Calculus+Aspects+of+Physic

=Calculus Aspects of Physics= include page="foo"

Calculus has been defined by many to be the study of change over time. In a physics class, this definition holds true and has several physical representations. Calculus concepts allow us to understand various relationships between particles in the form of change of something/change in time. toc =Limits= Before we can begin our discussion of calculus based physics, we must first understand the math behind them and in order to do this, we must start with the concept of limits. In the calculus sense, **limits** can be defined as the behavior of a function f(x) as x a certain value. The power of a limit is that a function does not need to exist at the value x approaches in order for the limit to be found. In cases such that f(x) does not exist at the specified value, there are a variety of techniques that can be employed in order solve the limit.

If the limit is of the form can be solved as follows



Now that we have an understanding of what limits are and how to solve them, the next step is to introduce the concept of the difference quotient. The **difference quotient** comes from the equation for slope of a line,. Instead of using x's and y's, this formula can be re-written as. For any non-linear function, this equation generates the slope of a secant line,a line intersecting with the function at more than one place.However, as demonstrated by the graphic below, as the distance h between the two points becomes smaller, the secant line becomes a better approximation for the line tangent to the graph at the point of interest. In order to find the slope of the tangent line, we can take the limit of the difference quotient as h approaches 0 (in more advanced environments, h is often replaced with, which represents the change in x)

Using the function, let's use the limit of the difference quotient to calculate the slope of the tangent line.



You will notice that the slope is defined by, once again, another equation and is not constant. Unlike linear functions, whose slopes are constant, non-linear functions have slopes that are not constant.

=Differentiating and Integrating=

. When differentiating, constants are reduced to zero.
 * Differentiating** is the process of calculating the slope of the tangent line for functions. As noted above, the limit of the difference quotient is one way to calculate the slope of the tangent line. There are other, sometimes simpler, methods that can be used to calculate the slopes as well. The simplest among these is the **Power Rule of Differentiation**. The Power rule states that

In the math world, is something can be done, it can also be undone. Anti-differentiation, a process called **Integration**, undoes differentiation and is used to calculate the area under curves. Similar to the power rule of differentiation, the **Power Rule of Integration** states that . The "C" at the end of the equation is the constant of integration and must be included in the answer to account for any constants that may have been there before differentiating. In order to find C, you must have an initial condition which would allow you to plug in a value for all variables other than C.

=Position, Velocity, and Acceleration=

Once we understand these basic calculus aspects, we can now apply them to a physics background. First we'll take the general motion equation and the definition of acceleration. If we re-arrange the definition of velocity a bit, we'll get .

The importance of this is that if we apply the **Power Rule of Differentiation** to the general motion equation and differentiate it in respect to time, we observe that the derivative of it is equal to our re-arrangement of the definition of acceleration. What this means is that velocity is the first derivative of position. Another way of saying that is if we graphed a particle's position on a position vs. time graph, the slope of that graph would be equal to the velocity of the particle at that point. If we use the power rule once more and differentiate the function again in respect to time, we observe that the second derivative of position and the first derivative of velocity is equal to acceleration. While normally in an AP Physics B course, acceleration is assumed to be constant, the reality is that it rarely is. Using differentiation, we can calculate the instantaneous acceleration of a particle at any point in time.