UNIT 1 | unit 1 differential calculus

MATHS 1

UNIT 1

Differential Calculus

1.1 Successive Differentiation

It is the process of differentiating the given function simultaneously many times and the result obtained are called successive derivative.

Let be a differentiable function.

First derivative is denoted by

Second derivative is

Third derivative is

Similarly the nth derivative is

Example:

Function

Derivaties

…………..

………..

…………

Example1: Find the nth derivative of

Since

Differentiating both side with respect to x

[

Again differentiating with respect to x

Similarly the nth derivative is

Example2: Find the nth derivative of

Let

]

Differentiating with respect to x we get

Again differentiating with respect to x we get

Similarly Again differentiating with respect to x we get

Example3: Find the nth derivative

Let

Differentiating with respect to x.

Again differentiating with respect to x.

Similarly the nth derivative with respect to x.

1.2 Taylor’s & Maclaurin’s series for one variable

TAYLOR’S

If can be expanded as an infinite series, then

If possesses derivative of all orders and the remainder .

Corollary: Taking and in equation (i) we get

Taking in above we get Maclaurin’s series.

Example1: Expand the polynomial in power of , by Taylor’s theorem.

Let .

Also

Then

Differentiating with respect to x.

Again differentiating with respect to x the above function.

Also the value of above functions at x=2 will be

By Taylor’s theorem

On substituting above values we get

Example2: Expand in power of

Let

Also

Differentiating f(x) with respect to x.

Again differentiating f(x) with respect to x.

Also the value of above functions at x=1 will be

By Taylor’s theorem

On substituting above values we get

Example3:Expand in power of. Hence find the value of correct to four decimal places.

Let

And .

Differentiating with respect to x.

Again differentiating with respect to x.

Also the value of above functions at will be

By Taylor’s theorem

On substituting above values we get

Maclaurin’s theorem:

This is a particular case of Taylor’s theorem in which a=0 and h=x in Taylor’s theorem.

If f(x) can be expanded as an infinite series, then

Where the remainder is

Example1:Ifusing Taylor’s theorem, show that for .

Deduce that

Let then

Differentiating with respect to x.

.Then

Again differentiating with respect to x.

Then

Again differentiating with respect to x.

Then

By Maclaurin’s theorem

Substituting the above values we get

Since

Hence

Example2: Prove that

Let

Differentiating with respect to x.

Again differentiating with respect to x.

and so on.

Putting , in above derivatives we get

so on.

By Maclaurin’s theorem

+………

Substituting the above values we get

Example3:Prove that

Let

Differentiating above function with respect to x.

Again differentiating above function with respect to x.

Putting , in above derivatives we get

so on.

By Maclaurin’s theorem

+………

Substituting the above values we get

Example: Expansion of Some standard series

Let and

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

Let and

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

Let and

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

Let and also

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

Let and also

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

Let and also

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

Let

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

++………

+………

Let

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

+………

Let

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

+………

+………)

10.

Let

Differentiating the above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

1.3 Indeterminate forms

In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then it is said to assume an indeterminate form. More specifically, an indeterminate form is a mathematical expression involving {\displaystyle 0}0,1 {\displaystyle 1} and {\displaystyle \infty }, obtained by applying the algebraic limit theorem in the process of attempting to determine a limit, which fails to restrict that limit to one specific value and thus does not yet determine the limit being sought. The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.

There are seven indeterminate forms which are typically considered in the literature:

{\displaystyle {\frac {0}{0}},~{\frac {\infty }{\infty }},~0\times \infty ,~\infty -\infty ,~0^{0},~1^{\infty },{\text{ and }}\infty ^{0}.}

The most common example of an indeterminate form occurs when determining the limit of the ratio of two functions, in which both of these functions tend to zero in the limit, and is referred to as "the indeterminate form {\displaystyle 0/0}0/0". For example, as {\displaystyle x}x approaches {\displaystyle 0}0, the ratios {\displaystyle x/x^{3}}, ({\displaystyle x/x}, and {\displaystyle x^{2}/x} go to {\displaystyle \infty }, {\displaystyle 1}1, and {\displaystyle 0}0 respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is ({\displaystyle 0/0}, which is undefined. In a loose manner of speaking,( {\displaystyle 0/0} can take on the values {\displaystyle 0}0,1 {\displaystyle 1}, or {\displaystyle \infty }, and it is easy to construct similar examples for which the limit is any particular value.

So, given that two functions {\displaystyle f(x)}f(x) and {\displaystyle g(x)}g(x) both approaching {\displaystyle 0}0 as {\displaystyle x}x approaches some limit point {\displaystyle c}c, that fact alone does not give enough information for evaluating the limit

{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}.}

Not every undefined algebraic expression corresponds to an indeterminate form. For example, the expression ({\displaystyle 1/0}) is undefined as a real number but does not correspond to an indeterminate form, because any limit that gives rise to this form will diverge to infinity.

Expressions that arise in other ways than by applying the algebraic limit theorem may assume the same form as one of the indeterminate forms. It is not appropriate, however, to call these expressions "indeterminate forms" outside the context of determining limits. The most common case is {\displaystyle 0/0} , which may, for example, arise from substituting {\displaystyle 0} 0 for {\displaystyle x} x in the equation {\displaystyle f(x)=|x|/(|x-1|-1)} f(x)= . This expression is undefined, as is division by zero in general. The other case is the expression {\displaystyle 0^{0}} . Whether this expression is left undefined, or is defined to equal {\displaystyle 1} 1, depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero. Note that {\displaystyle 0^{\infty }} and other expressions involving infinity are not indeterminate forms.

Example 1:

Solution:

= = 0

Solution:

=-2

1.4 Curvature and Radius of curvature

Curvature of a given curve:

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. The curvature of a curve at a point is normally a scalar quantity, that is, it is expressed by a single real number.

For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as depending on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.

Radius of curvature:

We can draw a circle that closely fits nearby points on a local section of a curve, as follows.

We say the curve and the circle osculate (which means "to kiss"), since the 2 curves have the same tangent and curvature at the point where they meet.

The radius of curvature of the curve at a particular point is defined as the radius of the approximating circle. This radius changes as we move along the curve.

How do we find this changing radius of curvature?

The formula for the radius of curvature at any point x for the curve y = f(x) is given by:

2y’’-4=0

Y’= =

y’’= = -

y’’.y2 = -2

y’’=-

Substitute y=4, consider y’ and y’’ to solve for R:

We know that,

Radius of convergence (R) =

Radius of convergence (R) =22.36

1.5 Circle of Curvature

Curvature of a given circle:

The curvature of a given curve at a particular point is the curvature of the approximating circle at that point. The curvature depends on the radius - the smaller the radius, the greater the curvature (approaching a point at the extreme) and the larger the radius, the smaller the curvature.

Let Cr be a circle of radius r centred at the origin.

A canonical parameterization of the curve is (counter clockwise)

g(s) =r