### Question 1

In class, we have emphasized that "rules" from linear algebra, calculus, etc. can almost always be given intuitive, geometric explanations. For instance, in today's lecture we saw that the chain rule arises from "repeated stretching" by consecutive maps.

Another "rule" we saw was the rule for the total derivative; for instance, for a function $f(t,x(t))$ we have

$$ \frac{df}{dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}\frac{dx}{dt}. $$

Try to give an intuitive explanation for why this relationship holds. Feel free to look anywhere you like online! Use, as much as possible, pictures and English prose rather than symbols and algebraic manipulation. *There is no one "correct" answer!* Do you think you could explain this idea to a random person walking down the street? (If so, try it!)

### Question 2

Today we also studied the "bead on the hoop," i.e., a particle constrained to move on a circle, ignoring the force of gravity. Earlier on in the semester, we derived the equations of motion for a pendulum using reduced coordinates (i.e., just the angle, rather than x and y). Add gravity to our "bead on a hoop" example, and show that the resulting equations of motion agree with the ones we got for the pendulum. (*Hint:* Almost all of the work is done for you already on these slides: [Lagrangian Mechanics] [Constrained Lagrangian Mechanics].)