The instantaneous rate of change, or derivative, describes the change in a value at a precise point. Derivatives show up in many types of problems including rates of change, tangents and velocity. A common derivative problem asks the student to find the change in an object's velocity at an arbitrary point in time. Derivatives are fundamental to mathematics, and students do not learn about them until calculus. Determining instantaneous rates of change is impossible without calculus.
Math teachers introduce Integrals toward the end of calculus one. Mathematicians call this process integration, and it describes the net change of a value or the area beneath a graph. This kind of problem works in reverse to derivatives. For example, to calculate acceleration, you find the derivative of velocity. To calculate velocity, you find the integral of acceleration. The process of integration requires a firm understanding of limits and derivatives, subjects specific to calculus.
A mathematical function is an equation that generates a specific image or curve onto a graph by drawing a point that corresponds to every solution. Optimization problems seek to find the smallest or largest point of a function. In real world applications, optimization problems attempt to find the area or volume of an object by using the least or most amount of material. Solving these problems requires knowledge of derivatives.
The growth rate of bacteria, an ecosystem or the human population describes the change in the population during an interval of time. Biologists describe growth rate systems with mathematical functions, but they need calculus to find the proper equation. Growing populations follow a rule that the equation of the population is equal to the derivative. By using calculus to find the derivative, the biologist can determine this equation and describe a population's growth patterns.