## Karnataka Class 12 Commerce Maths Unit III – Calculus Complete Details

Karnataka Class 12 Commerce Maths Unit III – Calculus : CBSE provides standard education to all and also promotes a state-of-the art environment that makes students vivacious and competent in all aspects. CBSE Syllabus is well-structured as several proficient subject experts are associated with this board. The syllabi of CBSE Maths, CBSE Science along with other syllabi are amended from time to time to make students up-to-date with current information so that they can meet all educational demands confidently.

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### Karnataka Class 12 Commerce Maths Unit III – Calculus Complete Details

Karnataka Class 12 Commerce Maths Unit III – Calculus : Here our team members provides you Karnataka Class 12 Commerce Maths Unit III – Calculus Complete Notes in pdf format. Here we gave direct links for you easy to download Karnataka Class 12 Commerce Maths Unit III – Calculus Complete Notes. Download these Karnataka Class 12 Commerce Maths Unit III – Calculus Complete Notes and read well. Karnataka Class 12 Commerce Maths Unit III – Calculus topics are :

**1. Continuity and Differentiability**

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

**2. Applications of Derivatives**

Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

**3. Integrals**

Integration as inverse process of differentiation.Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic propertiesof definite integrals and evaluation of definite integrals.

**4. Applications of the Integrals**

Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).

**5. Differential Equations**

Definition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given.Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

### Karnataka Class 12 Commerce Maths Unit III – Calculus Complete Details

Karnataka Class 12 Commerce Maths Unit III – Calculus : Karnataka Class 12 Commerce Maths Unit III – Calculus Five main Topics those explained bellow :

#### 1) Karnataka Class 12 Commerce Maths Continuity And Differentiability Complete Notes

Karnataka Class 12 Commerce Maths Unit III – Calculus : Definition of **continuity** for English Language Learners. : the quality of something that does not stop or change as time passes : a continuous quality. : something that is the same or similar in two or more things and provides a connection between them. Continuity is ‘the state of being continuous’ and continuous means ‘without any interruption or disturbance’. For example, the price of a commodity and its demand are inversely proportional. The graph of demand curve of a commodity is a continuous curve without any breaks or gaps.

**Continuity****, **in mathematics, rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable—say *x*—is associated with a value of a dependent variable—say *y*. Continuity of a function is sometimes expressed by saying that if the *x*-values are close together, then the *y*-values of the function will also be close. But if the question “How close?” is asked, difficulties arise.

For close *x*-values, the distance between the *y*-values can be large even if the function has no sudden jumps. For example, if *y* = 1,000*x*, then two values of *x* that differ by 0.01 will have corresponding *y*-values differing by 10. On the other hand, for any point *x*, points can be selected close enough to it so that the *y*-values of this function will be as close as desired, simply by choosing the *x*-values to be closer than 0.001 times the desired closeness of the *y*-values. Thus, continuity is defined precisely by saying that a function *f*(*x*) is continuous at a point *x*_{0} of its domain if and only if, for any degree of closeness ε desired for the *y*-values, there is a distance δ for the *x*-values (in the above example equal to 0.001ε) such that for any *x* of the domain within the distance δ from *x*_{0}, *f*(*x*) will be within the distance ε from *f*(*x*_{0}). In contrast, the function that equals 0 for *x* less than or equal to 1 and that equals 2 for *x* larger than 1 is not continuous at the point *x* = 1, because the difference between the value of the function at 1 and at any point ever so slightly greater than 1 is never less than 2.

Karnataka Class 12 Commerce Maths Unit III – Calculus : A function is said to be continuous if and only if it is continuous at every point of its domain. A function is said to be continuous on an interval, or subset of its domain, if and only if it is continuous at each point of the interval. The sum, difference, and product of continuous functions with the same domain are also continuous, as is the quotient, except at points at which the denominator is zero. Continuity can also be defined in terms of limits by saying that *f*(*x*) is continuous at *x*_{0} of its domain if and only if, for values of *x* in its domain,

A more abstract definition of continuity can be given in terms of sets, as is done in topology, by saying that for any open set of *y*-values, the corresponding set of *x*-values is also open. (A set is “open” if each of its elements has a “neighborhood,” or region enclosing it, that lies entirely within the set.) Continuous functions are the most basic and widely studied class of functions in mathematical analysis, as well as the most commonly occurring ones in physical situations.

#### CONTINUITY OF FUNCTIONS OF ONE VARIABLE

The following problems involve the CONTINUITY OF A FUNCTION OF ONE VARIABLE. Function *y* = *f*(*x*) is continuous at point *x*=*a* if the following three conditions are satisfied :

i.) *f*(*a*) is defined ,

ii.) exists (i.e., is finite) ,

and

iii.) .

Function *f* is said to be continuous on an interval *I* if *f* is continuous at each point *x* in *I*. Here is a list of some well-known facts related to continuity :

1. The SUM of continuous functions is continuous.

2. The DIFFERENCE of continuous functions is continuous.

3. The PRODUCT of continuous functions is continuous.

4. The QUOTIENT of continuous functions is continuous at all points *x* where the DENOMINATOR IS NOT ZERO.

5. The FUNCTIONAL COMPOSITION of continuous functions is continuous at all points *x* where the composition is properly defined.

6. Any polynomial is continuous for all values of *x*.

7. Function *e*^{x} and trigonometry functions and are continuous for all values of *x*.

Karnataka Class 12 Commerce Maths Unit III – Calculus :: Most problems that follow are average. A few are somewhat challenging. All limits are determined WITHOUT the use of L’Hopital’s Rule. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by using the above step-by-step definition of continuity at a point and the well-known facts, and by giving careful consideration to the indeterminate form during the computation of limits. Knowledge of one-sided limits will be required. For a review of limits and indeterminate forms click here.

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### 2) Karnataka Class 12 Commerce Maths Applications Of Derivatives Complete Notes

Karnataka Class 12 Commerce Maths Unit III – Calculus : 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.

Karnataka Class 12 Commerce Maths Unit III – Calculus : 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.**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.**Critical Points**In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.**Minimum and Maximum Values**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.**Finding Absolute Extrema**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.**The Shape of a Graph, Part I**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.**The Shape of a Graph, Part II**Here we will take a look at the Mean Value Theorem.**The Mean Value Theorem**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.**Optimization Problems**Here are even more optimization problems.**More Optimization Problems**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.**L’Hospital’s Rule and Indeterminate Forms**Here we will use derivatives to compute a linear approximation to a function. As we will see however, we’ve actually already done this.**Linear Approximations**We will look at differentials in this section as well as an application for them.**Differentials**With this application of derivatives we’ll see how to approximate solutions to an equation.**Newton’s Method**Here we will take a quick look at some applications of derivatives to the business field.**Business Applications**

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### 3) Karnataka Class 12 Commerce Maths Integrals

In mathematics, an **integral** assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [*a*, *b*] of the real line, the **definite integral**

- {\displaystyle \int _{a}^{b}\!f(x)\,dx}

is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines *x* = *a* and *x* = *b*. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.

Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the term *integral* may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an *indefinite integral* and is written:

- {\displaystyle F(x)=\int f(x)\,dx.}

The integrals discussed in this article are those termed *definite integrals*. It is the fundamental theorem of calculus that connects differentiation with the definite integral: if f is a continuous real-valued function defined on a closed interval [*a*, *b*], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by

- {\displaystyle \int _{a}^{b}\!f(x)\,dx=F(b)-F(a).}

The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [*a*, *b*] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space.

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4) Karnataka Class 12 Commerce Maths Applications Of The Integrals For Complete Details

In geometry, we have learnt formulae to calculate areas of various geometrical figures including triangles, rectangles, trapezias and circles. Such formulae are fundamental in the applications of mathematics to many real life problems. The formulae of elementary geometry allow us to calculate areas of many simple figures. However, they are inadequate for calculating the areas enclosed by curves. For that we shall need some concepts of Integral Calculus. In the previous chapter, we have studied to find the area bounded by the curve y = f (x), the ordinates x = a, x = b and x-axis, while calculating definite integral as the limit of a sum. Here, in this chapter, we shall study a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabolas and ellipses (standard forms only). We shall also deal with finding the area bounded by the above said curves.

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5) Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

A **differential equation** is a mathematical **equation**that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the **equation** defines a relationship between the two.

A **differential equation** is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

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### Karnataka Class 12 Commerce Maths Unit III – Calculus Complete Details

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