# Chapter Notes: Basic Maths Physics Class 11

Notes for mathematics for physics chapter of class 11 physics. Dronstudy provides free comprehensive chapterwise class 11 physics notes with proper images & diagram.

### What is aÂ Function?

A key idea in mathematical analysis and in Physics is the idea of dependence. One quantity depends on another if the variation of one of them is accompanied by a variation of other. Mathematicians speak of the ** independent variable **and the

**. In Physics, it is better to think in terms of**

*dependent variable***and**

*cause***or**

*effect***.**

*interdependent quantities*The dependence of one quantity on another can be quantitatively expressed in three different ways:

**(i) Tabular presentation**

**(ii) Graphical presentation**

**(iii) Mathematical equations**

Let us consider the distance covered by an automobile, moving at constant speed, as a function of time. Data for a particular example of such motion may be presented numerically, as in the following

**Table.**The exact mathematical relationship between the time and the distance in this example is not immediately obvious while examining the table. This is one of the

*disadvantages*of tabular presentation. Although the numerical values can be precisely specified, they do not at once convey the clear picture of

*how the variables are related*. A graph does this job much better.

**Table: **Time and Distance for a moving Automobile

Elapsed Time(min) |
Distance (km) |

0 | 0 |

2 | 1.5 |

4 | 3.0 |

6 | 4.5 |

8 | 6.0 |

10 | 7.5 |

Let us plot the same data on a graph as shown in the figure. The *independent variable *- ** time** is plotted horizontally; and the

*dependent variable*-

**is plotted vertically. Each pair of numbers in the table gives a single point on the graph. It is immediately obvious that the points may be joined by a single straight line.**

*distance*The equation that fits the above tabular and graphical data is s=0.75t.

Where

*s*represents the distance in kilometre and

*t*represents the time in minutes.

This equation, lacking dimensional consistency, is better replaced byÂ

*s*=

*v*

where v

_{o}t._{o}is a constant whose value in this example isÂ

*v*= 0.75 km/min

_{o}The equation provides the most concise expression of a functional relationship.

**Slope of a Line**

The ** slope** of a line in a graph is defined as the

**(measured in anticlockwise direction) that the line makes with the positive direction of the horizontal axis. This angle is designated by q in figure. That is**

*tangent of the angle*tan q =

*s/t = v*Â Â Â Â Â Â Â Â Â Â Â is the slope of the line given in the above figure.

_{o}**Note**that the quantity tanq is a

*dimensional quantity*in this case,

*length*divide by

*time*. We always measure

*slopes*as a

*vertical increment*divided by a

*horizontal increment*on a graph, each increment being measured in the appropriate unit for the quantity in question. With this understanding,

*the slope of a line is independent of the scales choosen to prepare the graph*.

**DerivativeÂ of a Function**

Slope has a simple physical meaning. *It is the rate of change of the quantity being plotted vertically with respect to the quantity being plotted horizontally*. Mathematically, ** slope is derivative of the function**.

If

*s = v*, the derivative of

_{o}t*s*with respect to

*t*is

**Mathematical Definition**

Let *y* be a function of *x*. If to a small increment Dx of x there corresponds a small increment Dy of y. ThenÂ Â is called the derivative of *y* with respect of *x* and is written asÂ orÂ orÂ

IfÂ .

then,Â

thus,Â

The derivative of *f(x) *at *x *= *a *is denoted by *f '**(a).*

**Geometrical Interpretation of Derivative**

Let us consider the graph of *y = f(x) *as shown in figure. Let P and Q be the two points on it. Then

is the slope of PQ asÂ along the curveÂ ,Â and PQ becomes tangents TPT' at P.

=Â at P.

Â is the slope of tangent at P.

**Rules of Differentiation**

The process of finding the derivative of a function is called *differentiating the function.* Differentiation obeys several simple rules that are worth committing to memory.

**(i)Â The derivative of a constant times a function is the constant times the derivative of the function.**

*Application 1*

*Find the derivative ofÂ y = 3x ^{2}*

**Solution:**

**(ii)Â The derivative of the sum of the functions is the sum of their derivatives.**

*Application 2*

**Find the derivative ofÂ Â y = x ^{3} + 3x^{2}**

**Solution:**

**(iii) Derivative of product of two functions is given as**

**(iv) Derivative of a quotient is given as**

**(v)Â The chain rule**Suppose

*f*is a function of

*u*, which in turn is a function of

*x*. The derivative can beÂ written as the product of two derivatives

*Application 3*

*Find the derivative of ***y = sin x^{2}**

*Solution:*

Let us assume u=x^{2}, then y= sin u. thenÂ = cos u andÂ

= (cos u)(2x) = 2x Cos u

2x cos x^{2}

**Applications of Derivative**

**(i) Increasing and Decreasing Function
**A function

*f*(

*x*) is said to be increasing if

*f*(

*x*) increases as x increases, and decreasing if

*f*(

*x*) decreases as

*x*increases.

In other words, ifÂ then f(x) is increasing

ifÂ then f(x) is decreasing.

As shown in the figure**, **when *f(x)* is increasing, the tangent to the curve at any point, say *P*, makes an acute angle with positive x-axis. The *slope of the tangent is positive*.

Thus,Â

As shown in the figure, when *f(x) *is decreasing, the tangent to the curve at any point, say *P*, makes an obtuse angle with positive *x*â€“axis. The *slope of the tangent is negative*.

Thus,

**(ii) Maximum and Minimum Values of a Function
**As shown in the figure, at the point of maximum and minimum of a function the

*slope of the tangent at the point is zero*.

Thus,Â

**Note
**(i)Â

*f(x)*is

**maximum**at a point x = a, if

(a)Â

*f'*

*(a)*= 0 and

(b)

*f'*

*(x)*changes in sign from

*positive*to

*negative*when

*xÂ*passes through the point

*x*=

*a*. In other words, the

*second derivative*of the function at x = a is

*negative.*i.eÂ

(ii)Â

*f (x)*is

**minimum**at a point

*x*=

*a*, if

(a)Â

*f'*

*(a)*= 0 and

(b)Â

*f'*

*(x)*changes in sign from

*negative*to

*positive*when

*x*passes through the point x = a. In other words, the

*second derivative*of the function at x = a is

*positive*. i.e.Â

**Integration
**

** Integration is the inverse operation of differentiation**. Integration of

*f(x)*consists in finding the

*function I*(

*x*)

*whose derivative is equal to f*(

*x*).

Mathematically,Â orÂ

In the above expression,

*f(x)*is called the

*integrand*;Â Â is the symbol of integration and

*dx*indicates the variable of integration. The symbols on the right side of equation together represent a single entity. It does not mean that

*f(x)*is multiplied by

*dx*. The function

*I(x)*is also known sometimes as the

**of**

*antiderivative**f(x)*.

*Application 4*

*If the derivative function is f(x) = 0. Find the integral function I(x).*

*Solution:*

Since a function whose derivative is zero is a constant, therefore I(x) = *c,Â *where *c *is any fixed number.

The integrals of some common functions are listed in the **Table : 2 **of integrals. Note that moving backward in this table (right column to left column) is equivalent to differentiating. In the same way, moving from right column to left column in the **Table : 1 **of derivatives is equivalent to integrating.

**General Significance of Integration
**

As we have learnt the graphical interpretation of differentiation as finding the *slope of a curve*. Integration also has a simple graphical meaning. It is related to finding the *area under a curve*.

If a function *f *(*x*) is expressed graphically in the form *f* (*x*) vs *x*, the ** area under the curve between the limits a and b **means the

*area bounded*by the curve of

*f*(

*x*), the

*x*-axis and two lines

*x*=

*a*and

*x*=

*b*.

The area under the graph of a *positive function *is defined to be *positive*. The area under (actually above) the graph of a *negative function* is defined to be *negative.* As shown in the figure(c) , *positive* and *negative* area add algebraically and *may cancel*.

** The total area between definite limits of x is called a definite integral. **The notation for the definite integral is

Area=

**Note
**A

*definite integral*between fixed limits is a fixed quantity,

*not a function*. It has a specific

*numerical*

*value*and generally has a

*unit*, which need not be a

*unit of area*. Just as the idea of slope is generalized from its purely geometric meaning and acquires a unit determined by the

*quotient of the vertically plotted quantity*and

*the horizontally plotted quantity*, the idea of areas is also generalized and acquires a

*unit determined by the product of the vertically and horizontally plotted quantities*
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