question id: ndK-1
Chap 6 Exercises
\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]
Exercise 1 Answer these questions. You can use the interactive R chunk (with domain from -5 to 5) or refer to the graphs in Chapter 5 of the textbook.
Choose from among these possibilities:
- Flat: not-concave
- Down: concave down on the whole domain
- Up: concave up on the whole domain
- DU: concave down on the left and up on the right
- UD: concave up on the left and down on the right
- UDU: concave down in the center and concave up on both flanks
- Which of these best describes the concavity of the gaussian function?
- Which of these best describes the concavity of the sigmoid function?
question id: ndK-2
- Which of these best describes the concavity of the second-order monomial \(m_2(x) \equiv x^2\)?
question id: ndK-3
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?
question id: pattern-book-domain-a
- Which pattern-book function has just a single value missing from its domain?
question id: pattern-book-domain-b
Exercise 3 A function’s domain is the set of possible inputs to the function. A function’s range is the set of possible outputs.
In each of the following questions, choose the correct answer from these possibilities:
- All: All
- Pos: The positive numbers (excluding zero)
- Neg: The negative numbers (excluding zero)
- NonNeg: The non-negative numbers
- Closed: Closed.
- What is the range of the pattern-book exponential function?
question id: nrlp2-1
- What is the range of the pattern-book sine function?
question id: nrlp2-2
- What is the range of the pattern-book logarithm function?
question id: nrlp2-3
- What is the range of the pattern-book square function?
question id: nrlp2-4
- What is the range of the pattern-book proportional function?
question id: nrlp2-5
- What is the range of the pattern-book sigmoid function?
question id: nrlp2-6
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.
- Calculate
log(-5)
. What is the result?
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.
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.
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: Excellent! ]
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 |