5.7: Summary.

The remarkable number \(e\), the exponential function \(E(t) = e ^t \), and the natural logarithm function \(L(t) = \ln {t}\) are the basic material for this chapter. The number \(e\) is defined by

\[\lim _{c \rightarrow 0}(1+c)^{\frac{1}{c}}=e\]

It is the only number, \(b\), for which

\[\lim _{h \rightarrow 0} \frac{b^{h}-1}{h}=1\]

Because \(\lim _{h \rightarrow 0}\left(e^{h}-1\right) / h=1\), the function \(E(t) = e ^t\) has the property that \(E ^{\prime} (t) = e ^t\). For any other base, \(b > 0\), the exponential function \(B(t) = b ^t\) has a derivative, but \(B ^{\prime} (t) = b ^{t} \ln {b}\) has the extra factor \(\ln {b}\). The natural logarithm function, \(L(t) = \log _{e} {t} = \ln {t}\) also has a simple derivative, \(1/t\), and its derivative is the simplest among all logarithm functions. For \(L(t) = \log _{b} {t}\), \(L ^{\prime} (t) = (1/t)/(\ln {b}\).

The function \(E(t) = e ^{k t}\) has the property

\[E^{\prime}(t)=e^{k t} k=k E(t)\]

Many mathematical models of biological and physical systems yield equations of the form \(y ^{\prime} (t) = k ~ y(t)\), and for that reason we frequently use the exponential function \(e ^{k t}\) to describe natural phenomena.

When analyzing data thought to be exponential, a semilog graph of the data will often signal whether the data is indeed exponential.

The exponential and logarithm chain rules

\[\left[e^{u(t)}\right]^{\prime}=e^{u(t)} u^{\prime}(t) \quad[\ln (u(t))]^{\prime}=\frac{1}{u(t)} u^{\prime}(t)\]

expand the class of functions for which we can compute derivatives, and the logarithm chain rule is used to extend the power chain rule for integer exponents to all values of the exponent.

 

Exercises for the Summary of Chapter 5.

Chapter Exercise 5.7.1 Compute \(P ^{\prime} (t)\) for:

1. \(P(t)=e^{5 t}\)
2. \(P(t)=\ln 5 t\)
3. \(P(t)=e^{t \sqrt{t}}\)
4. \(P(t)=e^{\sqrt{2 t}}\)
5. \(P(t)=\ln (\ln t)\)
6. \(P(t)=e^{\ln t}\)
7. \(P(t)=1 /\left(1+e^{t}\right)\)
8. \(P(t)=1 / \ln t\)
9. \(P(t)=1 /\left(1+e^{-t}\right)\)
10. \(P(t)=\left(1+e^{t}\right)^{3}\)
11. \(P(t)=\left(e^{\sqrt{t}}\right)^{3}\)
12. \(P(t)=\ln \sqrt{t}\)

Chapter Exercise 5.7.2 Use the logarithmic differentiation to compute \(y ^{\prime} (t)\) for

1. \(y(t)=10^{t}\)
2. \(y(t)=\frac{t-1}{t+1}\)
3. \(y(t)=(t-1)^{3}\left(t^{3}-1\right)\)
4. \(y(t)=(t-1)(t-2)(t-3)\)
5. \(y(t)=u(t) v(t) w(t)\)
6. \(y(t)=u(t) v(t)\)

Chapter Exercise 5.7.3 Use a semilog graph to determine which of the following data sets are exponential.

1. \(t\)	\(P(t)\)
   0	5.00
   1	3.53
   2	2.50
   3	1.77
   4	1.25
   5	0.88

2. \(t\)	\(P(t)\)
   0	5.00
   1	1.67
   2	1.00
   3	0.71
   4	0.55
   5	0.45

3. \(t\)	\(P(t)\)
   0	5.00
   1	3.63
   2	2.50
   3	1.63
   4	1.00
   5	0.63

 Chapter Exercise 5.7.4 The function

\[P(t)=\frac{10 e^{t}}{9+e^{t}}=\frac{10}{9 e^{-t}+1}\]

is an example of a logistic function. The logistic functions are often is used to describe the growth of populations. Plot the graphs of \(P(t)\) and \(P ^\{prime} (t)\). \(P ^{\prime}\) and Find at time at which \(P ^{\prime}\) is a maximum. Identify the point on the graph of \(P\) corresponding to that time.

Chapter Exercise 5.7.5 A pristine lake of volume 1,000,000 m³ has a river flowing through it at a rate of 10,000 m³ per day. A city built beside the river begins dumping 1000 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?

Chapter Exercise 5.7.6 Estimate the slope of the tangent to the graph of

\[y=\log _{10} x\]

at the point \((3, \log _{10} 3)\) correct to three decimal digits.

Chapter Exercise 5.7.7 Use logarithmic differentiation to show that \(y=t e^{3 t}\) satisfies \(y^{\prime \prime}-6 y^{\prime}+9 y=0\).

Chapter Exercise 5.7.8 Show that for any numbers \(C_1\) and \(C_2\), \(y=C_{1} e^{t}+C_{2} e^{-t}\) satisfies \(y^{\prime \prime}-y=0\).