Chap 23 Exercises

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Exercise 3 Recall the differentiation rules for three of the pattern-book functions as presented in XREF not implemented yet:

Function name	Formula	Formula for derivative	power-law exponent \(p\)
Identity	\(x\)	\(1\)	1
Square	\(x^2\)	\(2\, x\)	2
Reciprocal	\(1/x\)	\(-1/x^2\)	-1

All three of these pattern-book functions are members of the power-law family: \(x^p\). They differ only in the value of \(p\).

There is a differentiation rule for the power-law family generally. The next question offers several formulas for this rule, only one of which is correct. You can figure out which one by trying the pattern-book functions in the table above and seeing which formula gives the correct answer for the derivative.

Which of these formulas gives the correct differentiation rule for the power-law family \(x^p\)?

Exercise 4 Although we created an R function named slopeFun() for the purposes of demonstration, it is better to use the R/mosaic operator D() which calculates the derivative, sometimes using symbolic methods and sometimes using a finite-difference method.

As an example of the use of D(), here is some more R code that defines a function f() and finds \(\partial_x f()\), calling it d_f(). Then each function is plotted on the same domain.

Active R chunk 1

For each of the following functions, write a brief comparison of the function to it is differenced version. You can combine phrases such as “same shape”, “different shape. larger in amplitude”, “smaller in amplitude”, “same period”, “shorter period”, “longer period”, or whatever seems appropriate. For instance, for the original example in the sandbox, a reasonable comparison might be, “f() is concave down but Diff(f) is concave up.”

A. For the function \(f(x) \equiv 3 x\), compare \(f()\) to \(\partial_x f\).

question id: finch-trim-kayak-A

B. For the function \(f(x) \equiv x^2\), compare \(f()\) to \(\partial_x f\).

question id: finch-trim-kayak-B

C. For the function \(f(x) \equiv e^x\), compare \(f()\) to \(\partial_x f\).

question id: finch-trim-kayak-C

D. For the function \(f(x) \equiv e^{-0.3 x}\), compare \(f()\) to \(\partial_x f\).

question id: finch-trim-kayak-D

E. For the function \(f(x) \equiv \sin(x)\), compare \(f()\) to \(\partial_x f\).

question id: finch-trim-kayak-E

F. For the function \(f(x) \equiv \sin(2 \pi x)\), compare \(f()\) to \(\partial_x f\)).

question id: finch-trim-kayak-F

G. For the function \(f(x) \equiv \sin(\frac{2 \pi}{20} x)\), compare \(f()\) to \(\partial_x f\)).

question id: finch-trim-kayak-G

Exercise 5 In Exercise exr-goat-walk-window you constructed models \(D(t)\) of the availability of a drug in the bloodstream for three different pill-taking regimens: every six hours, every eight hours, and a double dose to start followed by a single dose every eight hours. The model for from a single, isolated pill is a zero before the pill is taken, then exponential decay from the level of the pill dose after the pill is taken. Like this:

pill <- makeFun(ifelse(t < 0, 0, exp(-k * t)) ~ t, k = log(2)/3)

The parameter \(k\) has been set to represent a drug with a half-life of three hours.

The model for the entire regiment is a linear combination of time-shifted single pills, e.g.

regimen8 <- makeFun(A*pill(t) + A*pill(t-8) + A*pill(t-16) + A*pill(t-24) + A*pill(t-32) ~ t, A=1)

From graphs of the functions themselves it is easy to check whether the availability ever falls below the therapeutic threshold (which we stipulated is 0.25). For instance, the eight-hour regiment with a dose of A=1 does fall below the threshold during the first day. So a larger dose is needed than A=1.

A primary objective in choosing a drug regime is to ensure that the availability stays above the therapeutic level. A secondary objective is to keep the elimination rate small so that the drug can be handled by patients with poorly functioning livers.

The derivative \(\partial_t \text{regimen8}(t)\) tells the instantaneous rate at which the drug is being administered to and eliminated from the patient’s body.

Active R chunk 2

For each of the three regimens—6-hour, 8-hour, 8-hour-double-first-dose—construct \(\partial_t \text{regimen}(t)\). Ignoring the glitches due to discontinuity at the times the pills are consumed, which of the three regimens has the lowest average rate of drug elimination?

question id: octopus-teach-sheet-text

Activities

Exercise 8 Confirm using algebraic manipulation the differentiation rule for a product of three functions:

\[\left[ u\cdot v\cdot w\right]' = \color{blue}{u'}\cdot v\cdot w\ +\ u\cdot \color{blue}{v'}\cdot w\ +\ u\cdot v\cdot \color{blue}{w}'\] Here, \(u\) is shorthand for \(u(x)\), and \(u'\) is shorthand for \(\partial_x u(x)\), and similarly for \(v\) and \(w\).

Hint: \(\left[ u\cdot v\cdot w\right] = u \cdot \left[ v \cdot w\right]\). So a product of three functions can be seen as a product \(u\cdot h\) where \(h \equiv v \cdot w\).

Exercise 9 Consider this function, defined piecewise:

\[h(x) \equiv \left\{ \begin{array}{ll}0 & \text{for}\ x < 0\\x^2 & \text{otherwise}\end{array}\right.\ .\]

Write the R command to create this function. (Hint: Remember ifelse from XREF not implemented yet.)
Using Active R chunk lst-camel-begin-piano, plot \(h(x)\) over the domain \(-1 \leq x \leq 1\), then sketch a copy of the graph on your paper.

Active R chunk 3

Create the function \(\partial_x h(x)\) by differentiating separately each piece of the piecewise function \(h()\). Write down \(\partial_x h(x)\) using mathematical notation similar to the definition of \(h(x)\) given above.
Sketch a graph of \(\partial_x h(x)\) over the domain \(-1 \leq x \leq 1\). You’re welcome to use a Active R chunk lst-camel-begin-piano, but you may be able to figure out the shape of the graph yourself.
The shape of the function you sketched in (4) has a name, given in the text in Section @ref(continuity). What is that name?

Now you will do much the same as in items (3), (4), and (5), but instead of the first derivative \(\partial_x h(x)\), create, sketch, and name the second derivative \(\partial_{xx} h(x)\).

Create and write down \(\partial_{xx} h(x)\) in mathematical notation.
Sketch \(\partial_{xx} h(x)\)
Classify the smoothness of \(h(x)\) using the following table:

Smoothness	Criterion
\(C^0\)	\(\partial_x h(x)\) is discontinuous
\(C^1\)	\(\partial_x h(x)\) is continuous
\(C^2\)	\(\partial_{xx} h(x)\) is continuous
\(C^3\)	\(\partial_{xxx} h(x)\) is continuous
\(\vdots\)	and so on.
\(C^\infty\)	All orders of derivative of \(h(x)\) are continous.

Exercise 10 Recall from Exercise exr-duck-see-socks the Lorenz curve used to describe income inequality. The Lorenz curve shows the fraction of total income versus population fraction.

Figure 4: A Lorenz curve (blue) fitted to income data from the US in 2009. (See Figure fig-lorenz-data.)

Since the population is arranged from poorest to richest along the horizontal axis, Lorenz curves must be both monotonically increasing and concave up. That is, any Lorenz function \(L(P)\), where \(P\) is the population fraction, must satisfy these criteria:

\(L(0) = 0\)
\(L(1) = 1\) that is, the aggregate fraction of income earned by the entire population is 100%.
\(\partial_P L(P) \ > \ 0\) that is, monotonically increasing
\(\partial_{PP} L(P) \ > \ 0\) that is, concave up.

Consider a function \(H(P) \equiv L_1(L_2(P))\) which is the composition of two Lorenz curves.

A. Use the composition rule to show that \(H(P)\) is monotonically increasing. (Hint, calculate \(\partial_P H(P)\) and show that it must be positive.)
B. Using both the composition and product rules, calculate \(\partial_{PP} H(P)\) and show that \(H(P)\) must be concave up.

Exercise 11 The formula for the function \(\dnorm(x)\) is \[\dnorm(x) \equiv \frac{1}{\sqrt{2 \pi}} \exp\left(\frac{x^2}{2}\right)\ .\]

A. Use the chain rule to find \(\partial_x \dnorm(x)\).

B. Confirm from your answer to (1) that there is another formula for \(\partial_x \dnorm(x)\), namely \[\partial_x \dnorm(x) = - x \dnorm(x)\ .\]

C. Use the product rule to find \(\partial_{xx} \dnorm(x)\).

D. From your answer to (3), compute the 3rd derivative \(\partial_{xxx} \dnorm(x)\):

E. Let’s generalize the pattern. Each of the previous derivatives has been a polynomial—let’s call it \(p_n(x)\) for the \(n\)th derivative—times \(\dnorm(x)\). Knowing \(p_n(x)\), we can easily find \(p_{n+1}(x)\): \[p_{n+1}(x) = -x p_n(x) + \partial_x p_n(x)\] We know \(p_1(x) = -x\) so \(p_2(x) = x^2 - 1\). In turn, this tells us \(p_3(x) = 3x - x^3\). Find:

\(p_4(x)\)
\(p_5(x)\)
\(p_6(x)\)

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