In a comment you asked for some examples of extra credit questions I've given.
Example 1:
Roxelle Porter-Knight is driving an early 23rd century gravity-pulsed car with $700$-factor inertial dampers along a one mile course laid out in the desert salt flats and manages to average only $28$ $\frac{\text{mi}}{\text{hr}}$ for the first half mile. How fast must she average over the next half mile so that her average speed for the entire one mile trip is $55$ $\frac{\text{mi}}{\text{hr}}?$ Give your answer to the nearest $0.1$ $\frac{\text{mi}}{\text{hr}}.$
I asked variations of this one in classes such as college algebra, precalculus, calculus 1, and occasionally others. I often asked a different version again in the same class, letting students know in advance that I would. I did this because I thought the underlying principle was important enough (average of velocities doesn't have to equal average velocity), especially in calculus 1, and it was an easy way to get most students to master something somewhat relevant to the material without putting in a lot of class time. Usually in college algebra and precalculus classes I would work it on the board for them the first time, and in calculus classes I left it up to them.
Example 2:
$$\lim_{n \rightarrow \infty}\frac{(2n^3 – n^2 + 6)^4(3n-7)^3}{n(1+7n)(2+3n^2)^2(3+4n^3)^3}$$
I often put something like this on a calculus 2 quiz on a day when I wanted to review limits of rational functions (e.g. for ratio test with series convergence), with the review being at a more mathematically mature level than was probably the case when they covered this in precalculus or calculus 1. By "more mathematically mature level", I mean knowing the idea of using dominate terms and being able to quickly find the dominate terms. I'd work the extra credit problem to demonstrate the idea:
$$\frac{(2n^3 – n^2 + 6)^4(3n-7)^3}{n(1+7n)(2+3n^2)^2(3+4n^3)^3} \;\; = \;\; \frac{(2^4n^{12} + \cdots)(3^3n^3 + \cdots)}{n(7n + \cdots)(3^2n^4 + \cdots)(4^3n^9 + \cdots)}$$
$$= \;\; \frac{2^43^3n^{15} + \cdots}{3^24^3\cdot7n^{15} + \cdots} \;\; \longrightarrow \;\; \frac{2^43^3}{3^24^3\cdot7} \;\; = \;\; \frac{2^43^3}{2^63^2\cdot7} \;\; = \;\; \frac{3}{2^2\cdot7} \;\; = \;\; \frac{3}{28}$$
This specific example came from a quiz given to a very strong high school class, by the way. Note that it is easy vary the difficulty of something like this.
Example 3:
Evaluate the following, where $a$ is a constant greater than $1.$ Your answer will be an expression involving $a.$ Show appropriate work! (A suitable graph along with some high school geometry area calculations is acceptable.)
$$\int_{0}^{a}|ax-a|\,dx$$
Example 4:
Expanding as much as possible using the chain rule, find the 2nd derivative of $f \circ f \circ f.$
Example 5:
$$\int (6x – 1)^2 \sqrt{2x-1}\,dx$$
This is a good type to use near the beginning of covering integration by $u$-substitution. It's a standard type, but students will remember the trick better if they struggle with it a bit on a quiz (or hour test). The idea, of course, is that the square root function is not an additive function, so you want to let $u = 2x-1$ (and not $u = 6x-1).$ Then you have to "force" the substitution by solving for $x$ in terms of $u$ in order to express $(6x – 1)^2$ in terms of $u.$
Example 6:
Find the unique solution to the following ODE that contains the point $\left( 0,\,1\right).$ Express your answer as $y$ solved in terms of $x,$ making use of the error function. The error function is defined as $\text{erf}\,(x) =\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt.$ [Footnote: Your text discusses the error function on pp. 84-85 and in Section 165 (pp. 511-518).]
$$y' - 2xy\;=\;1$$
I put this on a differential equations quiz on a day that I wanted to talk about using unfamiliar functions and indicated integrations to express solutions. I had planned to work this problem as one of my lecture examples, but then decided it would be worth the time to put it as an extra credit problem on that day's quiz. I'm certain the extra time I gave them for the quiz (I probably give them 20 minutes for what was otherwise a 10-minute quiz) was more than made up by their retention of the idea and by their attention to my working it out after the quiz.
