Chap 6 Exercises

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

Exercise 2 All but two of the pattern-book functions have a domain that runs over the whole number line: \(-\infty < x < \infty\).

Which pattern-book function has a domain that excludes zero and negative numbers as inputs?

Which pattern-book function has just a single value missing from its domain?

Exercise 4 The following R/mosaic code makes a plot.

Explain why the function graph doesn’t cover the whole domain indicated in the command.

question id: shark-rise-kitchen-1

Calculate log(-5). What is the result?

-5 -1.609 NaN 1.609 5

question id: shark-rise-kitchen-2

The answer to (2) demonstrates an important feature of modern computer numerics. When an input is outside of the domain of the function, the function value is undefined. This “undefined” value is represented by a special “number,” which has no value. The special “number” is written NaN standing for “not a number.”

Do some experimental arithmetic with NaN, for instance, adding 5 or multiplying by 3. Construct a simple rule that accounts for the results in your experiments.

question id: shark-rise-kitchen-3

Exercise 5 Suppose \(w(t)\) is your weight and \(h(t)\) is your height and that the domain of both functions encompasses your whole life span. It’s reasonable to consider each of these functions to be continuous. So the function \(h(t) - w(t)\) is continuous.

Is the following statement true? At some time since you were born your weight in pounds equaled your height in inches.

True False

question id: kVNeFw

Is the following statement true? At some time since you were born your weight in pounds equaled your height in inches.

```{mcq}
#| label: ewdy-1
#| show_hints: true
1. True [ correct hint: Right  ]
2. False [ hint: The value of this function when you were born was greater than zero. Calculate the value for $t=$today and you'll see that it is negative. A continuous function whose value goes from positive to negative must cross 0 at least once. ]
```

In-class activity

Exercise 6 We’re going to look at the pattern-book functions in terms of three properties

Slope: whether the graph has a slope that is consistently positive, negative, both, or neither, and
Concavity: whether the function being graphed is concave up, concave down, neither, or both (i.e., concave up in some regions of the domain and down for others).
Horizontal asymptote: whether the function becomes approximately constant as the input goes to either plus or negative infinity.

As a reminder, here’s a command to plot a function over the graphics domain \(-3 \leq x \leq 3\):

Fill in the following table based on the graphs you make of the pattern-book functions. We have already shown answers for the exponential and sinusoid

function name	R formula	slope	concavity	horiz. asymptote
exponential	`exp(x)`	positive	concave up	\(x \rightarrow -\infty\)
logarithm
sinusoid	`sin(x)`	both	both	neither
square
identity
constant
reciprocal
gaussian
sigmoid

Exercise 7 Run this graph-making command in your console:

In this exercise, you will be modifying the sandbox code to draw different functions, so you can examine their shapes.

Your task is to read and interpret the graphs of the basic modeling functions. Here, you will be looking for zero-crossings: the neighborhood of a point in the function’s domain where the output of the function is negative for inputs on one side and positive for inputs on the other side. If zero is touched but not crossed, we will call that “touched zero.”

The table below lists the pattern-book functions. For each function in the list, fill in the table. That is, say whether the function crosses zero, touches zero but does not cross, or does not touch at all in the part of the domain shown in the graphic: \(-3 \leq x \leq 3\). Also note if the value of the function appears to be reaching a horizontal asymptote at zero for very negative \(x\), for very positive \(x\), for both, or neither.

function name	R formula	zero in domain shown in graph	asymptotic zero
exponential	`exp(x)`	no zeros	for very negative \(x\)
logarithm
sinusoid
square
proportional
constant
reciprocal
gaussian
sigmoid

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