While free-response questions are generally preferred, sometimes the nature of a question lends itself to multiple choice.

Checkpoint1.6.1.Drop-down/Popup.

Note also that the solution to this problem uses an external link.

The number \(\sqrt{2}\)

is

is not

rational.

Answer.

\(\text{is not}\)

Solution.

If \(\sqrt{2}\) were rational, then \(\sqrt{2}=\frac{p}{q}\text{,}\) with \(p\) and \(q\) coprime. But then \(2q^2=p^2\text{.}\) By the Fundamental Theorem of Arithmetic^{ 1 }, the power of \(2\) dividing the left side is odd, while the power of \(2\) dividing the right side is even. This is a contradiction, so \(\sqrt{2}\) is not rational.

Checkpoint1.6.2.Choose one.

Which of the following suggest that differentiation and integration are inverse processes?

The Quadratic Formula

The Fundamental Theorem of Calculus

The Fundamental Theorem of Arithmetic

None of these

Answer.

\(\text{The Fundamental ... of Calculus}\)

Solution.

The correct answer is The Fundamental ... of Calculus.

Checkpoint1.6.3.Choose a Subset of Options.

Select all expressions that are equivalent to \(e^{x^2 + 1/x}\text{.}\) There may be more than one correct answer.

\(\displaystyle e^{x^2} + e^{1/x}\)

\(\displaystyle e^{ (x^3+1) / x}\)

\(\displaystyle \dfrac{ e^{x^2} }{ e^x }\)

\(\displaystyle e^{x^2} e^{1/x}\)

\(\displaystyle e^{x^2} e^{x^{-1}}\)

None of the above

Answer.

\(\text{Choice 2, Choice 4, Choice 5}\)

Solution.

The correct answer is Choice 2, Choice 4, Choice 5.

Checkpoint1.6.4.Choose a Subset of Options with Automated Labeling.

Select all expressions that are equivalent to \(e^{x^2 + 1/x}\text{.}\) There may be more than one correct answer.

\(\displaystyle e^{x^2} + e^{1/x}\)

\(\displaystyle e^{x^2} e^{x^{-1}}\)

\(\displaystyle e^{x^2} e^{1/x}\)

\(\displaystyle e^{ (x^3+1) / x}\)

\(\displaystyle \dfrac{ e^{x^2} }{ e^x }\)

None of the above

Answer.

\(\text{B, C, D}\)

Solution.

The correct answer is B, C, D.

Checkpoint1.6.5.Choose a Subset of Options with Explicit Labeling.

Select all expressions that are equivalent to \(e^{x^2 + 1/x}\text{.}\) There may be more than one correct answer.