Chap 18 Exercises

\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]

Exercise 8 As you know, given a function \(g(x)\) it is easy to construct a new function \({\cal D}_x g(x)\) that will be an approximation to the derivative \(\partial_x g(x)\). The approximation function, which we call the slope function, can be \[{\cal D}_x g(x) \equiv \frac{g(x + 0.1) - g(x)}{0.1}\]

In Active R chunk lst-fly-speak-canoe, use makeFun() to create a function \(g(x) \equiv \sin(x)\) and another that will be the slope function, called it slope_of_g().

Active R chunk 3

What’s the value of slope_of_g(1)?

Using Active R chunk lst-fly-speak-canoe, plot both g() and slope_of_g() (in blue) on a domain \(-5 \leq x \leq 5\). This can be done with slicePlot() in the following way:

slice_plot(g(x) ~ x, domain(x = -5:5)) |>
  slice_plot(slope_of_g(x) ~ x, color="blue")

Which of these statements best describes the graph of \(g()\) compared to slope_of_g()?

slope_of_g() is shifted left by about \(\pi/2\) compared to g(x).

slope_of_g() is shifted left by about \(\pi\) compared to g(x).

slope_of_g() has a larger amplitude than g()

The output of slope_of_g() is always positive

slope_of_g() is practically the same function as g(). That is, for any input the output of the two functions is practically the same.

question id: fsc1-2

Exercise 9 XREF not implemented yet discusses the dimension of quantities that are a derivative of some other quantity.

Tanks for bulk storage of natural gas are typically large cylinders with a cap that can move up and down. The volume of the tank is a function of the position of the cap. What is the dimension of the derivative of cylinder volume with respect to cap position?

Exercise 10 A common programming pattern in R/mosaic calculus is:

Operator(tilde_expression, [optional details])

Some of the operators we use in MOSAIC Calculus: slice_plot(), contour_plot, make_Fun(), D(), antiD(), Zeros()

For each of the following R/mosaic expressions, determine which kind of thing is being created. Feel free to run the expressions Active R chunk lst-tilde-function.

Active R chunk 4

Expression 1: makeFun(a*x - b ~ x)

Expression 2: D(a*x - b ~ x)

Expression 3: antiD(a*x - b ~ x)

Expression 4: slice_plot(a*x - b ~ x, bounds(x=0:5))

Expression 5.

f <- makeFun(a*x + b ~ x, a=2, b=-4)
slice_plot(f(x) ~ x, bounds(x=0:5))

Expression 6. Zeros(a*x - b ~ x, bounds(x=0:5))

Expression 7: a*x - b ~ x

Expression 8:

f <- makeFun(a*x + b ~ x, a=2, b=-4)`
Zeros(f(x) ~ x)

Suppose you create a function in the usual way, e.g. f <- makeFun(a*x + b ~ x, a=2, b=-4). Which of the following will plot a straight-line function with a slope of 5.

slice_plot(f(x) ~ x, bounds(x=-5:5))

slice_plot(f(x, b=2), bounds(x=-5:5)

slice_plot(f(x, a=5), bounds(x=-5:5)

question id: tilde-function-9