- MATHEMATICAL TOOLS
- FUNCTION
- TRIGONOMETRY
- MEASUREMENT OF POSITIVE AND NEGATIVE ANGLES
- SIX BASIC TRIGONOMETRIC FUNCTIONS
- The trigonometric function of a general angle q are defined in terms of x, y, and r.
- GENERAL TRIGONOMETRIC FORMULAS
- DIFFERENTIATION
- DEFINITION OF DIFFERENTIATION
- SLOPE OF A LINE
- AVERAGE RATES OF CHANGE
- THE DERIVATIVE OF A FUNCTION
- GEOMETRICAL MEANING OF DIFFERENTIATION
- RULES FOR DIFFERENTIATION
- THE PRODUCT RULE
- THE QUOTIENT RULE
- DERIVATIVES OF OTHER TRIGONOMETRIC FUNCTIONS
- CHAIN RULE OR “OUTSIDE INSIDE” RULE
- POWER CHAIN RULE
- RADIAN VS. DEGREES
- DOUBLE DIFFERENTIATION
- APPLICATION OF DERIVATIVES
- MAXIMA AND MINIMA
- INTEGRATION
- INTEGRAL FORMULAS
- RULES FOR INTEGRATION
- RULE NO. 3 : RULE OF SUBSTITUTION
- DEFINITE INTEGRATION OR INTEGRATION WITH LIMITS
- APPLICATION OF DEFINITE INTEGRAL : CALCULATION OF AREA OF A CURVE
- VECTOR
- DEFINITION OF VECTOR
- UNIT VECTOR
- MULTIPLICATION OF A VECTOR BY A SCALAR
- ADDITION OF VECTORS
- RESOLUTION OF VECTORS
- MULTIPLICATION OF VECTORS
- VECTOR PRODUCT
- PROPERTIES

MATHEMATICAL TOOLS

Read moreFunction is a rule of relationship between two variables in which one is assumed to be dependent and the

other independent variable, for example :

Read moreMEASUREMENT OF ANGLE AND RELATIONSHIP BETWEEN DEGREES AND

RADIAN

Read moreAn angle in the xy-plane is said to be in standard position if its vertex lies at the origin and its initial ray lies

along the positive x-axis (Fig.). Angles measured counterclockwise from the positive x-axis are assigned

positive measures ; angles measured clockwise are assigned negative measures.

Read moreThe trigonometric function of a general angle q are defined in terms of x, y, and r.

Read moreRULES FOR FINDING TRIGONOMETRIC RATIO OF ANGLES GREATER THAN 90°

Step 1 ® Identify the quadrant in which angle lies.

Read moresin 2q = 2 sin q cos q ; cos 2q = cos2 q – sin2q = 2cos2 q – 1 = 1 – 2sin2 q

cos2 q =

2

1+ cos2q ; sin2 q =

2

1– cos2q

4. sine rule for triangles 5. cosine rule for triangles

Read moreThe finite difference between two values of a physical quantity is represented by D notation.

For example

Read moreAnother name for differentiation is derivative. Suppose y is a function of x or y = f(x)

Differentiation of y with respect to x is denoted by symbol f ’(x)

Read moreIt is the tan of angle made by a line with the positive direction of x-axis, measured in anticlockwise

direction.

Read moreGiven an arbitrary function y = f(x) we calculate the average rate of change of y with respect to x over the

interval (x , x + Dx) by dividing the change in value of y, i.e. Dy = f(x + Dx) – f(x), by length of interval Dx over

which the change occurred.

Read moreWe know that, average rate of change of y w.r.t. x

Read moreThe geometrical meaning of differentiation is very much useful in the analysis of graphs in physics. To understand

the geometrical meaning of derivatives we should have knowledge of secant and tangent to a curve

Read moreRULE NO. 1 : DERIVATIVE OF A CONSTANT

Read moreWhile the derivative of the sum of two functions is the sum of their derivatives, the derivative of the product

of two functions is not the product of their derivatives. For instance,

Read moreJust as the derivative of the product of two differentiable functions is not the product of their derivatives, the

derivative of the quotient of two functions is not the quotient of their derivatives.

Read moreBecause sin x and cos x are differentiable functions of x , the related functions

Read moreWe now know how to differentiate sin x and x2 – 4, but how do we differentiate a composite like sin (x2 – 4)?

The answer is, with the Chain Rule, which says that the derivative of the composite of two differentiable

functions is the product of their derivatives evaluated at appropriate points

Read morePOWER CHAIN RULE

Read moreRADIAN VS. DEGREES

Read moreWe can interpret f ’’ (x) as the slope of the curve y = f ’(x) at the point (x, f ’(x)). In other words, it is the rate

of change of the slope of the original curve y = f (x)

Read moreAPPLICATION OF DERIVATIVES

DIFFERENTIATION AS A RATE OF CHANGE

Read moreSuppose a quantity y depends on another quantity x in a manner

shown in the figure. It becomes maximum at x1 and minimum at

x2. At these points the tangent to the curve is parallel to the x-

axis and hence its slope is tan q = 0. Thus, at a maximum or a

minimum,

Read moreIn mathematics, for each mathematical operation, there has been defined an inverse operation.

For example- Inverse operation of addition is subtruction, inverse operation of multiplication is division and

inverse operation of square is square root. Similarly there is a inverse operation for differentiation which is

known as integration

Read moreINTEGRAL FORMULAS

Indefinite Integral Reversed derivative formula

Read moreRULE NO. 1 : CONSTANT MULTIPLE RULE

Read moreRULE NO. 3 : RULE OF SUBSTITUTION

Read moreDEFINITE INTEGRATION OR INTEGRATION WITH LIMITS

Read moreFrom graph shown in figure if we divide whole area in infinitely small

strips of dx width.

Read moreIn physics we deal with two type of physical quantity one is scalar and other is vector. Each scalar quantities

has magnitude

Read moreIf a physical quantity in addition to magnitude -

has a specified direction.

Read moreUnit vector is a vector which has a unit magnitude and points in a particular direction. Any vector (A

)

can be written as the product of unit vector (A ˆ ) in that direction and magnitude of the given vector.

Read moreMULTIPLICATION OF A VECTOR BY A SCALAR

ADDITION OF VECTORS

Addition of vectors is done by parallelogram law or the triangle law

Read moreResolution along rectangular component :

It is convenient to resolve a general vector along axes of a rectangular

coordinate system using vectors of unit magnitude, which we call as unit

Read moreIt is always a scalar which is positive if angle between the vectors is acute (i.e. < 90º) and negative if angle between

them is obtuse (i.e. 90º < q £ 180º)

Read moreHere q is the angle between the vectors and the direction nˆ is given by the right-hand-thumb rule.

Read moreVector product of two vectors is always a vector perpendicular to the plane containing the two vectors i.e. orthogonal

Read more