When this <exercises> division is hosted on Runestone Academy, it will be enabled for group work. See group selection and submission features are at the end of the division. See [cross-reference to target(s) "worksheet-groupwork" missing or not unique] for more detail. (2024-07-24: experimental.)

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.

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.

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.