What does the unit circle represent and how is it useful?
The unit circle provides a foundation for the understanding of how trigonometry works. It is literally a visual representation of trigonometry. With a unit radius of 1 centered around the origin, the unit circle displays how sine and cosine vary with respect to one another in any application. Take the Pythagorean theorem and apply it to the unit circle. We obtain one of the most valuable trigonometric identities there is, x^2 + y^2 =1.... sin(x)^2 + cos(x)^2 = 1
How do I ensure that the crane does not fall over?
Draw a free body diagram modeling the crane. Sum the forces in the x-direction according to Newton's Second Law (F=ma) Sum the forces in the y-direction according to Newton's Second Law (F=ma) Did you remember every force? Don't forget about the force of friction acting against the legs of the crane. Don't forget the vertical force due to gravity acting in accordance with the mass of the crane. Don't forget to locate the center of mass of the crane. Don't forget to account for the moment arm created by the mass hanging off the end of the crane. Don't forget to quantify the maximum mass that the crane can handle. Don't forget to quantify the maximum distance the center of mass can move before it dominates and tips the crane over. Now we can begin developing our model.
Conceptually, explain the meaning behind the derivative and the integral. Explain how we use these tools in the real world and why they are important.
The derivative and the integral are tools used in calculus that help us to discover the non-obvious nature of given functions. These tools allow us to take a system described by a function (or functions) and obtain insight on its behavior. Take a system described by any random function, and calculus will tell you where its been and where it wants to be. This, in and of itself, is infinitely valuable in the real world.