Find CBSE Class 12th Mathematics notes for the chapter Application of Derivatives. every concept is followed by the solved numerical example. You can also find those questions which have been asked in previous year cbse board exam mathematics paper.
In the previous chapter we focused almost exclusively on the computation of derivatives. In this chapter will focus on applications of derivatives. It is important to always remember that we didn’t spend a whole chapter talking about computing derivatives just to be talking about them. There are many very important applications to derivatives.
The two main applications that we’ll be looking at in this chapter are using derivatives to determine information about graphs of functions and optimization problems. These will not be the only applications however. We will be revisiting limits and taking a look at an application of derivatives that will allow us to compute limits that we haven’t been able to compute previously. We will also see how derivatives can be used to estimate solutions to equations.
Here is a listing of the topics in this section.
The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter. Namely, rates of change.
In this section we will define critical points. Critical points will show up in many of the sections in this chapter so it will be important to understand them.
Minimum and Maximum Values :
In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.
Finding Absolute Extrema:
Here is the first application of derivatives that we’ll look at in this chapter. We will be determining the largest and smallest value of a function on an interval.
The Shape of a Graph, Part I:
We will start looking at the information that the first derivatives can tell us about the graph of a function. We will be looking at increasing or decreasing functions as well as the First Derivative Test.
The Shape of a Graph, Part II :
In this section we will look at the information about the graph of a function that the second derivatives can tell us. We will look at inflection points, concavity, and the Second Derivative Test.
Linear Approximations:
Here we will use derivatives to compute a linear approximation to a function. As we will see however, we’ve actually already done this.
Differentials:
We will look at differentials in this section as well as an application for them.
DERIVATIVE AS A RATE MEASURER:-
Derivatives can be used to calculate instantaneous rates of change. The rate of change of position with respect to time is velocity and the rate of change of velocity with respect to time is acceleration. Using these ideas, we'll be able to analyze one-dimensional particle movement given position as a function of time.
TANGENTS AND NORMALS.
A derivative at a point in a curve can be viewed as the slope of the line tangent to that curve at that point. Given this, the natural next question is what the equation of that tangent line is. Can calculus be used to figure out when a function takes on a local or global maximum value? Absolutely. Not only that, but derivatives and second derivatives can also help us understand the shape of the function (whether they are concave upward or downward). If you have a basic conceptual understanding of derivatives, then you can start applying that knowledge here to identify critical points, extreme, inflections points and even to graph functions.
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Critical points are places where the derivative of a function is either zero or undefined. These critical points are places on the graph where the slope of the function is zero. All relative maxima and relative minima are critical points, but the reverse is not true.
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Increasing and decreasing functions:
In this topic, we will use differentiation to find out whether a function is increasing or decreasing or none.
Consider the function f given by f(x)= x^{2}, x∈ R
The graph of this function is a parabola as given in fig given below:
First consider the graph to the right of the origin. Observe that as we move from left to right along the graph, the height of the graph continuously increases. For this reason, the function is said to be increasing for the real numbers x > 0.
Now consider the graph to the left of the origin and observe here that as we move from left to right along the graph, the height of the graph continuously decreases. Consequently, the function is said to be decreasing for the real numbers x < 0.
We shall now give the following analytical definitions for a function which is increasing or decreasing on an interval.
Definition 1 Let I be an open interval contained in the domain of a real valued function f. Then f is said to be
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