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.
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 |
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Hence by Rolle’s theorem, there exist |
f’(c) = 0 |
i.e. 2c = 0 |
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Thus |
f'(c) = 0 |
Hence Rolle’s Theorem is verified. |
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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 |
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 |
Hence f(x) is differentiable in |
Consider |
Also, |
Thus |
Hence all the conditions of Rolle’s theorem are satisfied, so there exist |
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 |
Hence all conditions of LMVT are satisfied Hence at least |
Such that |
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i.e. |
i.e. |
i.e. |
i.e. |
since e = 2.7183 |
Clearly c = 1.7183 |
Hence LMVT is verified. |
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Q6) Verify mean value theorem for f(x) = tan-1x in [0, 1]
S6)
Here |
Clearly |
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. |
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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, |
i.e. |
Cast I: |
If |
i.e. |
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Case II: |
If |
i.e. |
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Thus for x2 > x1 |
Thus f(x) is increasing function is (a, b) |
Case III: |
If |
Then from equation (1) |
i.e. |
since |
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Q7) Prove that
And hence show that
S7)
Let |
Clearly |
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 |
i.e. |
since a < c < b |
i.e. |
i.e. |
i.e. |
i.e. |
Hence the result |
Put |
i.e. |
i.e. |
i.e. |
i.e. |
Hence the result |
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, |
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Clearly both f’(x) and g’(x) exist & finite in |
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Hence by Cauchy mean value theorem, there exist at least |
i.e. |
i.e. 1 = cot c |
i.e. |
clearly |
Hence Cauchy mean value theorem is verified. |
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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 |
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Clearly f(x) and g(x) are derivable in (a, b) |
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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 |
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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 |
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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, |
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Since |
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3)
Then
Proof: |
Let f(x) = cos x |
Then by Maclaurin’s series, |
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Since |
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4) then
Proof: |
Here f(x) = tan x |
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Since |
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5) Then
Proof:- |
Here f(x) = sin hx. |
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6) . Then
Proof:- |
Here f(x) = cos hx |
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7) f(x) = tan hx
Proof: |
Here f(x) = tan hx |
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8) then
Proof:- |
Here f(x) = log (1 + x) |
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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 |
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Note that in above expansion if we replace m = -1 then we get, |
Now replace x by -x in above we get, |
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Expand by, Maclaurin’s theorem
Q1)
S1)
Here f(x) = log (1 + sin x) |
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Expand by Maclaurin’s theorem,
log sec x
Solution:
Let f(x) = log sec x |
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Prove that
Solution:
Here f(x) = x cosec x |
= |
Now we know that |
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Expand upto x6
Solution:
Here |
Now we know that |
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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, |
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Using Taylors series method expand
in powers of (x + 2)
Solution:
Here |
a = -2 |
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Since |
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Thus equation (1) becomes |
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Expand in ascending powers of x.
Solution:
Here |
i.e. |
Here h = -2 |
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Thus |
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Expand in powers of x using Taylor’s theorem,
Solution:
Here |
i.e. |
Here |
h = 2 |
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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.
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
L – Hospital rule for |
Statement: |
If |
Indeterminate form, then |
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If |
Then |
We continue the procedure until the limit is exist. |
Q1) Evaluate
S1)
Let |
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Q2) Evaluate
S2)
Let |
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Q3) Evaluate
S3)
Let |
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Q4) Find the value of a, b if
S4)
Let |
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But |
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Q5) Evaluate
S5)
Let |
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Q6) Evaluate
S6)
Let |
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Taking log on both sides we get, |
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i.e. |
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Q7) Evaluate
S7)
Let |
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i.e. |
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Q8) Evaluate
S8)
Let |
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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)