SB Interactive Exercises

Exercises 5.6 Interactive Exercises

A sample of interactive problem types.

1. True/False.

Every vector space has finite dimension.

True.
The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.
False.
The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.

\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)

2. Multiple-Choice, Not Randomized, One Answer.

What color is a stop sign?

Green
Green means “go!”.
Red
Red is universally used for prohibited activities or serious warnings.
White
White might be hard to see.

Hint 1.

What did you see last time you went driving?

Hint 2.

Maybe go out for a drive?

3. Multiple-Choice, Not Randomized, Multiple Answers.

Which colors might be found in a rainbow? (Note that the radio buttons now allow multiple buttons to be selected.)

Red
Red is a definitely one of the colors.
Yellow
Yes, yellow is correct.
Black
Remember the acronym…ROY G BIV. “B” stands for blue.
Green
Yes, green is one of the colors.

Hint.

Do you know the acronym…ROY G BIV for the colors of a rainbow, and their order?

4. Multiple-Choice, Randomized, One Answer.

What color is a stop sign? [Static versions retain the order as authored.]

Green
Green means “go!”.
Red
Red is universally used for prohibited activities or serious warnings.
White
White might be hard to see.

Hint 1.

What did you see last time you went driving?

Hint 2.

Maybe go out for a drive?

5. Multiple-Choice, Randomized, Multiple Answers.

Which colors might be found in a rainbow? (Note that the radio buttons now allow multiple buttons to be selected.) [Static versions retain the order as authored.]

Red
Red is a definitely one of the colors.
Yellow
Yes, yellow is correct.
Black
Remember the acronym…ROY G BIV. “B” stands for blue.
Green
Yes, green is one of the colors.

Hint.

Do you know the acronym…ROY G BIV for the colors of a rainbow, and their order?

6. Mathematical Multiple-Choice, Not Randomized, Multiple Answers.

Which of the following is an antiderivative of \(2\sin(x)\cos(x)\text{?}\)

\(\sin^2(x)+832\)
Remember that when we write \(+C\) on an antiderivative that this is the way we communicate that there are many possible derivatives, but they all “differ by a constant”.
\(\sin^2(x)\)
The derivative given in the statement of the problem looks exactly like an application of the chain rule to \(\sin^2(x)\text{.}\)
\(-\cos^2(x)\)
Take a derivative on \(-\cos^2(x)\) to see that this answer is correct. Extra credit: does this answer “differ by a constant” when subtracted from either of the other two correct answers?
\(-2\cos(x)\sin(x)\)
The antiderivative of a product is not the product of the antiderivatives. Use the product rule to take a derivative and see that this answer is not correct.

Hint.

You can take a derivative on any one of the choices to see if it is correct or not, rather than using techniques of integration to find a single correct answer.

7. Parsons Problem, Mathematical Proof.

Create a proof of the theorem: If \(n\) is an even number, then \(n\equiv 0\mod 2\text{.}\)

Suppose \(n\) is even.
---
Then there exists an \(m\) so that \(n = 2m\text{.}\)
---
Then \(n\) is a prime number. #paired
---
Then there exists an \(m\) so that \(n = 2m + 1\text{.}\) #paired
---
Click the heels of your ruby slippers together three times. #distractor
---
So \(n = 2m + 0\text{.}\)
 This is a superfluous second paragraph in this block.
---
Thus \(n\equiv 0\mod 2\text{.}\)
---
And a little bit of irrelevant multi-line math

\begin{align*}
c^2&a^2+b^2\\
&x^2+y^2\text{.}
\end{align*}
 #distractor

Hint.

Dorothy will not be much help with this proof.

8. Parsons Problem, Programming.

The Sieve of Eratosthenes computes prime numbers by starting with a finite list of the integers bigger than 1. The first member of the list is a prime and is saved/recorded. Then all multiples of that prime (which not a prime, excepting the prime itself!) are removed from the list. Now the first number remaining in the list is the next prime number. And the process repeats.

The code blocks below can be rearranged to form one of the many possible programs to implement this algorithm to compute a list of all the primes less than \(250\text{.}\) [Ed. this version of this problem requires the reader to provide the necessary indentation.]

This reprises Exercise 2.5.1.

n = 250
---
primes = []
candidates = list(range(2,n))
---
candidates = []
primes = list(range(2,n)) #paired
---
primes = candidates + [p] #distractor
---
while candidates:
---
    p = candidates[0]
    primes.append(p)
---
    for nonprime in range(p, n, p):
---
        if nonprime in candidates:
            candidates.remove(nonprime)
---
print(primes)

9. Parsons Problem, Programming.

The code blocks below can be rearranged to form one of the many possible programs to implement this algorithm to compute a list of all the primes less than \(250\text{.}\) [Ed. this version of this problem does not require the reader to provide the necessary indentation, which is the default.]

This reprises Exercise 2.5.1.

n = 250
---
primes = []
candidates = list(range(2,n))
---
candidates = []
primes = list(range(2,n)) #paired
---
primes = candidates + [p] #distractor
---
while candidates:
---
    p = candidates[0]
    primes.append(p)
---
    for nonprime in range(p, n, p):
---
        if nonprime in candidates:
            candidates.remove(nonprime)
---
print(primes)

10. Parsons Problem, Mathematical Proof, Numbered Blocks.

Create a proof of the theorem: If \(n\) is an even number, then \(n\equiv 0\mod 2\text{.}\) [Ed. This version has numbered blocks, online they are on the right end of the block.]

Suppose \(n\) is even.
---
Then there exists an \(m\) so that \(n = 2m\text{.}\)
---
Then \(n\) is a prime number. #paired
---
Then there exists an \(m\) so that \(n = 2m + 1\text{.}\) #paired
---
Click the heels of your ruby slippers together three times. #distractor
---
So \(n = 2m + 0\text{.}\)
 This is a superfluous second paragraph in this block.
---
Thus \(n\equiv 0\mod 2\text{.}\)

Hint.

Dorothy will not be much help with this proof.

11. Parsons Problem, Programming.

The code blocks below can be rearranged to form one of the many possible programs to implement this algorithm to compute a list of all the primes less than \(250\text{.}\) [Ed. This version has numbered blocks, online they are on the left end of the block.]

This reprises Exercise 2.5.1.

n = 250
---
primes = []
candidates = list(range(2,n))
---
candidates = []
primes = list(range(2,n)) #paired
---
primes = candidates + [p] #distractor
---
while candidates:
---
    p = candidates[0]
    primes.append(p)
---
    for nonprime in range(p, n, p):
---
        if nonprime in candidates:
            candidates.remove(nonprime)
---
print(primes)

12. Matching Problem, Dates.

Match each event in United States history with the year it happened.

Review Encyclopedia Brittania, 25 Decade-Defining Events in U.S. History ¹url.

Monroe Doctrine
1823
Haymarket Riot
1886
Louisiana Purchase
1803
Battle of Gettysburg
1863

13. Matching Problem, Derivatives.

Match each function with its derivative.

Did you compute the derivative of each function in the premises (left column)?

\(x^3-6x^2+5\)
\(3x^2-12x\)
\(x^{-3}\)
\(-3x^{-4}\)
\((x+1)^2\)
\(2x+2\)

14. Matching Problem, Linear Algebra.

Match each subspace with a basis for that subspace. (You may assume that each set is really a basis for at least one of the subspaces.)

Each putative basis is a subset of exactly one of the three subspaces. So for each subspace, two of the three sets can be ruled out by simply testing that the vectors of the basis are members of the subspace, via the membership criteria.

\(\left\{\langle x,y,z\rangle\mid - y + z = 0\right\}\)
\(\left\{\langle -4, 3, 3\rangle, \langle 3, -2, -2 \rangle\right\}\)
\(\left\{\langle x,y,z\rangle\mid -3x - 5y + z = 0\right\}\)
\(\left\{\langle -4, 3, 3\rangle, \langle 5, -4, -5 \rangle\right\}\)
\(\left\{\langle x,y,z\rangle\mid -2x - 5y + 2z = 0\right\}\)
\(\left\{\langle 3, -2, -2 \rangle, \langle 5, -4, -5 \rangle\right\}\)

Hint.

For openers, a basis for a subspace must be a subset of the subspace.

15. Clickable Areas, “Regular” Text.

Identify (by clicking, or by circling) all of the nouns in this quotation by Eleanor Roosevelt.

The incorrect words are pronouns.

“The future belongs to those who believe in the beauty of their dreams.”

16. Clickable Areas, Code.

Identify (by clicking, or by circling) all of the assignment statements in this Python function.

Remember, the operator = is used for assignment.

def main():
    x = 4
    for i in range(5):
        y = i
        if y > 2:
            print(y)

17. Clickable Areas, Text in a Table.

A two-dimensional array was created in Python with the list comprehension:

[[0 for x in range(3)] for y in range(2)]

Then the values were (mostly) changed from zeros and the final array is shown below.

Identify (by clicking, or by circling) all of the boolean values in the array.

Python boolean variables are True and False. A value in quotation marks is a string, not a boolean.

`42`	`True`	`'towel'`
'true'	0	False

This second table has no <area>, in order to test CSS for tables.

`42`	`True`	`'towel'`
'true'	0	False

Hint.

Python boolean variables begin with capital latters.

18. Short Answer.

This sample book is configured to make some simple questions interactive on a capable platform, by adding a <response> element as a signal.

19. Fill-In, Integer Answer.

The game of bowling uses pins that you try to knock down. (This answer blank has been set to be very wide.)

20. Fill-In, String and Number Answers.

Complete the following line of a Python program so that it will declare an integer variable age with an initial value of 5. (These two answer blanks have been set to be very short.)

age = ;

21. Fill-In, Case-Insensitive Answer.

The word is the opposite of “yes”. (Try a mixture of upper and lower-case letters.)

22. Fill-In, Decimal Answer.

The decimal number is an approximation of \(\sfrac{1}{3}\) to within three significant figures. (Wikipedia ²).

www.britannica.com/list/25-decade-defining-events-in-us-history

en.wikipedia.org/wiki/Significant_figures

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Exercises 5.6 Interactive Exercises

1. True/False.

2. Multiple-Choice, Not Randomized, One Answer.

3. Multiple-Choice, Not Randomized, Multiple Answers.

4. Multiple-Choice, Randomized, One Answer.

5. Multiple-Choice, Randomized, Multiple Answers.

6. Mathematical Multiple-Choice, Not Randomized, Multiple Answers.

7. Parsons Problem, Mathematical Proof.

8. Parsons Problem, Programming.

9. Parsons Problem, Programming.

10. Parsons Problem, Mathematical Proof, Numbered Blocks.

11. Parsons Problem, Programming.

12. Matching Problem, Dates.

13. Matching Problem, Derivatives.

14. Matching Problem, Linear Algebra.

15. Clickable Areas, “Regular” Text.

16. Clickable Areas, Code.

17. Clickable Areas, Text in a Table.

18. Short Answer.

19. Fill-In, Integer Answer.

20. Fill-In, String and Number Answers.

21. Fill-In, Case-Insensitive Answer.

22. Fill-In, Decimal Answer.