If the position function for a particle is given by a quadratic function, what can be said about the force acting on the particle?
Newton's second law states that the net force acting on an object is proportional to the acceleration of the object, with the objects mass as the constant of proportionality. In other words, F=ma. The acceleration is the derivative function (rate of change) of the velocity function of the given particle. The velocity function of the particle is the the derivative of the position function of the particle. Thus, the acceleration of the particle is given by the second derivative of the position function. The second derivative of a quadratic function is always constant. Thus, the force exerted on the particle must be constant because the mass of the particle and the acceleration of the particle is constant.
Given the prime decomposition of two positive integers, how can we find the greatest common divisor of the two numbers?
We know that every prime number in the prime factorization of a number, divides that number. In fact, so does any power of a rime number in the factorization, as well as any product of primes in the factorization. To find the gcd then, we must take the highest power of each prime which divides both integers and multiply them together. This number then divides both integers, and any other number which also divides both integers must also divide our chosen product. Thus, this product is the gcd of the given integers.
What is the significance and meaning of the interval of convergence for a power series of function, f?
The interval of convergence (IOC) for funciton, f, tells us all of the values in the domain of f, for which the power series representation of f converges. This means that for any value in the IOC, we can approximate the function, f, at that value, as close as we'd like with the power series representation of f. Power series are infinite series, so practically, as we include more and more terms of that power series representation in our approximation, we get closer and closer to the true value of f.