The value of this variable changes with respect to a change in the dependent variable. In a mathematical model, the variable that gets manipulated is the independent variable.ĭependent variable - The variable in a function that represents the output is known as the dependent variable. Independent variable - In a function, the variable that acts as the input is known as the independent variable. Here, x (input) is the independent variable, and y (output) is the dependent variable. Some important terms associated with differential calculus are listed below:įunction - A function is defined as a binary relation where each input is mapped to exactly one output. If y = f(x) is the function that is differentiated then, according to differential calculus, the notation is given as f'(x) = dy / dx. For example, the speed of a moving object can be interpreted as the rate of change of distance with respect to time. 4.4: The Mean Value Theorem The Mean Value Theorem is one of the most important theorems in calculus.Differential calculus is the study of the rate of change of a dependent quantity with respect to a change in an independent quantity.
#Application of differential calculus pdf how to
In this section, we look at how to use derivatives to find the largest and smallest values for a function. 4.3: Maxima and Minima Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach.
#Application of differential calculus pdf series
In addition, the ideas presented in this section are generalized later in the text when we study how to approximate functions by higher-degree polynomials Introduction to Power Series and Functions. Linear functions are the easiest functions with which to work, so they provide a useful tool for approximating function values. 4.2: Linear Approximations and Differentials In this section, we examine another application of derivatives: the ability to approximate functions locally by linear functions.In this section, we consider several problems in which two or more related quantities are changing and we study how to determine the relationship between the rates of change of these quantities.
For example, if a balloon is being filled with air, both the radius of the balloon and the volume of the balloon are increasing.
Being able to solve this type of problem is just one application of derivatives introduced in this chapter.
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