## Exercises 3.6 Interactive Exercises

A sample of interactive problem types.

### 1. True/False.

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.

Every vector space has finite dimension.

\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by TheoremÂ 1.2.16. [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.

Green

Green means â€śgo!â€ť.

Red

Red is universally used for prohibited activities or serious warnings.

White

White might be hard to see.

What color is a stop sign?

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

Red

Red is a definitely one of the colors.

Yellow

Yes, yellow is correct.

Black

Remember the acronymâ€¦ROY G BIV. â€śBâ€ť stands for blue.

Green

Yes, green is one of the colors.

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

Do you know the acronymâ€¦ROY G BIV for the colors of a rainbow, and their order?

### 4. Multiple-Choice, Randomized, One Answer.

Green

Green means â€śgo!â€ť.

Red

Red is universally used for prohibited activities or serious warnings.

White

White might be hard to see.

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

### 5. Multiple-Choice, Randomized, Multiple Answers.

Red

Red is a definitely one of the colors.

Yellow

Yes, yellow is correct.

Black

Remember the acronymâ€¦ROY G BIV. â€śBâ€ť stands for blue.

Green

Yes, green is one of the colors.

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.]

Do you know the acronymâ€¦ROY G BIV for the colors of a rainbow, and their order?

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

\(\sin^2(x)+832\)

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 â€śdiffer by a constantâ€ť.\(\sin^2(x)\)

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 â€śdiffer by a constantâ€ť 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.

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

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 \(n\equiv 0\mod 2\text{.}\)

### 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 ExerciseÂ I.2.5.1.

### 9. 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 does not require the reader to provide the necessary indentation, which is the default.]

This reprises ExerciseÂ I.2.5.1.

### 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.]

### 11. 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 has numbered blocks, online they are on the left end of the block.]

This reprises ExerciseÂ I.2.5.1.

### 12. Matching Problem, Dates.

### 13. Matching Problem, Derivatives.

### 14. Matching Problem, Linear Algebra.

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

### 15. Clickable Areas, â€śRegularâ€ť Text.

### 16. Clickable Areas, Code.

### 17. Clickable Areas, Text in a Table.

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 â€śyesâ€ť. (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^{â€‰2â€‰}).

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

`en.wikipedia.org/wiki/Significant_figures`