Velocity and Acceleration I'm currently studying applications of calculus to the physical world in mathematics. I don't understand how a particle with zero velocity can have a positive acceleration. If the particle is at rest, then how can it be accelerating? 

The key fact that makes this possible is the fact that position, velocity and acceleration are all vector quantities. Even if the magnitudes of these quantities is zero, the direction in a given coordinate system is not zero. Abel hints at this in his comment above-when a ball is thrown up in the air and reaches it's maximum height, the velocity is 0, but the ball has begun to change direction at the inflection point of the parabolic arc. As a result,acceleration is not 0 as the ball has begun to change direction downwards without changing velocity. In mathematical terms,think about the second derivative test for constructing the graph in one dimension. Critical points occur where the first derivative with respect to the independent variable is 0 at the given point,yet the second derivative is nonzero since whether it's positive or negative determines the direction of the graph. It's the exact same thing in this case-the velocity is the first derivative of position with respect to time and acceleration is the second derivative. 
