3.4: Summary

1. If we "zoom in" enough on a point \(x_{0}\) on a graph of a function (with "smooth" behavior), we see a straight line. This straight line is the tangent line at that point. The slope of this line is the derivative (instantaneous rate of change) at that point, \(x_{0}\).
2. Given the graph of a function \(f(x)\), the derivative \(f^{\prime}(x)\) can be sketched by approximating the slopes of the tangent lines of \(f(x)\), and plotting those slopes as points.
3. A function is continuous at \(x=a\) in its domain if \(\lim _{x \rightarrow a} f(x)=f(a)\). Discontinuous functions might have a hole (removable discontinuity), a jump, or blow up.
4. Computing derivatives requires the use of limits: \[f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \nonumber \] Limits are detailed further in Appendix D. In the absence of analytical methods, or in the presence of only data, a numerical derivative calculus can be used to approximate: \[f^{\prime}(x)_{\text {numerical }} \approx \frac{\Delta f}{\Delta x} . \nonumber \]
5. Derivatives that were computed in this chapters are summarized in Table 3.1.
6. The applications we encountered in this chapter included:
   1. molecular motors and vesicle transport; and
   2. Michaelis-Menten and Hill function kinetics for reaction speeds.

Table 3.1: Some computed derivatives.

\(\mathbf{f}(\mathbf{x})\)	\(\mathbf{f}^{\prime}(\mathbf{x})\)
\(C\)	0
\(B x\)	\(B\)
\(B x+C\)	\(B\)
\(K x^{3}\)	\(3 K x^{2}\)
\(\frac{1}{x}\)	\(-\frac{1}{x^{2}}\)
\(\sqrt{x}\)	\(\frac{1}{2 \sqrt{x}}\)

Quick Concept Checks

1. What geometric characteristic might stop a function from having a derivative?
2. Find the derivative of \(y=2 x+1\) using a
   1. geometric argument, and
   2. algebraic argument.
3. Draw the derivative of the function \(f(x)\) at \(x=1\), depicted on the graph below.

4. Define a continuous function.