Ways to Use Calculus

Differentiation and Slopes

• Differentiation is the study of rates of change. If a graph of a function is plotted, for example, as y = 4x + 2, then you can differentiate that function in order to find the slope of the graph at any point. There are many different rules of differentiation, but the one associated with powers can be stated as follows:

  If y = x^n, then dy/dx = n x^(n-1)

  Here, dy/dx is the derivative of the function y. Following the example, if y = 4x + 2, then dy/dx = 4. Hence, the slope of the function is constant.

Integration and Areas Under Curves

• Integration is the inverse function of differentiation. Again using the example y = 4x + 2, you can integrate the function in order to find the area under the curve. There are many different rules of integration, but the one associated with powers is:

  If y = x^n, the integral of y is x(n+1) / n

  Following the example, if y = 4x+2, then the integral is 2x^2 + 2x.

Differentation and Speed

• Because differentiation leads to the rate of change or slope of a quantity, it can be used to calculate the graph of how speed varies with time, given a graph of how position varies with time. For example, if the position has the function s = 3t, where s is the distance and t is the time, then to find the speed, you'll find the rate of change of s with t. To do this, differentiate the function. Following the example, if s = 3t, then ds/dt = 3. Hence, the speed is constant.

Differentiation and Acceleration

• The rate of change of velocity with time is known as acceleration, and you can obtain this rate by differentiating the velocity with respect to time. For example, if the velocity of a particle is described as v = 3t + 4, then the acceleration is dv/dt = 3. Hence, the acceleration is constant.