1.1 Rolle’s Theorem . | unit 1 differential calculus - Goseeko

Back to Study material

M1

UNIT 1

Differential Calculus

Continuous Function

A function f(x) is said to be continuous at x = a if

f(a) exist is function must exist at x = a

Differentiation

Let y = f(x) be any function A function f(x) is said to be differentiable at x = a if

is exist and it is denoted by

Note:-

A differentiable function is always continuous but converse need not be true.

1.1 Rolle’s Theorem:- .

If

f(x) is continuous in the closed [a, b]

f(x) is differentiable in (a, b) &

f(a) = f(b)

Then there exist at least one value ‘c’ in (a, b) such that f’(c) = 0.

Q1)Verify Rolle’s theorem for the function f(x) = x2 for

S1)

Here f(x) = x2;

Since f(x) is algebraic polynomial which is continuous in [-1, 1]

Consider f(x) = x2

a) Diff. w.r.t. x we get

b) f'(x) = 2x

Clearly f’(x) exist in (-1, 1) and does not becomes infinite.

Clearly

f(-1) = (-1)2 = 1

f(1) = (1)2 = 1

f(-1) = f(1).

Hence by Rolle’s theorem, there exist such that

f’(c) = 0

i.e. 2c = 0

c = 0

Thus such that

f'(c) = 0

Hence Rolle’s Theorem is verified.

Q2) Verify Rolle’s Theorem for the function f(x) = ex(sin x – cos x) in

S2)

Here f(x) = ex(sin x – cos x);

Clearly ex is an exponential function continuous for every also sin x and cos x are Trigonometric functions Hence (sin x – cos x) is continuous in and Hence ex(sin x – cos x) is continuous in .

Consider

f(x) = ex(sin x – cos x)

diff. w.r.t. x we get

f’(x) = ex(cos x + sin x) + ex(sin x + cos x)

= ex[2sin x]

Clearly f’(x) is exist for each & f’(x) is not infinite.

Hence f(x) is differentiable in .

Consider

Also,

Thus

Hence all the conditions of Rolle’s theorem are satisfied, so there exist such, that

i.e.

i.e. sin c = 0

But

Hence Rolle’s theorem is verified.

Q3) Verify whether Rolle’s theorem is applicable or not for

S3)

Here f(x) = x2;

Clearly x2 is an algebraic polynomial hence it is continuous in [2, 3]

Consider

Clearly f’(x) is exist for each

Consider

Thus .

Thus all conditions of Rolle’s theorem are not satisfied Hence Rolle’s theorem is not applicable for f(x) = x2 in [2, 3]

Q4)

a) Show that between any two real root of equation , is at least one real root of .

b) Discuss the applicability of Rolle’s theorem for the function

Lagrange’s Mean value Theorem:-

Statement:- If

f(x) is continuous in [a, b]

f(x) is differentiable in (a, b) then there exist at least one value such that

Q5) Verify the Lagrange’s mean value theorem for

S5)

Here

Clearly f(x) = log x is logarithmic function. Hence it is continuous in [1, e]

Consider f(x) = log x.

Diff. w.r.t. x we get,

Clearly f’(x) is exist for each value of & is finite.

Hence all conditions of LMVT are satisfied Hence at least

Such that

i.e.

i.e.

i.e.

i.e.

since e = 2.7183

Clearly c = 1.7183

Hence LMVT is verified.

Q6) Verify mean value theorem for f(x) = tan-1x in [0, 1]

S6)

Here ;

Clearly is an inverse trigonometric function and hence it is continuous in [0, 1]

Consider

diff. w.r.t. x we get,

Clearly f’(x) is continuous and differentiable in (0, 1) & is finite

Hence all conditions of LMVT are satisfied, Thus there exist

Such that

i.e.

i.e.

i.e.

i.e.

Clearly

Hence LMVT is verified.

Meaning of sign of Derivative:

Let f(x) satisfied LMVT in [a, b]

Let x1 and x2 be any two points laying (a, b) such that x1 < x2

Hence by LMVT, such that

i.e. … (1)

Cast I:

If then

i.e.

is constant function

Case II:

If then from equation (1)

i.e.

means x2 - x1 > 0 and

Thus for x2 > x1

Thus f(x) is increasing function is (a, b)

Case III:

If

Then from equation (1)

i.e.

since and then hence f(x) is strictly decreasing function.

Q7) Prove that

And hence show that

S7)

Let ;

Clearly is an logarithmic function and hence it is continuous also

Consider

diff. w.r.t. x we get,

Clearly f’(x) exist and finite in (a, b) Hence f(x) is continuous and differentiable in (a, b). Hence by LMVT

Such that

i.e.

i.e.

since

a < c < b

i.e.

i.e.

i.e.

i.e.

Hence the result

Now put a = 5, b = 6 we get

Hence the result

Q8) Prove that , use mean value theorem to prove that,

Hence show that

S8)

Let f(x) = sin-1x;

Clearly f(x) is inverse trigonometric function and hence it is continuous in [a, b]

Consider f(x) = sin-1x

diff. w.r.t. x we get,

Clearly f’(x) is finite and exist for . Hence by LMVT, such that

i.e.

since a < c < b

i.e.

i.e.

i.e.

i.e.

Hence the result

Put we get

i.e.

i.e.

i.e.

i.e.

Hence the result

1.2 Mean Value Theorems

Cauchy’s Mean Value Theorem:

Statement:-

If f(x) and g(x) are any two functions such that

f(x) and g(x) are continuous in (a, b)

both f(x) and g(x) are derivable in (a, b)

Then for any value of , at least such that

Q9)

Verify Cauchy mean value theorems for & in

S9)

Let & ;

Clearly f(x) and g(x) both are trigonometric functions. Hence continuous in

Since &

diff. w.r.t. x we get,

&

Clearly both f’(x) and g’(x) exist & finite in . Hence f(x) and g(x) is derivable in and

Hence by Cauchy mean value theorem, there exist at least such that

i.e.

i.e. 1 = cot c

i.e.

clearly

Hence Cauchy mean value theorem is verified.

Q10) Considering the functions ex and e-x, show that c is arithmetic mean of a & b.

Solution:

Clearly f(x) and g(x) are exponential functions Hence they are continuous in [a, b].

Consider &

diff. w.r.t. x we get

and

Clearly f(x) and g(x) are derivable in (a, b)

By Cauchy’s mean value theorem such that

i.e.

i.e.

i.e.

i.e.

i.e.

i.e.

Thus

i.e. c is arithmetic mean of a & b.

Hence the result

Q11) Show that

Prove that if

and Hence show that

Verify Cauchy’s mean value theorem for the function x2 and x4 in [a, b] where a, b > 0

If for then prove that,

[Hint:, ]

Expansions of functions

In this topic we learn two important series expansions namely

a) Maclaurin’s series

b) Taylors Series

Maclaurin’s Series Expansions

Statement:-

Maclaurin’s series of f(x) at x = 0 is given by,

Expansion of some standard functions

1) f(x) = ex then

Proof:-

Here

By Maclaurin’s series we get,

i.e.

Note that

1) Replace x by –x we get

2) f(x) = sin x then

Proof:

Let (x) = sin x

Then by Maclaurin’s series,

… (1)

Since

By equation (i) we get,

3)

Then

Proof:

Let f(x) = cos x

Then by Maclaurin’s series,

… (1)

Since

From Equation (1)

4) then

Proof:

Here f(x) = tan x

By Maclaurin’s expansion,

… (1)

Since

…..

By equation (1)

5) Then

Proof:-

Here f(x) = sin hx.

By Maclaurin’s expansion,

(1)

By equation (1) we get,

6) . Then

Proof:-

Here f(x) = cos hx

By Maclaurin’s expansion

(1)

By equation (1)

7) f(x) = tan hx

Proof:

Here f(x) = tan hx

By Maclaurin’s series expansion,

… (1)

By equation (1)

8) then

Proof:-

Here f(x) = log (1 + x)

By Maclaurin’s series expansion,

… (1)

By equation (1)

9)

In above result we replace x by -x

Then

10) Expansion of tan h-1x

We know that

Thus

11)Expansion of (1 + x)m

Proof:-

Let f(x) = (1 + x)m

By Maclaurin’s series.

… (1)

By equation (1) we get,

Note that in above expansion if we replace m = -1 then we get,

Now replace x by -x in above we get,

Expand by, Maclaurin’s theorem

Q1)

S1)

Here f(x) = log (1 + sin x)

By Maclaurin’s Theorem,

… (1)

……..

equation (1) becomes,

1.3 Taylor's Series and Maclaurin's Series,

Expand by Maclaurin’s theorem,

log sec x

Solution:

Let f(x) = log sec x

By Maclaurin’s Expansion’s,

(1)

By equation (1)

Prove that

Solution:

Here f(x) = x cosec x

=

Now we know that

Expand upto x6

Solution:

Here

Now we know that

… (1)

… (2)

Adding (1) and (2) we get

Show that

Solution:

Here

Thus

Taylor’s Series Expansion:-

a) The expansion of f(x+h) in ascending power of x is

b) The expansion of f(x+h) in ascending power of h is

c) The expansion of f(x) in ascending powers of (x-a) is,

d) Using the above series expansion we get series expansion of f(x+h) or f(x).

Expansion of functions using standard expansions

Expand in power of (x – 3)

Solution:

Let

Here a = 3

Now by Taylor’s series expansion,

… (1)

equation (1) becomes.

Using Taylors series method expand

in powers of (x + 2)

Solution:

Here

a = -2

By Taylors series,

… (1)

Since

, , …..

Thus equation (1) becomes

Expand in ascending powers of x.

Solution:

Here

i.e.

Here h = -2

By Taylors series,

… (1)

equation (1) becomes,

Thus

Expand in powers of x using Taylor’s theorem,

Solution:

Here

i.e.

Here

h = 2

By Taylors series

… (1)

By equation (1)

Exercise

a) Expand in powers of (x – 2)

b) Expand in powers of (x + 2)

c) Expand in powers of (x – 1)

d) Using Taylors series, express in ascending powers of x.

e) Expand in powers of x, using Taylor’s theorem.

1.4 Indeterminate forms

Consider

Then limit of f(x) and g(x) both are zero when then

L becomes form.

This form is called indeterminate form. The other indeterminate formal are , , , , OO, etc.

To evaluate limit in this case we use L – Hospital rule

1.5 L' Hospital's Rule Evaluation of Limits and Applications.

L – Hospital rule for and

Statement:

If takes either or

Indeterminate form, then

Provided limit is exist

If again takes either or .

Then ; limit is exist

We continue the procedure until the limit is exist.

Q1) Evaluate

S1)

Let

…

By L – Hospital rule,

Q2) Evaluate

S2)

Let

…

By L – Hospital rule

Q3) Evaluate

S3)

Let

…

By L – Hospital rule

…

…

Q4) Find the value of a, b if

S4)

Let

…

By L – Hospital rule

…

…

… (1)

…

But

From equation (1)

Q5) Evaluate

S5)

Let

…

…

(By L – Hospital Rule)

Q6) Evaluate

S6)

Let

… 0o form

Taking log on both sides we get,

…

…

By L – Hospital Rule

i.e.

Q7) Evaluate

S7)

Let

…

Taking log on both sides,

…

By L – Hospital rule,

i.e.

Q8) Evaluate

S8)

Let

…

Taking log on both sides, we get

…

By L – Hospital Rule,

Exercise

Evaluate the following limits.

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

a) Find the value of a, b, c if

b) If is finite then find the value of p and hence the value of the limit.

c) Find the value of a, b if,

d) Find the value of a and b if,

e) Find the value of a and b if,

Reference Books:

1. Advanced Engineering Mathematics by Erwin Kreyszig (Wiley Eastern Ltd.)

2. Advanced Engineering Mathematics by M. D. Greenberg (Pearson Education)

3. Advanced Engineering Mathematics by Peter V. O’Neil (Thomson Learning)

4. Thomas’ Calculus by George B. Thomas, (Addison-Wesley, Pearson)

5. Applied Mathematics (Vol. I & Vol. II) by P.N.Wartikar and J.N.Wartikar Vidyarthi Griha Prakashan, Pune.

6. Linear Algebra –An Introduction, Ron Larson, David C. Falvo (Cenage Learning, Indian edition)

Index

Notes

Highlighted

Underlined

:

Browse by Topics

:

Notes

Highlighted

Underlined