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

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

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

**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

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

Where

This equation, lacking dimensional consistency, is better replaced byÂ

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

tan q =

**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

**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

*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

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

ifÂ f({x_2})" /> 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 0" />

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

Thus,Â

**Note
**(i)Â

(a)Â

(b)

(ii)Â

(a)Â

(b)Â

**Integration
**

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

Mathematically,Â orÂ

In the above expression,

*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

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

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