Chap 5 Exercises

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

Exercise 1 These questions ask you to find the output of a pattern-book function for a specified input value.

Answer the questions using Active R chunk 1.

Active R chunk 1
  1. What is the exact output of the pattern-book exponential function when the input is \(x=0\)?
0       0.3989423       1/2       1      

question id: eval-functions-at-input-1

  1. What is the exact output of the pattern-book sine function when the input is \(x=0\)?
0       0.3989423       1/2       1      

question id: eval-functions-at-input-2

  1. What is the exact output of the pattern-book sigmoid function when the input is \(x=0\)?
0       0.3989423       1/2       1      

question id: eval-functions-at-input-3

  1. What is the output (to several digits) of the pattern-book Gaussian function when the input is \(x=0\)?
0       0.3989423       1/2       1      

question id: eval-functions-at-input-4

  1. What is the exact output of the pattern-book constant function?
0       0.3989423       1/2       1      

question id: eval-functions-at-input-5

  1. For question (5), why was it OK that I didn’t tell you what the input value is?

question id: eval-functions-at-input-6

Exercise 2  

  1. Which of the pattern book functions has an output value very near zero over all of its domain except for a finite interval?
\(\text{dnorm()}\)       \(\sin()\)       \(e^x\)       \(\ln()\)      

question id: output-values-1

  1. Which of these basic pattern-book functions has value very near zero over almost half of its domain?
\(\text{pnorm()}\)       \(\ln()\)       \(x^{-1}\)       \(x^{2}\)      

question id: output-values-2

  1. Which of these pattern-book functions has value very near zero over almost half of its domain?
\(e^x\)       \(\ln()\)       \(x^{-1}\)       \(x^{2}\)      

question id: output-values-3

  1. Which of these pattern-book functions is not a monomial?
\(\sqrt{x}\)       \(x^2\)       \(x\)      

question id: output-values-4

  1. Which value(s) is missing from the domain of \(x^{-1}\)?

Zero

All non-positive numbers

Negative numbers

No numbers are missing.

question id: output-values-5

  1. Which value(s) is missing from the domain of \(x^{1/2}\)?

Zero

All non-positive numbers

Negative numbers

No numbers are missing.

question id: output-values-6

  1. Which value(s) is missing from the domain of \(\ln(x)\)?

Zero

All non-positive numbers

Negative numbers

No numbers are missing.

question id: output-values-7

Which value(s) is missing from the domain of \(\sin(x)\)?

Zero

All non-positive numbers

Negative numbers

No numbers are missing.

question id: output-values-8

  1. Which value(s) is missing from the domain of \(x^{-1/2}\)?

Zero

All non-positive numbers

Negative numbers

No numbers are missing.

question id: output-values-9

Exercise 3  

  1. Which of the following values is in the domain of the function \(x^{0.429}\)?
-2       -1       0       All of them      

question id: in-the-domain-1

  1. Is zero in the domain of the function \(x^{-1}\)?
Yes       No      

question id: in-the-domain-2

  1. Which of the following values is in the domain of the function \(\sin(x)\)?
-2       -1       0       All of them      

question id: in-the-domain-3

  1. What is the output from the command log(-1) ?
0       An error message.       -Inf       NaN      

question id: in-the-domain-4

  1. What is the output from the command log(0) ?
0       -Inf       NaN      

question id: in-the-domain-5

  1. Which of the following values is in the domain of the function \(\ln()\)?
-1       0       1       All of them      

question id: in-the-domain-6

  1. Which of the following values is in the domain of the function \(x^2\)?
-1       0       1       All of them      

question id: in-the-domain-7

Exercise 4  

  1. By how much should you increase the input to the exponential function to produce a doubling of the output?
0.5       0.693       1       1.386       2       none of these      

question id: doubling-exp-1

  1. By how much should you increase the input to the exponential function to produce a quadrupling of the output?
0.5       0.693       1       1.386       2       none of these      

question id: doubling-exp-2

  1. By how much should you decrease the input to the exponential function to cut the output value by a factor of one-half?
0.5       0.693       1       1.386       2       none of these      

question id: doubling-exp-3

Exercise 5 Active R chunk 2 defines a function \(h()\) that is a variation on the exponential function, but not written in the form \(e^t\):

Active R chunk 2
  1. Judging from the output when you run the code, what is the doubling time for this function? (Choose the closest answer.)
0.517       0.63090       0.749       1.000       1.342      

question id: variations-on-exp-1

Here is another form of exponential function, again written in an unconventional form.

  1. Using trial and error, find the doubling time for \(g()\).
0.517       0.631       0.749       1.000       1.342      

question id: variations-on-exp-2

Exercise 6 The R/mosaic commands in Active R chunk 3 draw a plot of the pattern-book sinusoidal function and evaluate the sinusoid for two inputs. The sinusoid is a periodic function, that is, the output repeats itself over and over as the input increases. The time that it takes for one cycle is called the period of the function.

Active R chunk 3

Using trial and error in Active R chunk 3, find the period of the sinusoid.

Exercise 7  

  1. True or false: \(2^x\) is a power-law function.
TRUE       FALSE      

question id: cat-lend-futon1

  1. True or false: \(3/x^2\) is a power-law function.
TRUE       FALSE      

question id: cat-lend-futon2

  1. True or false: \(5\sqrt{x}\) is a power-law function.
TRUE       FALSE      

question id: cat-lend-futon3

  1. The gravitational force, F, between two bodies is inversely proportional to the square of the distance \(d\) between them. Then …
\(F = k d^{2}\)       \(F = kd^{-2}\)       \(F = k d^{1/2}\)       \(F = k d^{-1/2}\)      

question id: cat-lend-futon4

Exercise 8 The following traditional-style notation is intended to define a function that is 2 times the pattern-book sinusoid. But something is wrong.

\[g(t) \equiv 2 \sin(x)\] What’s wrong with the definition? (Hint: It’s subtle.)

\(g()\) isn’t an appropriate name

The formula should be written \(2 \times \sin(x)\)

\(t\) is not a good choice for the input name.

The input name in the formula does not match the input name on the left side of \(\equiv\).

question id: OCSP2

:::

Exercise 9 The following exponential function y() is written in terms of a parameter b. You can set the parameter by giving its value as a named argument to y(). For instance, in the following, b is being set to 1.6.

Active R chunk 4
  1. How can you tell from the output from Active R chunk 4 that 7 is not the doubling time of y() when b=1.6?

question id: expb-1

  1. Using trial and error with Active R chunk 4, find a value for b that corresponds to a doubling-time of 7. Choose the closest answer. (Remember, you will be changing b.)
0.8234       1.1045       1.6943       1.7381      

question id: expb-2

Active R chunk 5
  1. What about the output from Active R chunk 5 makes it clear that 3 is not the doubling time for the pattern-book exponential function exp()?

question id: expb-3

  1. Using trial and error in Active R chunk 5, find the doubling time for the exp() function. (Pick the closest answer.)
0.312       0.693       0.971       1.21      

question id: exp-double

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