What is Mean value theorem: Definition and 150 Discussions
In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
More precisely, the theorem states that if
f
{\displaystyle f}
is a continuous function on the closed interval
[
a
,
b
]
{\displaystyle [a,b]}
and differentiable on the open interval
(
a
,
b
)
{\displaystyle (a,b)}
, then there exists a point
c
{\displaystyle c}
in
(
a
,
b
)
{\displaystyle (a,b)}
such that the tangent at
c
{\displaystyle c}
is parallel to the secant line through the endpoints
Suppose f:]a,b[\to\mathbb{R} is some differentiable function. Then it is possible to define a new function
]a,b[\to [a,b],\quad x\mapsto \xi_x
in such way that
f(x) - f(a) = (x - a)f'(\xi_x)
for all x\in ]a,b[. Mean Value Theorem says that these \xi_x exist.
One question that sometimes...
Can we have two tangents (two turning points) within the given two end points just asking? I know the theorem holds when there is a tangent to a point ##c## and a secant line joining the two end points.
Or Theorem only holds for one tangent point. Cheers
I don't need an answer (although I don't have sadly, it's from a test).
I need just a tip on how to start it...
i cannot use Taylor in here (##\ln(x)## is not Taylor function), therefore, its only MVT, but I don't know which point I should try... since I must get the annoying ##\ln(x)##...
$\tiny{3.2.15}$
Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function the secant line through the endpoints, and the tangent line at $(c,f(c))$.
$f(x)=\sqrt{x} \quad [0,4]$
Are the secant line and the tangent line parallel...
a) Proof: By theorem above, there exists a ##a \in \mathbb{R}## such that for all ##x \in I## we have ##f'(x) = a##. Let ##x, y \in I##. Then, by Mean Value Theorem,
$$a = \frac{f(x) - f(y)}{x - y}$$
This can be rewritten as ##f(x) = ax - ay + f(y)##. Now, let ##g(y) = -ay + f(y)##. Then...
Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0.
(a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0.
(b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0.
If you use a major theorem, then cite the theorem...
I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ...
I am focused on Section 3.2 The Cauchy Riemann Equations ...
I need help in fully understanding the Proof of Theorem 3.4 ...The start of Theorem 3.4 and its...
Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?
Hey! :o
Let $D=\left \{x=(x_1, x_2)\in \mathbb{R}^2: x_2>\frac{1}{x_1}, \ x_1>0\right \}$.
We have the function $f: D\rightarrow \left (0,\frac{\pi}{2}\right )$ with $f(x)=\arctan \left (\frac{x_2}{x_1}\right )$.
I want to show using the mean value theorem in $\mathbb{R}^2$ that for all...
Homework Statement
Find the point "c" that satisfies the Mean Value Theorem For Derivatives for the function
## f(x) = \frac {x-1} {x+1}## on the interval [4,5].
Answer - c = 4.48
Homework Equations
##x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a}##
##f'(c) = \frac { f(b) - f(a)} {b-a}##
The Attempt...
Homework Statement
Find all the numbers c that satisfy the conclusion of the Mean Value Theorem for the functions
f(x)=\dfrac{1}{x-2} on the interval [1, 4]
f(x)=\dfrac{1}{x-2} on the interval [3, 6]
I don't need help solving for c, I just want to know how I can verify that the hypotheses of...
Obviously ##\mathbb{R^2}## is convex, that is, any points ##a,b\in\mathbb{R^2}## can be connected with a line segment. In addition, ##f## is differentiable as a composition of two differentiable functions. Thus, the conditions of mean value theorem for vector functions are satisfied. By applying...
Homework Statement
I and J are open subsets of the real line. The function f takes I to J, and the function g take J to R. The functions are in C1. Use the mean value theorem to prove the chain rule.
Homework Equations
(g o f)' (x) = g'(f (x)) f'(x)
MVT
The Attempt at a Solution
[/B]
I know...
Homework Statement
Let f is differentiable function on [0,1] and f^{'}(0)=1,f^{'}(1)=0. Prove that \exists c\in(0,1) : f^{'}(c)=f(c).
Homework Equations
-Mean Value Theorem
The Attempt at a Solution
The given statement is not true. Counter-example is f(x)=\frac{2}{\pi}\sin\frac{\pi}{2}x+10...
Homework Statement
the original function is ##−6 x^3−3x−2 cosx##
##f′(x)=−2x^2−3+2sin(x)##
##−2x^2 ≤ 0## for all x
and ##−3+2 sin(x) ≤ −3+2 = −1##, for all x
⇒ f′(x) ≤ −1 < 0 for all x
The Attempt at a Solution
this problem is part of a larger problem which says
there is a cubic...
Hello,
1. Homework Statement
1) Let f(x) continuous for all x and (f(x)2)=1 for all x. Prove that f(x)=1 for all x or f(x)=-1 for all x.
2) Give an example of a function f(x) s.t. (f(x)2)=1 for all x and it has both positive and negative values. Does it contradict (1) ?
2. The attempt at a...
Homework Statement
for ##0<\alpha,\beta<2##, prove that ##\int_0^4f(t)dt=2[\alpha f(\alpha)+\beta f(\beta)]##
Homework Equations
Mean value theorem: ##f'(c)=\frac{f(b)-f(a)}{b-a}##
The Attempt at a Solution
I got the answer for the question but I have made an assumption but I don't know if...
Homework Statement
Let ###f## be double differentiable function such that ##|f''(x)|\le 1## for all ##x\in [0,1]##. If f(0)=f(1), then,
A)##|f(x)|>1##
B)##|f(x)|<1##
C)##|f'(x)|>1##
D)##|f'(x)|<1##
Homework Equations
MVT: $$f'(c)=\frac{f(b)-f(a)}{b-a}$$
The Attempt at a Solution
I first tried...
Homework Statement
Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem?
Homework EquationsThe Attempt at a Solution
f(x) = 1 - x2/3.
f(-1) = 1 - 1 = 0
f(1) = 1 - 1 = 0
f' = 2/3 x -1/3.
I don't...
Mean Value Theorem
Suppose that ##f## is a function that is continuous on ##[a,b]## and differentiable on ##(a,b)##. Then there is at least one ##c## in ##(a,b)## such that:
$$f'(c) = \frac{f(b) - f(a)}{b - a}$$
My question is: wouldn't it be better to state that ##c## is in ##[a,b]## rather...
Homework Statement
Prove the Mean Value Theorem for integrals by applying the Mean Value Theorem for derivatives to the function
F(x) = \int_a^x \, f(t) \, dt
Homework Equations
[/B]
Mean Value Theorem for integrals: If f is continuous on [a, b], then there exists a number c in [a, b]...
Homework Statement
Verify Lagrange's MVT for
## f(x)= sinx - sin2x ## in [ 0, π ]
Homework Equations
## f'(c) = \frac{f(b)-f(a)}{b-a} ##
The Attempt at a Solution
Got on solving cosx= 2cos2x
How to find c lies in [0, π ]?
Solved it using quadratic equation but it gives a complicated value...
Homework Statement
Homework Equations
Lagrange's mean value theorem
The Attempt at a Solution
Applying LMVT,
There exists c belonging to (0,1) which satisfies f'(c) = f(1)-f(0)/1 = -f(0)
But this gets me nowhere close to the options... :(
Homework Statement
Let f: R -> R be a function such that \lim_{z\to 0^+} zf(z) \gt 0 Prove that there is no function g(x) such that g'(x) = f(x) for all x in R.
Homework Equations
Supposed to use the mean value theorem. If f(x) is continuous on [a,b] and differentiable on (a,b) then...
Need help with this exercise been stuck on it for a while i think i get the gist of what i am supposed to do but can't seem to get it to work i am definitely missing something. I set h(x)= x-2/3 - g(x) and tried using the mean value theorem on [a,b] and then tried finding the minimum value of...
Definition/Summary
The mean value theorem states that if a real-valued function f is continuous and differentiable on an open interval (a,b), then there is a point c in that interval such that f'(c) \ =\ (f(b) - f(a))/(b - a).
It also applies if the condition of differentiability is...
Using second mean value theorem in Bonnet's form show that there exists a
p in [a,b] such that
\int_a^b e^{-x}cos x dx =sin ~p
I know the theorem but how to show this using that theorem .
Homework Statement
http://i.minus.com/jX32eXvLm6FGu.png
Homework Equations
The MVT applies if
1) The function is continuous on the closed interval [a,b] such that a<b.
2) The function is differentiable on the open interval (a,b)
And if the above two conditions are fulfilled...
Homework Statement
Use the mean value theorem to show that \frac{b^3-a^3}{b-a} = \sum_{j=1}^{n}d_j^2 (x_j - x_{j-1}) \text{where} x_{j-1} < d_j < x_j .
Homework Equations
The mean value theorem states that if f is continuous on [a,b] and differentiable on (a,b) then there exists a c in...
Homework Statement
Suppose that f(0)=-3 and f'(x)<=5 for all values of x. The the largest value of f(2) is
A)7
B)-7
C)13
D)8Homework Equations
The Attempt at a Solution
The problem can be easily solved using the mean value theorem but solving it in a different way doesn't give the right answer...
Hello all,
I have a couple of questions.
First, about the mean value theorem for integrals. I don't get it. The theorem say that if f(x) is continuous in [a,b] then there exist a point c in [a,b] such that
\[\int_{a}^{b}f(x)dx=f(c)\cdot (b-a)\]
Now, I understand what it means (I think), but...
Hi all,
I'm having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven't been able to prove it - so maybe it isn't.
Is the following true?
If F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n}...
Homework Statement
State the Mean Value Theorem and find a point which satisfies the conclusions of the Mean Value Theorem for f(x)=(x-1)3 on the interval [1,4].
2. The attempt at a solution
Mean Value Theorem:states that there exists a c∈(a,b) such that f'(c)=\frac{f(b)-f(a)}{b-a}...
Homework Statement
Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Using the generalised mean value theorem, prove that:
f(x)=f(c) + (x-c)f'(c)+\frac{(x-c)^2}{2}f''(\theta) for some \theta \in (c, x)
Homework Equations
Hints given suggest consdiering F(x) =...
Here is the question:
Here is a link to the question:
Calculus 1 Help on Mean Value Theorem? - Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Homework Statement
I have function f which is defined upon an interval [a,b]. I have calculated the mean value using the theorem
\frac{1}{b-a} \int_{a}^b f(x) dx
What I would like to do is to plot in Maple the mean value rectangle. Where the hight of this rectangle represents the mean value...
Homework Statement
For every x in the interval [0,1] show that:j
\frac{1}{4}x+1\leq\sqrt[3]{1+x}\leq\frac{1}{3}x+1
The Attempt at a Solution
Well i subtracted 1 from all sides and divided by x and I got:
\frac{1}{4}\leq\frac{\sqrt[3]{1+x}-1}{x}\leq\frac{1}{3}
But now I need to find a...
Homework Statement
1) Let f be a function differentiable two times on the open interval I and a and b two numbers in I
Prove that: \exists c\in]a,b[:\frac{f(b)-f(a)}{b-a}=f'(a)+\frac{b+a}{c}f''(c)
2) Let f be a function differentiable three times on the open interval I and a and b two...
Homework Statement
Let f and g be two continuous functions on [a,b] and differentiable on ]a,b[ such that for every x in ]a,b[ : f'(x)<g'(x)
Homework Equations
Show that f(b)-f(a)<g(b)-g(a)
The Attempt at a Solution
So I said that there exists a c in ]a,b[ such that f'(c)=(f(b)-f(a))/(b-a)...
Homework Statement
Find a function f on [-1,1] such that :-
(a) there exists c \in (-1,1) such that f'(c) = 0 and
(b) f(a) \neq f(b) for any a\neq b \in [-1,1]
Homework Equations
Lagrange's Mean Value Theorem (LMVT), which states that if f:[a,b]-->ℝ is a function which is...
Homework Statement
Let f: R->R be a function which satisfied f(0)=0 and |df/dx|≤ M. Prove that |f(x)|≤ M*|x|.
Homework Equations
Mean value theorem says that if f is continuous on [a,b] and differentiable on (a,b), then there is a point c such that f'(c)=[f(b)-f(a)]/(b-a).
The...
I am supposed to use the mean-value theorem to show that lim_x→infty(√(x+5)-√(x))=0.
Can anyone help me solving this problem?
I have tried to set up the mean value theorem, but i just do not know how to proceed.
Given f(n) = (1 - (1/n))n
I calculate that the limit as n -> infinity is 1/e.
Also given that x/(1-x) > -log(1-x) > x with 0<x<1 (I proved this in an earlier part of the question) I want to show that:
1 > (f(60)/f(infinity)) > e-1/59 > 58/59
I have tried using my value for f...
Homework Statement
Let f(x)=log(x)+sin(x) on the positive real line. Use the mean value theorem to assure that for all M>0, there exists positive numbers a and b such that f(b)-f(a)/b-a=MHomework Equations
f'(x)=1/x+cos(x)
The Attempt at a Solution
I know that as x→0, f'(x) gets arbitrarily...
I am reading the proof for the M.V.T, mostly understanding it all, except for this one step. Here is the link to it: http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx#Extras_DerAppPf_MVT
It's near the bottom of the page.
What I don't precisely is why they create a new...