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Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes

Karnataka Class 12 Commerce Maths Applications Of Derivatives : CBSE board has been offering a robust, holistic school education to students since its inception. The board first analyzes students’ learning requirements and according to that, it prepares suitable syllabus for each class. Additionally, the board designs appropriate question papers to evaluate students’ subject knowledge at the end of each academic session. Moreover, to keep students stress free during exams, the board also designs sample papers for each class. The CBSE board conducts research to get to know the current educational requirements and based on that, it chooses suitable subjects and their relevant topics. Hence, students, who are pursuing their studies under this board, get updated information and keep them prepared for any competitive exams.

Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes

Karnataka Class 12 Commerce Maths Applications Of Derivatives : Here we provides you Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes in PDF Format.Karnataka Class 12 Commerce Maths Applications Of Derivatives topic covers those are Tangents and normal: Equations of tangent and normal to the curves at a point and problems Derivative as a Rate of change: derivative as a rate measure and problems Increasing/decreasing functions and problems Maxima and minima : introduction of extrema and extreme values, maxima and minima in a closed interval, first derivative test, second derivative test. Simple problems restricted to 2 dimensional figures only Approximation and problems.

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Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes

Karnataka Class 12 Commerce Maths Applications Of Derivatives : Applications Of  Derivatives are also used in theorems. Even though this chapter is titled “Applications of Derivatives”, the following theorems will only serve as much application as any other mathematical theorem does in relation to the whole of mathematics. The following theorems we will present are focused on illustrating features of functions which are useful in an identification sort-of-sense. Since graphical analysis is constructed using a different set of analyses, the theorems presented here will instead be applicable to only functions. However, all of what this chapter will discuss on has a graphical component, which this chapter may make reference to in order to more easily bridge a connection. In Real Analysis, graphical interpretations will generally not suffice as proof.

Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes

Karnataka Class 12 Commerce Maths Applications Of Derivatives : Applications Of Derivatives is the rate of change of one quantity in terms of another quantity. So, for example, if you want to know how something changes in time, you usually need to invoke a derivative. This could be something like the position of an object, or the strength of an electric field, or quantity of water in a container, or the amount of money in your bank account…

Suffice it to say that it would be essentially impossible to do any physics without derivatives, which would mean no modern engineering, which would mean essentially no technology developed after the 1700s.

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.

  1. Rates of Change  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.
  2. Critical Points  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.
  3. 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.
  4. 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.
  5. 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/decreasing functions as well as the First Derivative Test.
  6. 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.
  7. The Mean Value Theorem  Here we will take a look at the Mean Value Theorem.
  8. Optimization Problems  This is the second major application of derivatives in this chapter.  In this section we will look at optimizing a function, possibly subject to some constraint.
  9. More Optimization Problems  Here are even more optimization problems.
  10. L’Hospital’s Rule and Indeterminate Forms  This isn’t the first time that we’ve looked at indeterminate forms.  In this section we will take a look at L’Hospital’s Rule.  This rule will allow us to compute some limits that we couldn’t do until this section.
  11. 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.
  12. Differentials  We will look at differentials in this section as well as an application for them.
  13. Newton’s Method  With this application of derivatives we’ll see how to approximate solutions to an equation.
  14. Business Applications  Here we will take a quick look at some applications of derivatives to the business field.

Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes

Karnataka Class 12 Commerce Maths Applications Of Derivatives :  The derivative of a function of a real variable measures the sensitivity to change of the function (output) value with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object’s velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the “instantaneous rate of change”, the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Higher Order Derivatives

To begin our construction of new theorems relating to functions, we must first explicitly state a feature of differentiation which we will use from time to time later on in this chapter.

Suppose {\displaystyle f:\mathbb {R} \to \mathbb {R} }{\displaystyle f:\mathbb {R} \to \mathbb {R} } be differentiable

Let {\displaystyle f'(x)}f'(x) be differentiable for all {\displaystyle x\in \mathbb {R} }{\displaystyle x\in \mathbb {R} } . Then, the derivative of {\displaystyle f'(x)}f'(x) is called the second derivative of {\displaystyle f}f and is written as {\displaystyle f”(a)}{\displaystyle f''(a)} .

What we have stated is that there may exist a second derivative, which is a derivative of a derivative. We will further define second derivative at a to refer to a derivation of a derivative only at the value a.

Similarly, we can define the nth-derivative of {\displaystyle f}f , written as {\displaystyle f^{(n)}(x)}f^{{(n)}}(x)

Foundation Theorems

This short section will first introduce some intriguing properties that differentiation has to offer. Concepts here will help shed insight into the latter theorems. In the previous chapter, you have been introduced to the concept that being able to take the derivative of a function implies continuity at that point. This is important to remember as each theorem will use concepts that continuity offers to justify their proof.

In this section, we will provide more groundwork.

Minimum and Maximum Points

We will now introduce two new concepts about functions, both of which you may be familiar with. We will then justify its existence by creating a theorem to go along with it.

Definitions

We will define a maximum point for a function {\displaystyle f}f on an interval A as such:

{\displaystyle \exists \alpha \in A:f(\alpha )\geq f(x)\quad \forall x\in A}{\displaystyle \exists \alpha \in A:f(\alpha )\geq f(x)\quad \forall x\in A}

We will define a minimum point for a function {\displaystyle f}f on an interval A as such:

{\displaystyle \exists \zeta \in A:f(\zeta )\leq f(x)\quad \forall x\in A}{\displaystyle \exists \zeta \in A:f(\zeta )\leq f(x)\quad \forall x\in A}

Both definitions complement each other, as they refer to opposing inequalities.

Existence of Minimum and Maximum Points Theorem[

This proof will justify the minimum and maximum point definition by relating it to differentiation, namely through this statement

TheoremGiven a function {\displaystyle f}f defined on {\displaystyle (a,b)}(a,b) which is both differentiable at {\displaystyle x}x and it is a maximum or minimum point, its derivative at {\displaystyle x}x is equal to 0

The proof is straightforward: it invokes the definition of differentiation to assert the result of the theorem.

Proof that a Maximum Point has a derivative of 0 for any Given Defined Function
Application of definition. Note that the variable {\displaystyle h}h is any number such that the addition of it with α is still within the interval {\displaystyle (a,b)}(a,b) . {\displaystyle f(\alpha )\geq f(\alpha +h)}{\displaystyle f(\alpha )\geq f(\alpha +h)}
Algebraic Manipulations {\displaystyle f(\alpha +h)-f(\alpha )\leq 0}{\displaystyle f(\alpha +h)-f(\alpha )\leq 0}
Divide by h, which means two cases because it may be negative (and that means an inequality change!) {\displaystyle h>0}{\displaystyle h>0} {\displaystyle h<0}{\displaystyle h<0}
{\displaystyle {\frac {f(\alpha +h)-f(\alpha )}{h}}\leq 0}{\displaystyle {\frac {f(\alpha +h)-f(\alpha )}{h}}\leq 0} {\displaystyle {\frac {f(\alpha +h)-f(\alpha )}{h}}\geq 0}{\displaystyle {\frac {f(\alpha +h)-f(\alpha )}{h}}\geq 0}
One sided limits are applied, which is algebraically valid if applied to both sides (which it is; limit of 0 is 0) {\displaystyle \lim _{h\to 0^{+}}{\frac {f(\alpha +h)-f(\alpha )}{h}}\leq 0}{\displaystyle \lim _{h\to 0^{+}}{\frac {f(\alpha +h)-f(\alpha )}{h}}\leq 0} {\displaystyle \lim _{h\to 0^{-}}{\frac {f(\alpha +h)-f(\alpha )}{h}}\geq 0}{\displaystyle \lim _{h\to 0^{-}}{\frac {f(\alpha +h)-f(\alpha )}{h}}\geq 0}
Merge the one-sided limits together to form a full limit, which demands an equal limit to be valid. Only one value is equal: 0. {\displaystyle \lim _{h\to 0}{\frac {f(\alpha +h)-f(\alpha )}{h}}=0}{\displaystyle \lim _{h\to 0}{\frac {f(\alpha +h)-f(\alpha )}{h}}=0}
{\displaystyle f'(\alpha )=0}{\displaystyle f'(\alpha )=0}
{\displaystyle \blacksquare }\blacksquare

Download here  Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes In PDF Format 

Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes

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Important Note – Preparing for XI & XII Commerce?
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