Chap 4 Exercises

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

Exercise 2 Which of the following R expressions creates a graph of the function \(\sin(z)\) with a graphics domain from \(-4\leq z \leq 5\)?

slice_plot(sin(x), domain(x=-4:5))

slice_plot(sin(z) ~ x, domain(x=-4:5))

slice_plot(sin(z), domain(z=-4:5))

slice_plot(sin(z) ~ z, domain(z=-4:5))

question id: gspf-01

Hint: You can try copying and running each expression in Active R chunk lst-girl-shut-pan to check your answer.

Active R chunk 1

Exercise 4 Consider the two commands in the following interactive chunk:

The first line defines a function named f().

The second line attempts to plot f() over the graphics domain -10 to 10. But it doesn’t work.

What’s wrong?

f() is not a proper name for a function.

The first argument should be a tilde expression like f ~ x.

The first argument should be a tilde expression like f(x) ~ x.

question id: first-argument-a

Fix the code in the chunk so that the command works.

Exercise 5 Each of the following commands is intended to graph a given function over a graphic domain running from zero to ten. But none of them work, as is. Fix each command so that it works. (Suggestion: Try running the command, as is. Then look at the error message.)

Exercise 6 The R/mosaic command in Active R chunk lst-constant-fun will draw a graph of the function specified by the tilde expression.

Active R chunk 3

Why is the graph a flat line?

Because the vertical scale has been chosen inappropriately.

Because the output of the function is always 15, regardless of the value of the input \(y\).

Because the graphics domain is not long enough.

question id: constant-fun-a

The vertical scale of the graph is very limited, covering a range of only \(15 \pm 0.05\). This scale was selected automatically by slice_plot() because the function itself does not give any context for what the vertical scale should be. You can set the vertical scale manually by piping the graphics command into this expression, gf_lims(y = c(0, 20)). Make this change to the R-chunk and verify that the scale is indeed what is indicated.

Exercise 7 The function f() produced by doodle_fun() has two inputs. A contour plot is an appropriate way to visualize the function. In this exercise, you’re going to play with the location of contours. The contours_at argument tells what levels of the output should be marked with a contour.

Place a new contour inbetween the two contours at output levels 0 and 5.
By trial and error, find the output level whose contour goes through the origin of the (x,y) input space.

question id: draw-contours-essay

Exercise 8 Look through all the contour plots in this document (except Figure fig-carbon-phases) to see if it ever happens that two contours at different output levels intersect.

Chemists are familiar with “phase diagrams” that show the different configurations for a mass of molecules as a function of temperature and pressure. Figure fig-carbon-phases is for carbon-dioxide, CO_2_. You read off the form of the substance in exactly the same way you read a contour plot.

Figure 2: Phase diagram for CO_2_. Source Ben Finney, Mark Jacobs, CC0, via Wikimedia Commons

In chemistry, the “contours” are called “phase transitions.” But unlike genuine contour plots, the different phase transitions can intersect. In chemistry, such an intersection point is called a “critical point.”

Practice with contour plots

It may take some practice to learn to read contour plots fluently but it is a skill that is worthwhile. Note that the graphics frame is the Cartesian space of the two inputs. The output is presented as contour lines. Each contour line has a label giving the numerical value of the function output. Each of the input value pairs on a given contour line corresponds to an output at the level labeling that contour line. To find the output for an input pair that is not on a contour line, you interpolate between the contours on either side of that point.

Exercise 9 {{ <include ./Preliminaries/gradient-field-pink.qmd >}}

Exercise 10 Each of these attempts to define a mathematical function using R leads to an error message. Modify each statement to work properly.

Exercise 11 Consider this graph of a function \(g(x)\):

What is the domain of \(g(x)\)?

What is the range of \(g(x)\)?

Exercise 13 Figure fig-squirrel-hold-bulb displays a function, but we don’t show you any formula for the function, just the graph.

Figure 3: An interactive contour plot. As you place the cursor on a point on the plot, a box displays the \((x,y,z)\) coordinates.

Find three points on the surface where \(f(x, y) \approx 15\). (It does not have to be exactly 15, just close.)

question id: squirrel-hold-bulb-1

Find a point where \(f(x=2, y) =\approx 8\).

question id: squirrel-hold-bulb-2

Exercise 14 Often, the specific numerical value at a point is not of primary interest. Instead, we may be interested in how steep the function is at a point, which is indicated by the spacing between contours. When contours are closely spaced, the hillside is steep. Where contours are far apart, the hillside is not steep, perhaps even locally flat.

Another common task for interpreting contour plots is to locate the input pair that is at a local high point or low point: the top of a hill or the bottom of a hollow. Such points are called local argmax or local argmin respectively. The output of the function at a local argmax is called the local maximum; similarly for a local argmin, where the output is called a local minimum.

The word “argmax” is a contraction of “argument of the maximum.” We will tend to use the word “input” instead of “argument”, but it means the same thing to say “the inputs to a function” as saying “the arguments of a function.”

Answer the following questions For the function shown in Figure fig-steepness.

Figure 4: The function for Exercise exr-steepness.

Find an input coordinate where the function is steepest.
Find input coordinates for the high and low points of the function .

question id: steepness-essay

Answer for Exercise exr-steepness

A function is steepest where contour lines are spaced closely together, that is, where the function output changes a lot with a small change in input. This is true near inputs \((x=0, y=1)\). But notice that steepness involves a direction. Near \((x=0,y=1)\), changing the \(x\) value does not lead to a big change in output, but a small change in the value of \(y\) leads to a big change in output. In other words, the function is steep in the y-direction but not in the x-direction.

The highest output value explicitly marked in the graph is 8. We can imagine from the shapes of the contours surrounding the 8 contour that the function reaches a peak somewhere in the middle of the region enclosed by the 8 contour, near the input coordinate \((x=0, y=-1.5)\).

Similarly, the lowest output value marked is -10. In the middle of the area enclosed by the -10 contour is a local low point. That there are two such regions, one centered near input coordinate \((x=-0.5, y=3.3)\), the other at \((x=1.5, y=3.1)\).

Exercise 15 Consider this interactive contour plot:

Find a value for the (x, y) inputs where the function output is about -0.35.

question id: frog-swim-closet-1

Find a value for the (x, y) inputs where the function output is about 0.9.

question id: frog-swim-closet-2

One contour goes through the input A=(x=1.53, y=1.59). The neighboring contour goes through B=(x=1.53, y=0.90).
1. What is the function output half-way between these two points, at input C=(x=1.53, y=1.25)?
2. How does the output at C relate to the output at A and B?

question id: frog-swim-closet-3

Choose one of the contours and track the output as you move the cursor along that contour. What pattern do you see in the output?

question id: frog-swim-closet-4

Exercise 17 IN-CLASS ACTIVITY

Using the blank grids provided, draw (by hand) a slice plot along the path indicated.

a slice plot of f() along the slice y = 0.
a slice plot of f() along the slice x = 0.5

Exercise 18 In-class activity

There are nine pattern-book functions: constant, identity, square, reciprocal, sinusoid, exponential, logarithm, gaussian, sigmoid.

On a piece of paper, sketch from memory a graph of each of the pattern-book functions.

Exercise 19

For the function in Figure fig-drl-reading-graph-1, find the output values for each of these inputs. Your answers will necessarily be approximate, but be as close as feasible.

Input	Output
-1
0
3
5