Explain, using the Unit Circle, how cos^2 (x) + sin^2 (x) = 1. Give an example
The Pythagorean theorum states a^2 + b^2 = c^2. Knowing the radius of the unit circle is always 1, it is possible to take any point along the circle and square the x-coordinate (cos(x)) as well as the y-coordinate (sin(x)) and it will sum to 1, if you make a triangle from that point to the origin of the coordinate system. Take, for example, the point in the first quadrant, 30 degrees above the horizontal, (sqrt(3)/2, 1/2). If you add (sqrt(3)/2)^2 and (1/2)^2, that simplifies to 3/4 + 1/4, which is 1. This can be done at any and all points because a triangle can be drawn to the origin at any point on the circle with the same results.
The integral and dreivative are two of the most important concepts to understand in Calculus. Explain what a derivative and integral are. Not the literal definitions of them mathematically, but what they mean in the physical world. Give an example of each as well
A derivative is, basically, the rate of change of something. Take a car, for example. If what we are looking at is the distance the car travels, the derivative of that is the velocity of the car. The velocity is the rate at which the car changes its position. The higher the velocity, the greater the rate of change. The opposite of the derivative is the integral. This is the total summation of the change of the property that is being looked at, over a certain amount of time. With the car example, the integral of velocity would be the distance the car travels in a certain time. That is, the total sum of the distances traveled at each velocity.
A 5 kg cannonball is shot out of a cannon, angled at 45 degrees and with an initial velocity of 10 m/s. How will its flight path compare to a ball that weights 1 kg that is shot out of the same cannon (assuming the same angle and initial velocity as the first shot, and neglecting wind and air resistance)? Which will hit the ground first if they are shot at the same time?
The flight paths should be identical. Since the vertical acceleration of each will be gravity and only gravity, they should both rise and fall at the same rate. The horizontal velocities will both be equal and constant. Since they will both have the same initial horizontal velocities, and there are no horizontal forces acting on either ball, they will not change. Mass does not come into account at all in these types of projectile motion problems (ones with no horizontal forces acting on the object and just gravity in the vertical direction). They will both hit the ground at the same time if shot simultaneously.