Mean value theorem Definition and 150 Threads

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



(
a
,
f
(
a
)
)


{\displaystyle (a,f(a))}
and



(
b
,
f
(
b
)
)


{\displaystyle (b,f(b))}
, that is,

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  1. K

    Prove Inequality using Mean Value Theorem

    Homework Statement Essentially, the question asks to use the mean value theorem(mvt) to prove the inequality: abs(sina - sinb) \leq abs(a - b) for all a and b The Attempt at a Solution I do not have a graphing calculator nor can I use one for this problem, so I need to prove that the...
  2. R

    How Does the Mean Value Theorem Prove an Inequality Involving Tan^-1?

    Use the mean value theorem to show that (abs. value of tan^-1 a) < (abs. value a) for all a not equal to 0. And use this inequality to find all solutions of the equation tan^-1 x = x. I have no idea how to do this.
  3. W

    Geometric meaning of Mean Value Theorem

    I'd like the geometric meaning of the Mean Value Theorem. Say for instance I had a function of velocity that varied as t^{3} + 3t^{2} + 3t +1. I consider the interval [0,4]. So by MVT, I have a number c in [0,4] such that f'(c)(4) = f(4) - f(0). What does that mean? That there is an...
  4. W

    Mean value theorem and indefinate intgral

    Homework Statement http://img241.imageshack.us/img241/7753/scan0001io9.th.jpg Homework Equations The Attempt at a Solution i completed the first part fine- knowing the function makes a u shape with min point being 0 and max being 1/2 at both +/- 1. i can't see how using the...
  5. 1

    Solve Mean Value Theorem Problem on [1,4]

    Homework Statement Given the function f(x)= x(x^2-8)-5 satisfies the hypothesis of the Mean Value Thereom on the interval [1,4], find a number C in the interval (1,4) which satisfies this thereom. Homework Equations f'(c) = f(b)-f(a) / b-a The Attempt at a Solution 1) Expand...
  6. M

    Solve Sin((pi*x)/6) >= (x/2) with Mean Value Theorem

    Mean Value Theorem Please Help! Hi there, hopefully someone can help me I'm completely lost! I'm trying to solve sin((pi*x)/6) >= (x/2) for o<= x <= 1 using the mean value theorem. I think I need to show that f ' (c) >= 0, however this f ' (c) is negative for some values in the...
  7. F

    Mean Value Theorem to calculate solids of revolution?

    Mean Value Theorem to calculate solids of revolution? Ive been studying calculus on my own because my school doesn't offer it and i came across solids of revolution tonight. In one of the problems it says "What is the volume of the solid formed by rotating y=e^x across the x-axis between...
  8. M

    Creative Application of Mean Value Theorem

    Homework Statement Assume that f is twice differentiable on the entire real line. Show that f(-1) + f(1) - 2f(0) = f"(c) for some c in [-1,1] Homework Equations I'm thinking the mean value theorem will be helpful here -- the MVT states that, given a function f differentiable on [a,b]...
  9. A

    Mean Value Theorem exercise (Analysis)

    Homework Statement Let a>b>0 and let n \in \mathbb{N} satisfy n \geq 2. Prove that a^{1/n} - b^{1/n} < (a-b)^{1/n}. [Hint: Show that f(x):= x^{1/n}-(x-1)^{1/n} is decreasing for x\geq 1, and evaluate f at 1 and a/b.] Homework Equations I assume, since this exercise is at the end of...
  10. S

    Applying the Mean Value Theorem

    Hey, I know the basic definition of the MVT, but I'm having a lot of trouble applying it to this problem. I would greatly appreciate any kind of help or guidance. A graph of the derivative of f(x) is displayed below. Information about the function f(x) is known only for -2.5 < x < 3.5. Also...
  11. S

    Solve the Mean Value Theorem Problem with Graph

    Hey, I know the basic definition of the MVT, but I'm having a lot of trouble applying it to this problem. I would greatly appreciate any kind of help or guidance. A graph of the derivative of f(x) is displayed below. Information about the function f(x) is known only for -2.5 < x < 3.5. Also...
  12. 2

    Mean Value Theorem and Rolle's Theorem: Conditions and Examples

    Homework Statement a. If f is defined on an interval [x,y], its differentiable on open interval (x,y), and f(x)=f(y) then there is a number c in (x,y) where f'(c)=0 b. Does the absolute value of x, |x|, satisfy Rolle's Theorem on [-1, 1]? The attempt at a solution For the first one...
  13. T

    Mean value theorem for integrals

    Hi, I have this question which asks for a 2*pi periodic function on the reals, that is integrable on [-pi, pi] but fails to satisfy the mean value theorem for integrals. The question also says to help answer the above question, you may wish to show that the function: g: [0,1] -> reals given...
  14. K

    Mean Value Theorem Applications

    Q: Prove that if f: R^n -> R is defined by f(x)=arctan(||x||), then |f(x)-f(y)| <= ||x-y|| FOR ALL x,y E R^n. [<= means less than or equal to] Theorem: (a corollary to the mean value theorem) Suppose f is differentiable on an open, convex set S and ||gradient [f(x)]|| <= M for all x E S...
  15. A

    Solving for Real Roots of a Polynomial Equation: Using the Mean Value Theorem

    [SOLVED] Mean value theorem First I just want to say that my professor hasn't gotten up to teaching us this so I may be a little slow in understanding this material and want to thank you for being patient with me. The question asks to show that the equation X^4 -4X + c = 0 has at most two...
  16. dontdisturbmycircles

    Is the Mean Value Theorem Applicable to Prove sin x < x for x > 0?

    Homework Statement Show that sin x < x for all x > 0 The Attempt at a Solution I thought I was pretty good at calculus so I have kinda been shifting my calc class onto the bottom of my todo list, but this mean value theorem problem is giving me some problems. For x > 1, sin x \leq 1 <...
  17. P

    What is the Generalised Mean Value Theorem?

    Does anyone know the mean value theorem associated with the Taylor series. Representing the Taylor series a finite sum and an end term? I don't get how they get it to look that way.
  18. R

    Verify Mean Value Theorem: f(x)=3x^2+6x+7 on [-2,3]

    Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers that satisfy the conclusion of the Mean Value Theorem. f(x)=3x^2+6x+7 on [-2,3] so f(3)-f(-2)=f'(x)(3+2) i get 52-7=(6x+6)(5) then 45=30x+30 then 5=30x...
  19. K

    How Does the Mean Value Theorem Apply to Square Root Functions?

    1) By applying the Mean Value Theorem to f(x)=sqrt x, show that 1/11 < (sqrt 102) -10 < 1/10. This is a sample problem in my course, and how to start this problem is the key thing. The solution says "f is continuous and differentiable for all x>0, so by the mean value theorem, there exists c...
  20. S

    Proving One-to-One Function & Solutions with Mean Value Theorem

    Use the Mean Value Theorem to show that: a)Suppose f is a diferentiable function on the interval a < b, and suppose f '(x) is not equal to 0 for all x Element Symbol (a,b). Show that f is one-to-one on the interval (a,b). b) Assume that |f ' (x)| < C < 1 for all x. Show that f (x) = x has at...
  21. S

    Using the Mean Value Theorem: Showing One-to-One Behavior

    Homework Statement Use the Mean Value Theorem to show that: a)Suppose f is a diferentiable function on the interval a < b, and suppose f '(x) is not equal to 0 for all x Element Symbol (a,b). Show that f is one-to-one on the interval (a,b). b) Assume that |f ' (x)| < or equal to C < 1...
  22. H

    Can we find at most two real roots for the equation x^4 + 4x + c = 0?

    This theorem is confusing me even though it is sittin right in front of me.. I am given an equation x^4 + 4x + c = 0 and asked to find at most two real roots?? I know we need to take the derivative, but from there I am lost.
  23. C

    Does G(x) = x^2 - e^{1/(1+x)} Assume a Value of 0 for 0 < x < 2?

    Use the Intermediate Value Theorem and/or the Mean Value Theorem and/or properties of G'(x) to show that the function G(x) = x^2 - e^{\frac{1}{1+x}} assumes a value of 0 for exactly one real number x such that 0 < x < 2 . Hint: You may assume that e^{\frac{1}{3}} < 2 . So I'm completely lost...
  24. O

    Proof of the mean value theorem

    the step i don't understand about the proof in this page http://en.wikipedia.org/wiki/Mean_value_theorem is the last one, where they assume that the x at which f'(x) is equal to the secant slope is the same x at which g'(x) = 0. can someone explain to me why this is so in a formal way?
  25. B

    Mean value theorem for integrals.

    find c such that (f)average=f(c) f(x)=7sin(x)-sin(2x), [0,pi] ok so i found (f)average= 14/pi then i tried to compute: 7sin(c)-sin(2c)= (14/pi) but the answers i got were wrong. please if someone could reply with some helpful information on how to solve this. thank you
  26. K

    Mean Value Theorem for Integrals: Does it Hold?

    Let be F(x)=\int dx f(x) my question is if F(x) is continuous (but not differentiable ) does the Mean-value theorem for integrals hold?.. a<c<b and f(c)(b-a)=\int_{a}^{b} dx f(x) ?... :rolleyes: :confused:
  27. C

    Does the Mean Value Theorem Guarantee a Function Value of 4?

    If f is continuous and \int^{3}_{1} f(x) dx = 8 , show that f takes on the value 4 at least once on the interval [1,3] . I know that the average value of f(x) is 4. So does this imply that f_{ave} = f(c) = 4 and f(x) takes on the value of 4 at least once on the interval [1,3] ?
  28. X

    Does the Converse of the Mean Value Theorem Hold?

    hi... was wondering, does the converse of the mean value theorem hold? that is, given any function f(x), and a tangent to the graph of y = f(x) at any point, can we always construct two points on the graph (with the tangent lying between) such that the line joining them is parallel to the...
  29. N

    Mean value theorem for integration

    The problem states: Find all values of c such that \sqrt(1+\sqrt(x)) satisfies the statement of the mean value theorem for integration on the interval [0. 1]. Also express the result in exact form completely simplified. I am a little confused. I'm just finding the definite inegral? I...
  30. E

    Prove Inequality with Mean Value Theorem: |\sin a - \sin b| \leq |a - b|

    Use the Mean Value Theorem to prove the inequality |\sin a - \sin b| \leq |a - b| for all a and b. I know by the Mean Value Theorem, I can say: \sin a - \sin b = \cos c(a - b) I've been trying to figure it out for awhile, but could not, so I peeked at my solution's manual. They assumed b < a...
  31. J

    Proving roots using mean value theorem

    Prove x^4 + 4x + c = 0 has at most two real roots My thinking is that to prove this I would assume that it has three real roots and look for a contradiction. So I set f(x) = x^4 + 4x + c and assume three real roots x_1, x_2, x_3 such that f(x_1) = f(x_2) = f(x_3) = 0 By MVT I...
  32. M

    Mean Value Theorem for Integrals

    Can someone please explain to me how to use this and give an example that we can walk through please? My book doesn't give an example of what it is talking about. ~Kitty
  33. P

    Mean Value Theorem of Electrostatics

    In "Classical Electrodynamics - 3rd Ed.," J.D. Jackson has an exercise, 1.10, to derive the mean value theorem of electrostatics. Does anyone know of a derivation which is located on the web? Pete
  34. P

    Little confussed (Mean Value Theorem)

    Hello All I am a bit confussed with this question I have. Show that the equation 2x - 1 - sin x = 0 has exactly one root. So this apears in the Mean Value Theorem section of my book. If some one can help it would be great. I believe I need to use the Intermediate Value Theorem to...
  35. B

    Gradient and mean value theorem

    Hi please , can someone help me with this problem. I need to know the procedure. Let f(x,y)=x^3-xy. set a(0,1) and b(1,3). Find a point c on the line sement[ab] for which f(b)-f[a]= gradient(f[c]) * (b-a) Thank you B.
  36. K

    Mean Value Theorem: Proving Inequalities with f'(x)

    Please help me on this. If f`(x) is continuous on [a,b],apply the Mean Value Theorem to prove the inequalities min[f`] \leq f(b)-f(a)/_ b-a \leq [Max f`]
  37. S

    Mean Value Theorem for Nonlinear Equations in R^n

    Can someone help me... i need to show, that a system of 2 nonlinear equations has a root. I think it is possible to use something like "mean value theorem". But i can not find any mean value theorem for R^n -> R^n.
  38. W

    Need help with a Mean Value Theorem Problem

    Hi all, I've been thinking on this a lot but couldn't come up with an answer so I need your help. I've seen this in Thomas' Calculus 10th edition. Anyway here goes the problem A marathoner ran the 26.2 mi New York City Marathon in 2.2 h. Show that at least twice the marathoner was running at...
  39. J

    Differential calculus question (Mean value theorem)

    You are given the following information about the function f(x): i) There is an x-value x* approximately equal to 0.8 such that f(x*)=0 ii) f(0.7) = C is negative iii) m1 < f'(x) < m2 for 0.7 < x < 0.9 where m1 and m2 are positive constants Apple the Mean Value Theorem to f(x) on the...
  40. S

    What is the intuitive meaning of the Mean Value Theorem?

    so by the defn: suppose that f is continuous on a closed I:= [a,b] and that f has a derivative in the open interval (a,b). then tehre exists at least one point c in (a,b) st f(b) - f(a) = f'(c)(b - a). ok, so what if I put this in terms of f'(c)? isn't that the definition of the...
  41. V

    Finding Values for Mean Value Theorem in Integrals

    I'm having a some difficulty with this problem: Fidn the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the indicated interval: f(x)=x - 2(square root of x) Interval: [0,2] I found the integral to be: x^2/2 - 4/3 x^3/2, then I solved for the...
  42. C

    Understanding the Mean Value Theorem in Calculus

    Why is it that if you have \frac{f(x_1) - f(x_2)}{x_1-x_2} = f'(\xi) then \xi = x_1 + \theta(x_2-x_1) where 0<\theta<1 ? Thanks
  43. C

    How Can the Mean Value Theorem Prove the Derivative of an Indefinite Integral?

    Hello all Using the Mean Value Theorem, prove that the derivative of the indefinite integral \int f(x) \ dx is f(x) So do I just use the fact that \int^b_a f(x) \ dx = f(\xi)(b-a) ? Thanks
  44. E

    Prove Mean Value Theorem: Min/Max on Bounded, Closed Set

    Prove: If f : U → R is continuous on U, and E ⊂ U is closed and bounded, then f attains an absolute minimum and maximum on E. I have no idea even where to start on this. Intuitively it's so obvious that i don't know what to do. The definitions given by the teachers that I have to work with...
  45. T

    Proving with mean value theorem

    Suppose that g(a) = g'(a)=0 and |g''(x)| < M for all x in [a, a+h] (for some positive constant M). Show that |g(a+h)| < Mh^2. (Hint: Let k be any number such that 0<= k <= h and apply Mean Value Theorem to g' on [a,a+k].)
  46. B

    Calulus Help: Mean Value Theorem

    Calculus Help: Mean Value Theorem Let f(x) = x^3 - x on the interval [2,3]. Find all numbers c in the interval (2,3) that satisfy the conclusion of the mean-value theorem. Here's what I did: f '(c) = f(b) - f(a) / (b - a) f '(c) = f(3^3 - 3) - f(2^3 - 2) / (3-2) = 18, but it's...
  47. R

    Mean Value Theorem - f(x) = f(a)+(x-a)f'(u)

    Hi, If f is continuous in [a,b] and differentiable in (a,b), and if xE(a,b), then there exists u in (a,b) such that f(x) = f(a)+(x-a)f'(u) What's wrong if i state : (i)If f is continuous in [a,b] and differentiable in [a,b], and if xE[a,b] or...
  48. H

    Mean Value Theorem and electrostatic potential

    Prove that for charge-free two-dimensional space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any circle centered on that point. Do this by considering the electrostatic potential as the real part of an analytic function...
  49. S

    Apply Mean Value Theorem to Show arctan x - x = 0 at x = 0

    oks...im having problems applying the mean value theorem...i understand the concept behind it but whenever i try 2 do a question i have no idea wat 2 do...heres 1 question I've been trying: Showing all your work, apply the Mean Value Theorem to show that the function arctan x - x is equal to...
  50. I

    Cauchy Mean Value Theorem Proof for Continuous and Integrable Functions

    Hi, I really need some help in sovling this proof! Prove the Cauchy Mean Value Theorem: If f,g : [a,b]->R satisfy f continuous, g integrable and g(x)>=0 for all x then there exists element c is a member of set [a,b] so that int(x=b,a)f(x)g(x)dx=f(c)int(x=b,a)g(x)dx. Thanks for your help :D
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