3.8: Summary of Chapter 3, The Derivative.

You now have an introduction to the concept of rate of change and to the derivative of a function. The derivative and its companion, the integral that is studied in Chapter 9, enabled¹⁰ an explosion in science and mathematics beginning in the late seventeenth century, and remain at the core of science and mathematics today. Briefly, for a suitable function, \(P\), the derivative of \(P\) is the function \(P ^{\prime}\) defined by

\[P^{\prime}(t)=\lim _{h \rightarrow 0} \frac{P(t+h)-P(t)}{h} \label{3.40}\]

We also wrote \(P ^{\prime} (t)\) as \([ P(t) ]^{\prime}\) and \(\frac{dP}{dt}\) . A helpful interpretation of \(P ^{\prime} (a)\) is that it is the slope of the tangent to the graph of \(P\) at the point \((a, P(a))\).

The derivatives of three functions were computed (for \(C\) a number and \(n\) a positive integer) and we wrote what we call primary formulas:

\[\left.\begin{array}{l}
P(t)=C \Longrightarrow P^{\prime}(t)=0 \quad \text { or }[C]^{\prime}=0 \\
P(t)=t \quad \Longrightarrow P^{\prime}(t)=1 \quad \text { or } \quad[t]^{\prime}=1 \\
P(t)=t^{n} \quad \Longrightarrow P^{\prime}(t)=n t^{n-1} \quad \text { or }\left[t^{n}\right]^{\prime}=n t^{n-1}
\end{array}\right\} \label{3.41}\]

Two combination formulas were developed:

\[\left.\begin{array}{l}
P(t)=C u(t) \quad \Longrightarrow P^{\prime}(t)=C \times u^{\prime}(t) \\
P(t)=u(t)+v(t) \Longrightarrow P^{\prime}(t)=u^{\prime}(t)+v^{\prime}(t)
\end{array}\right\} \label{3.42}\]

They can also be written

\[\begin{aligned}\
[C u(t)]^{\prime} &=C[u(t)]^{\prime} \\
[u(t)+v(t)]^{\prime} &=[u(t)]^{\prime}+[v(t)]^{\prime}
\end{aligned}\]

We will expand both the list of primary formulas and the list of combination formulas in future chapters and thus expand the array of derivatives that you can compute without explicit reference to the Definition of Derivative Equation \ref{3.40}.

We saw that derivatives describe rates of chemical reactions, and used the derivative function to find optimum values of spider web design and the height of a pop fly in baseball. We examined two cases of dynamic systems, mold growth and falling objects, using rates of change rather than the average rates of change used in Chapter 1. A vast array of dynamical systems and optimization problems have been solved since the introduction of calculus. We will see some of them in future chapters.

Finally we defined and computed the second derivative and higher order derivatives and gave some geometric interpretations (concave up and concave down) and some physical interpretation (acceleration).

 

Exercises for Chapter 3, The Derivative.

Chapter Exercise 3.8.1 Use Definition of the Derivative 3.3.3, \(\lim _{b \rightarrow t} \frac{P(b)-P(t)}{b-t}\), to compute the rates of change of the following functions, \(P\).

1. \(P(t) = t^3\)
2. \(P(t) = 5t ^2\)
3. \(P(t) = \frac{t^3}{4}\)
4. \(P(t) = t^{2} + t^{3}\)
5. \(P(t) = 2 \sqrt{t}\)
6. \(P(t) = \sqrt{2t}\)
7. \(P(t) = 7\)
8. \(P(t) = 5 − 2t\)
9. \(P(t) = \frac{1}{1 + t}\)
10. \(P(t) = \frac{1}{3t}\)
11. \(P(t) = 5t^7\)
12. \(P(t) = \frac{1}{t^2}\)
13. \(P(t) = \frac{1}{(3t + 1)^2}\)
14. \(P(t) = \sqrt{t + 3}\)
15. \(P(t) = \frac{1}{\sqrt{t + 1}}\)

Chapter Exercise 3.8.2 Data from David Dice of Carlton Comprehensive High School in Canada¹¹ for the decrease in mass of a solution of 1 M HCl containing chips of \(\mathrm{CaCO}_3\) is shown in Table 3.8.0. The reaction is

\[\mathrm{CaCO}_{3}(\mathrm{~s})+2 \mathrm{HCl}(\mathrm{aq}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l})+\mathrm{CaCl}_{2}(\mathrm{aq}) .\]

The reduction in mass reflects the release of \(\mathrm{CO}_2\).

1. Graph the data.
2. Estimate the rate of change of the mass at each of the times shown.
3. Draw a graph of the rate of change of mass versus the mass.

Table for Exercise 3.8.0 Data for Ex. 3.8.2.

Time (min)	Mass (g)
0.25	248.46
0.50	247.95
1.00	246.83
1.50	245.95
2.00	245.22
2.50	244.67
3.00	244.27
3.50	243.95
4.00	243.72
4.50	243.52
5.00	243.37

Chapter Exercise 3.8.3 Use derivative formulas \ref{3.41} and \ref{3.42} to compute the derivative of \(P\). Use Primary formulas only in the last step. Assume the \(t^n\) rule \([t ^{n}] ^{\prime} = n t^{n-1}\). to be valid for all numbers \(n\), integer, rational, irrational, positive, and negative. In some cases, algebraic simplification will be required before using a derivative formula.

1. \(P(t) = 3t ^{2} − 2t + 7\)
2. \(P(t) = t + \frac{2}{t}\)
3. \(P(t) = \sqrt{2t}\)
4. \(P(t) = (t ^{2} + 1 )^3\)
5. \(P(t) = 1/ \sqrt{2t}\)
6. \(P(t) = 2t ^{-3} - 3t ^{-2}\)
7. \(P(t) = \frac{5 + t}{t}\)
8. \(P(t) = 1 + \sqrt{t}\)
9. \(P(t) = ( 1 + 2t)^2\)
10. \(P(t) = ( 1 + 3t)^3\)
11. \(P(t) = \frac{1 + \sqrt[3] {t}}{\sqrt{t}}\)
12. \(P(t) = \sqrt[3] {2t}\)

Chapter Exercise 3.8.4 Find an equation of the tangent to the graph of \(P\) at the indicated points. Draw the graph \(P\) and the tangent.

1. \(P(t) = t ^{4} \text{ at } (1, 1)\)
2. \(P(t) = t ^{12} \text{ at } (1, 1)\)
3. \(P(t) = t ^{1/2} \text{ at } (4, 2)\)
4. \(P(t) = \frac{5}{2} \text{ at } (5, 1)\)
5. \(P(t) = \sqrt{1 + t} \text{ at } (8, 3)\)
6. \(P(t) = \frac{1}{2t} \text{ at } (\frac{1}{2} , 1)\)

 

¹⁰ It might also be argued that the explosion in science enabled or caused the creation of the derivative and integral. The two are inseparable.

¹¹ http://www.carlton.paschools.ps.sk.ca/chemical/chem