x}\) m. \(y(x)=e^{5}\) n. \(y(x)=\sqrt{e}\) o. \(y(x)=10^{x}\) p. \(y(x)=10^{-x}\) q. \(y(x)=x^{2}+2^{x}\) r. \(y(x)=\left(e^{5 x}+e^{-3 x}\right)^{5}\)

We found a base \(e\) so that \(E(t)=e^{t}\) has the property that the rate of change of \(E\) at 0 is \(1 .\) Suppose we had searched for a number \(B\) so that the average rate of change of \(E_{B}(t)=B^{t}\) on [0,0.01] is 1: $$m_{0,0.01}=\frac{E_{B}(0.01)-E_{B}(0)}{0.01}=\frac{B^{0.01}-B^{0}}{0.01}=\frac{B^{0.01}-1}{0.01}=1$$ a. Solve the last equation for \(B\). b. Solve for \(B\) in each of the equations: $$\frac{B^{0.001}-1}{0.001}=1 \quad \frac{B^{0.00001}-1}{0.00001}=1 \quad \frac{B^{0.0000001}-1}{0.0000001}=1$$

Find (approximately) equations of the lines tangent to the graphs of a. \(y=1.5^{t}\) at the points \((-1,2 / 3), \quad(0,1),\) and \((1,3 / 2)\) b. \(y=2^{t} \quad\) at the point \(\quad(-1,1 / 2), \quad(0,1),\) and (1,2) c. \(y=3^{t} \quad\) at the point \(\quad(-1,1 / 3), \quad(0,1),\) and (1,3) d. \(y=5^{t} \quad\) at the point \(\quad(-1,1 / 5), \quad(0,1), \quad\) and \(\quad(1,5)\)

Estimate the slope of the tangent to the graph of $$y=\log _{10} x$$ at the point \(\left(3, \log _{10} 3\right)\) correct to three decimal digits.

Use the logarithm chain rule to prove that for all numbers, \(n\) : Power chain rule for all \(n \quad\left[(u(t))^{n}\right]^{\prime}=n(u(t))^{n-1} u^{\prime}(t)\) Assume that \(u\) is a positive increasing function and \(u^{\prime}(t)\) exists.

Interpret \(e^{t^{2}}\) as \(e^{\left(t^{2}\right)}\). Argue that $$\lim _{b \rightarrow a} \frac{e^{\left(b^{2}\right)}-e^{\left(a^{2}\right)}}{b^{2}-a^{2}}=e^{\left(a^{2}\right)}$$ What is the ambiguity in the notation \(e^{a^{2}}\). (Consider \(4^{3^{2}}\).) Use parenthesis, they are cheap. However, common practice is to interpret \(e^{t^{2}}\) as \(e^{\left(t^{2}\right)}\).

On a popular television business news channel, an analyst exclaimed that "XXX stock has gone parabolic." Is there some sense in which this exclamation is more exuberant than "XXX stock has gone exponential?"

Suppose a bacterium Vibrio natriegens is growing in a beaker and cell concentration \(C\) at time \(t\) in minutes is given by $$C(t)=0.87 \times 1.02^{t} \quad \text { million cells per ml }$$ a. Approximate \(C(t)\) and \(C^{\prime}(t)\) for \(t=0,10,20,30,\) and 40 minutes. b. Plot a graph of \(C^{\prime}(t)\) vs \(C(t)\) using the five pairs of values you just computed.

Use logarithmic differentiation to show that \(y=t e^{3 t}\) satisfies \(y^{\prime \prime}-6 y^{\prime}+9 y=0\).

Use the logarithmic differentiation to compute \(y^{\prime}(t)\) for a. \(y(t)=t^{\pi}\) b. \(y(t)=t^{e}\) c. \(y(t)=\left(1+t^{2}\right)^{\pi}\) d. \(y(t)=t^{3} e^{t}\) e. \(y(t)=e^{\sin t}\) f. \(y(t)=t^{t}\)

Argue that $$\lim _{b \rightarrow a} \frac{e^{\sqrt{b}}-e^{\sqrt{a}}}{\sqrt{b}-\sqrt{a}}=e^{\sqrt{a}}$$

Let \(y(x)=e^{x}\). Compute \(y^{\prime}(x), y^{\prime \prime}(x)=\left(y^{\prime}\right)^{\prime},\) and \(y^{\prime \prime \prime}(x)\).

Suppose penicillin concentration in the serum of a patient \(t\) minutes after a bolus injection of \(2 \mathrm{~g}\) is given by $$P(t)=200 \times 0.96^{t} \quad \mu \mathrm{g} / \mathrm{ml}$$ a. Approximate \(P(t)\) and \(P^{\prime}(t)\) for \(t=0,5,10,15,\) and 20 minutes. b. Plot a graph of \(P^{\prime}(t)\) vs \(P(t)\) using the five pairs of values you just computed.

Show that for any numbers \(C_{1}\) and \(C_{2}, y=C_{1} e^{t}+C_{2} e^{-t}\) satisfies \(y^{\prime \prime}-y=0\).

Use the logarithmic differentiation to compute \(y^{\prime}(t)\) for a. \(y(t)=\frac{(t-1)(t+1)}{t-2}\) b. \(y(t)=t e^{t}\) c. \(y(t)=e^{-\frac{t^{2}}{2}}\) d. \(y(t)=\sqrt{1+t^{2}}\) e. \(y(t)=\frac{t^{2}}{t^{2}+1}\) f. \(y(t)=2^{t}\) g. \(y(t)=b^{t} \quad b>0\) h. \(y(t)=\frac{e^{t}-e^{-t}}{e^{t}+e^{-t}}\) i. \(y(t)=\frac{\ln t}{e^{t}}\)

A patient takes \(10 \mathrm{mg}\) of coumadin once per day to reduce the probability that he will experience blood clots. The half-life of coumadin in the body is 40 hours. What level, \(H,\) of coumadin will be accumulated from previous ingestion of pills and what will be the daily fluctuation of coumadin in the body.

We introduced the power chain rule \(\left[(u(x))^{n}\right]^{\prime}=n(u(x))^{n-1}[u(x)]^{\prime}\) for fractional and negative exponents, \(n\), in Section 4.3 .1 (see Exercises 4.3 .3 and 4.3 .4 ). Use these rules when necessary in the following exercise. Compute \(y^{\prime}(x)\) and \(y^{\prime \prime}(x)\) for a. \(y(x)=x^{2}+e^{x}\) b. \(y(x)=3 x^{2}+2 e^{x}\) c. \(y(x)=\left(1+e^{x}\right)^{2}\) d. \(y(x)=\left(e^{x}\right)^{2}\) e. \(y(x)=e^{2 x}=\left(e^{x}\right)^{2}\) f. \(y(x)=e^{-x} \quad=\left(e^{x}\right)^{-1}\) g. \(y(x)=e^{3 x} \quad=\left(e^{x}\right)^{3}\) h. \(y(x)=e^{x} \times e^{2 x}\) i. \(y(x)=\left(5+e^{x}\right)^{3}\) j. \(y(x)=\frac{1}{1+e^{x}} \quad=\left(1+e^{x}\right)^{-1}\) k. \(y(x)=\sqrt{e^{x}} \quad=\left(e^{x}\right)^{\frac{1}{2}}\) l. \(y(x)=e^{\frac{1}{2} x}\) m. \(y(x)=e^{0.6 x}=\left(e^{x}\right)^{0.6}\) n. \(y(x)=e^{-0.005 x}\)

Consider the kinetics of penicillin that is taken as a pill in the stomach. The diagram in Figure Ex. \(5.4 .9(\) a \()\) may help visualize the kinetics. We will find in Chapter 17 that a model of plasma concentration of antibiotic \(t\) hours after ingestion of an antibiotic pill yields an equation similar to $$C(t)=5 e^{-2 t}-5 e^{-3 t} \quad \mu \mathrm{g} / \mathrm{ml}$$ A graph of \(C\) is shown in Figure Ex. \(5.4 .9 .\) At what time will the concentration reach a maximum level, and what is the maximum concentration achieved? As we saw in Section 3.5 .2 and may be apparent from the graph in Figure Ex. 5.4 .9 , the highest concentration is associated with the point of the graph of \(C\) at which \(C^{\prime}=0 ;\) the tangent at the high point is horizontal. The question, then, is at what time \(t\) is \(C^{\prime}(t)=0\) and what is \(C(t)\) at that time?

Identify the errors in the following derivative computations. \(\begin{array}{l}\text { a. }\left[\left(t^{4}+t^{-1}\right)^{7}\right]^{\prime} \\ & 7\left(t^{4}+t^{-1}\right)^{6}\left[t^{4}+t^{-1}\right]^{\prime} \\ & 7\left(t^{4}+t^{-1}\right)^{6}\left[t^{4}\right]^{\prime}+\left[t^{-1}\right]^{\prime} \\\ & 7\left(t^{4}+t^{-1}\right)^{6} 4 t^{3}+(-1) t^{-2} \\ & 28 t^{3}\left(t^{4}+t^{-1}\right)^{6}-t^{-2}\end{array}\) \(\begin{array}{l}\text { b. }\left[5 t^{7}+7 t^{-5}\right]^{\prime} \\\ {\left[5 t^{7}\right]^{\prime}+\left[7 t^{-5}\right]^{\prime}} \\ & 5\left[t^{7}\right]^{\prime}+7\left[t^{-5}\right]^{\prime} \\ & 5 \times 7 t^{6}+7 \times(-5) t^{-4} \\ & 35\left(t^{6}-t^{-4}\right)\end{array}\) c. \(\left[10 t^{8}+8 e^{5 t}\right]^{\prime}\) \(\left[10 t^{8}\right]^{\prime}+\left[8 e^{5 t}\right]^{\prime}\) \(10\left[t^{8}\right]^{\prime}+8\left[e^{5 t}\right]^{\prime}\) \(10 \times 8 t^{7}+8 \times 5 e^{4 t}\) \(40\left(2 t^{7}+e^{4 t}\right)\)

Plasma penicillin concentration is $$P(t)=5 e^{-0.3 t}-5 e^{-0.4 t}$$ \(t\) hours after ingestion of a penicillin pill into the stomach. A small amount of the drug diffuses into tissue and the tissue concentration, \(C(t)\), is $$ C(t)=-e^{-0.3 t}+0.5 e^{-0.4 t}+0.5 e^{-0.2 t} \quad \mu \mathrm{g} / \mathrm{ml} $$ a. Use your technology (calculator or computer) to find the time at which the concentration of the drug in tissue is maximum and the value of \(C\) at that time. b. Compute \(C^{\prime}(t)\) and solve for \(t\) in \(C^{\prime}(t)=0\). This is really bad, for you must solve for \(t\) in $$ 0.3 e^{-0.3 t}-0.2 e^{-0.4 t}-0.1 e^{-0.2 t}=0 $$ Try this: $$ \text { Let } \quad Z=e^{-0.1 t} \quad \text { then solve } \quad 0.3 Z^{3}-0.2 Z^{4}-0.1 Z^{2}=0 . $$ c. Solve for the possible values of \(Z\). Remember that \(Z=e^{-0.1 t}\) and solve for \(t\) if possible using the possible values of \(Z\). d. Which value of \(t\) solves our problem?

In Section 1.3 we found from the discrete model of light extinction, \(I_{d+1}=I_{d}-0.18 I_{d}\) that the solution \(\quad I_{d+1}=0.82 I_{d}\) Light decrease in water is continuous, however. Find a value of \(k\) for which the solution to the continuous model, \(I^{\prime}(x)=-k I(x),\) matches the data.

Argue that if \(S_{1}\) and \(S_{2}\) are two sets of numbers and every number is in either \(S_{1}\) or \(S_{2}\) and every number in \(S_{1}\) is less than every number in \(S_{2}\) then it is not true that there are numbers \(L_{1}\) and \(L_{2}\) such that \(L_{1}\) is the greatest number in \(S_{1}\) and \(L_{2}\) is the least number in \(S_{2}\). Is this a contradiction to the Completeness Axiom?

Let \(S_{2}\) denote the points of the \(X\) -axis that have positive \(x\) -coordinate and \(S_{1}\) denote the points of the \(X\) -axis that do not belong to \(S_{2} .\) Does \(S_{2}\) have a left most point?

Suppose \(S_{2}\) is the set of numbers to which \(x\) belongs if and only if \(x\) is positive and \(x^{2}>2\) and \(S_{1}\) consists of all of the other numbers. 1\. Give an example of a number in \(S_{2}\). 2\. Give an example of a number in \(S_{1}\). 3\. Argue that every number in \(S_{1}\) is less than every number in \(S_{2}\). 4\. Which of the following two statements is true? (a) There is a number \(C\) which is the largest number in \(S_{1}\). (b) There is a number \(C\) which is the least number in \(S_{2}\). 5\. Identify the number \(C\) in the correct statement of the previous part.

Suppose your number system is that of Early Greek mathematicians and includes only rational numbers. Does it satisfy the Axiom of Completion?

Suppose solar radiation striking the ocean surface is \(1250 \mathrm{~W} / \mathrm{m}^{2}\) and 20 percent of that energy is reflected by the surface of the ocean. Suppose also that 20 meters below the surface the light intensity is found to be \(800 \mathrm{~W} / \mathrm{m}^{2}\). a. Write an equation descriptive of the light intensity as a function of depth in the ocean. b. Suppose a coral species requires \(100 \mathrm{~W} / \mathrm{m}^{2}\) light intensity to grow. What is the maximum depth at which that species might be found?

In two bodies of water, \(L_{1}\) and \(L_{2},\) the light intensities \(I_{1}(x)\) and \(I_{2}(x)\) as functions of depth \(x\) are measured simultaneously and found to be $$I_{1}(x)=800 e^{-0.04 x} \quad \text { and } \quad I_{2}(x)=700 e^{-0.05 x}$$ Explain the differences in the two formulas in terms of the properties of water in the two bodies.

An egg is covered by a hen and is at \(37^{\circ} \mathrm{C}\). The hen leaves the nest and the egg is exposed to \(17^{\circ} \mathrm{C}\) air. a. Draw a graph representative of the temperature of the egg \(t\) minutes after the hen leaves the nest. Mathematical Model 5.5.6 Egg cooling. During any short time interval while the egg is uncovered, the change in egg temperature is proportional to the length of the time interval and proportional to the difference between the egg temperature and the air temperature. b. Let \(T(t)\) denote the egg temperature \(t\) minutes after the hen leaves the nest. Consider a short time interval, \([t, t+\Delta t],\) and write an equation for the change in temperature of the egg during the time interval \([t, t+\Delta t]\). c. Argue that as \(\Delta t\) approaches zero, the terms of your previous equation get close to the terms of $$T^{\prime}(t)=-k(T(t)-17)$$ d. Assume \(\mathrm{T}(0)=37\) and find an equation for \(T(t)\). e. Suppose it is known that eight minutes after the hen leaves the nest the egg temperature is \(35^{\circ} \mathrm{C}\). What is \(k ?\) f. Based on that value of \(k,\) if the coldest temperature the embryo can tolerate is \(32^{\circ} \mathrm{C},\) when must the hen return to the nest?