Class notes
Mean value theorem
- Course
- Institution
It is about mean value 's very helpful for bsc,msc maths students, engineering students etc
[Show more]
Seller
Follow
Anjalivarun
Content preview
BCA 21 / IMCA 21 / Mathematics SVT 87MEAN VALUE THEOREMS
Rolle's Theorem
Statement : If f (x) is a function
(i) Continuous in the closed interval a ≤ x ≤ b
(ii) differentiable in the open interval a < x < b and
(iii ) f (a ) = f (b) then there exists at least one value c of x such that f ′(c) = 0 for a < c < b.
P
Geometrical Meaning y = f (x )
A & B are points on the curve such that f (a ) = f (b). Tangent at P is parallel to A B
AB such that the slope of the tangent is f ′(c) = 0
f (a ) f (b )
Lagrange's Mean Value Theorem a c b
Statement : If f (x) is a function
(i) Continuous in the closed interval a ≤ x ≤ b
(ii) differentiable in the open interval a < x < b
f (b) − f (a)
then there exists at least one value c of x in the interval such that = f ′(c)
b−a
Note : - If the interval is (a, a + h) then f (a + h) − f (a ) = f ′(c)
ie f (a + h) = f (a ) + hf ′(c)
P
Geometrical Meaning
B
A & B are points on the curve corresponding to x = a & x = b. Join AB There
A
will be a tangent at P, parallel to AB so that scope of the tangent is f (b )
f (a )
f (b ) − f ( a )
which is f ′(c) a c b
b−a
Cauchy's Mean Value Theorem
Statement : If f ( x) & g ( x) are two functions which are
(i) continuous in the closed interval [a, b]
(ii) differentiable in the open interval (a, b) and
(iii) g ′( x) ≠ 0 for any value of x in (a, b)
f (b) − f (a ) f ′(c)
then there exists at least one value c of x in (a, b) such that =
g (b) − g (a ) g ′(c)
Taylor's Theorem
If f ( x) is a function such that (i) f ( x) and its (n − 1) derivatives are continuous in the closed interval [a, a + h]
ie a ≤ x ≤ a + h and (ii) n th derivative f ( n) ( x) exists in the open interval (a, a + h).
, 88 KSOU Differential Calculus
Then there exists at least one number θ (0 < θ < 1) such that
h2 h n ( n)
f (a + h) = f (a) + hf ′(a) + f ′′(a) + L + L + f (a + θh)
21 n!
x2 x n ( n)
Note : - put a = 0 & h = x, then f ( x) = f (0) + xf ′(0) + f ′′(0) + L + L + f (θx), where 0 < θ < 1
21 n!
xn
This expression for f ( x) is called Maclaurin' s Expansion and further if f (θx) tends to zero as n → ∞.
n!
x2
Then f ( x) = f (0) + xf ′(0) + f ′′(0) + LL to ∞. This is called Maclaurin' s Series for f ( x).
2!
Examples
(1) Verify Rolle' s Theorem for f ( x) = ( x + 2) 3 ( x − 3) 4 in (−2, 3) and find c.
Solution : f ( x) is continuous in the closed interval [−2, 3] & differentiable in (−2, 3) and further f (−2) = 0, f (3) = 0
ie f (−2) = f (3)
∴ There exists a value c in (−2, 3) such that f ′(c) = 0
f ′( x) = 3( x + 2) 2 ( x − 3) 4 + 4( x + 2) 3 ( x − 3) 3
f ′(c) = 3(c + 2) 2 (c − 3) 4 + 4(c + 2) 3 (c − 3) 3 = (c + 2) 2 (c − 3) 3 [3(c − 3) + 4(c + 2)] = (c + 2) 2 (c − 3) 2 (7c − 1)
1
f ′(c) = 0 ⇒ 7c − 1 = 0 ⇒ c =
7
( 2) Find ' c' of the Lagrange' s Mean Value Theorem for the function f ( x ) = ( x − 1)( x − 2)( x − 3) in (0, 4)
Solution : f (0) = (−1)(−2)(−3) = −6, f (4) = 3 × 2 × 1 = 6
f ′( x) = ( x − 2)( x − 3) + ( x − 1)( x − 3) + ( x − 1)( x − 2)
f ′(c) = (c − 2)(c − 3) + (c − 1)(c − 3) + (c − 1)(c − 2) = c 2 − 5c + b + c 2 − 4c + 3 + c 2 − 3c + 2 = 3c 2 − 12c + 11
f (b) − f (a )
By Lagrange' s Theorem = f ′(c)
b−a
f ( 4) − f ( 0)
∴ = f ′(c)
4−0
6+6
∴ = 3c 2 − 12c + 11
4
ie 12c 2 − 48c + 44 = 12
ie 12c 2 − 48c + 32 = 0 ⇒ 3c 2 − 12c + 8 = 0
12 ± 144 − 96 12 ± 48 12 ± 4 3
∴ c= = =
6 6 6
2 2 2
∴ c = 2± ie c = 2 − & 2+
3 3 3
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller Anjalivarun. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.39. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
79223 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now