## Exercises3.6Interactive Exercises

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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.

Hint.

$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ by Theorem 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.

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?

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?

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 $$n\equiv 0\mod 2\text{.}$$

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 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 Exercise 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.]

Hint.
Dorothy will not be much help with this proof.

### 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 Exercise I.2.5.1.

### 14.Matching Problem, Linear Algebra.

Hint.

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

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

Hint.

Python boolean variables begin with capital latters.

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

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 = ;

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