Mean value theorem review (article) | Khan Academy (2024)

Review your knowledge of the mean value theorem and use it to solve problems.

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  • meghna rana

    7 years agoPosted 7 years ago. Direct link to meghna rana's post “How is mean value theorem...”

    How is mean value theorem applied in real life?

    (12 votes)

    • jude4A

      7 years agoPosted 7 years ago. Direct link to jude4A's post “Mean value theorem can be...”

      Mean value theorem review (article) | Khan Academy (4)

      Mean value theorem review (article) | Khan Academy (5)

      Mean value theorem can be used to determine the speed of something, like a policeman using a speedometer.

      (33 votes)

  • Lori Feng

    8 years agoPosted 8 years ago. Direct link to Lori Feng's post “In the question, the inte...”

    In the question, the interval is a closed one, but in the explanation, the interval is open, thus excluding 3 as an answer. Why does the explanation have an open interval?

    (7 votes)

  • Urikhan

    9 months agoPosted 9 months ago. Direct link to Urikhan's post “Why c would be only in op...”

    Why c would be only in open interval? Because a and b are differentiable over open interval?

    (1 vote)

    • Venkata

      9 months agoPosted 9 months ago. Direct link to Venkata's post “Correct. We do need to fi...”

      Correct. We do need to find f'(c) after all. So, c needs to be a point where the derivative exists (i.e. a differentiable point) and the interval which is differentiable is (a,b), not [a,b]

      (3 votes)

  • Avin

    3 months agoPosted 3 months ago. Direct link to Avin's post “I'm a bit confused on the...”

    I'm a bit confused on the graph given by the initial explanation; the MVT says that the function must be continuous on the endpoints as well, though on the graph, the right sided limit does not exist for x=6, making it discontinuous over the interval [0,6]. Is that correct or am I confusing the concept?

    (1 vote)

    • WaGyu

      3 months agoPosted 3 months ago. Direct link to WaGyu's post “Yeah, you have some confu...”

      Yeah, you have some confusion about continuity.
      A discontinuous point means that either you have no defined value at that point or it jumps to a different point. The graph you're talking doesn't show a jump at the point nor an undefined point (which is typically shown as an empty circle).
      MVT doesn't care about differentiability at the endpoints as it states that it "must be differentiable over the open interval (a,b) and continuous over the closed interval [a,b]." Hope it helped.

      (2 votes)

  • mdabney

    3 years agoPosted 3 years ago. Direct link to mdabney's post “The question in practice ...”

    The question in practice problem 1 specifically asks for a value for c in the closed interval [0,3], but marks one of the correct solutions as incorrect. I understand why it wouldn't want to accept 3 as a correct answer as it's the answer to one of the intermediate steps toward the solution it's looking for, but it is a correct answer to the question as it's currently posed. The question wording should be changed to ask for c in the open interval (0,3) to prevent that mistake. As Lori Feng mentions below, it's even described to be incorrect in the explanation because it assumes the question is asking for a value in the open interval rather than the closed interval listed in the question.

    (1 vote)

    • loumast17

      3 years agoPosted 3 years ago. Direct link to loumast17's post “in the definition of the ...”

      in the definition of the mean value theorem given at the top it says that the interval [a,b] is for the function, then the interval (a,b) is for the derivative, which is where c would be.

      It is kinda crummy it's like that, but technically yeah, c would only be in the open interval

      (2 votes)

  • Ray2017

    6 months agoPosted 6 months ago. Direct link to Ray2017's post “Why do we exclude a and b...”

    Why do we exclude a and b when we want to find the possible values for c?

    (1 vote)

    • pufferfish56

      4 days agoPosted 4 days ago. Direct link to pufferfish56's post “MVT only cares if the fun...”

      MVT only cares if the function is continuous at the endpoints, not any range outside of that (and thus doesn't care if the endpoints are differentiable), so if MVT holds true, there is at least one solution that is not one of the endpoints between a and b. That doesn't mean that the endpoints couldn't have a parallel secant line and tangent line, but it means it's not part of the range where it's guaranteed.

      (1 vote)

  • zubair

    5 years agoPosted 5 years ago. Direct link to zubair's post “if this man stop for some...”

    if this man stop for some minutes and continue his journey then?

    (1 vote)

  • jagetiyaareen5

    4 years agoPosted 4 years ago. Direct link to jagetiyaareen5's post “Sir what about roll's and...”

    Sir what about roll's and lagrance's theorem
    and thank u sir

    (1 vote)

  • No_Solution

    3 months agoPosted 3 months ago. Direct link to No_Solution's post “What would be an example ...”

    What would be an example where, if f is differentiable between a CLOSED interval of [a, b] instead of an OPEN interval (a,b), the MVT isn't valid?

    (1 vote)

    • kubleeka

      3 months agoPosted 3 months ago. Direct link to kubleeka's post “(a, b) is a subset of [a,...”

      (a, b) is a subset of [a, b], so if f is differentiable on [a, b], then it is automatically differentiable on (a, b) as well.

      We use an open interval for that part of the statement because we generally like to use the weakest assumptions possible, so that we can apply the theorem in as many cases as possible.

      If I have a function that equals sin(x) for 0<x<π and 0 otherwise, then we can apply the mean value theorem on the interval [0, π], even though our function isn't differentiable at the endpoints. If we insisted on using the closed interval in the statement of the theorem, then we couldn't apply the theorem here, and for no good reason.

      (1 vote)

  • Maha Rahman

    6 years agoPosted 6 years ago. Direct link to Maha Rahman's post “If a=b, (the two endpoint...”

    If a=b, (the two endpoints have the same y value) can mean value theorem be applied?

    (0 votes)

    • kubleeka

      6 years agoPosted 6 years ago. Direct link to kubleeka's post “Yes, it would imply that ...”

      Yes, it would imply that the function has a point in the interval where the derivative is 0. This special case is called Rolle's Theorem.

      (2 votes)

Mean value theorem review (article) | Khan Academy (2024)

FAQs

What is the answer to the mean value theorem? ›

The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].

What is MVT used for? ›

What is the mean value theorem? The mean value theorem connects the average rate of change of a function to its derivative. It says that for any differentiable function ‍ and an interval ‍ (within the domain of ‍ ), there exists a number ‍ within ‍ such that ‍ is equal to the function's average rate of change over ‍ .

Can a function be differentiable on a closed interval? ›

So the answer is yes: You can define the derivative in a way, such that f′ is also defined for the end points of a closed interval. Note that for some theorem like the mean value theorem you only need continuity at the end points of the interval.

Why do you need differentiability to apply the mean value theorem? ›

The mean value theorem guarantees, for a function ‍ that's differentiable over an interval from ‍ to ‍ , that there exists a number ‍ on that interval such that ‍ is equal to the function's average rate of change over the interval.

How to prove MVT? ›

Mean Value Theorem Proof

h(a) = h(b) = 0 and h(x) is continuous on [a, b] and differentiable on (a, b). Thus applying the Rolles theorem, there is some x = c in (a, b) such that h'(c) = 0. Thus the mean value theorem is proved.

What is the conclusion of the mean value theorem? ›

The conclusion is that there exists a point in the interval such that the tangent at the point c , f c is parallel to the line that passes through the points a , f a and b , f b .

Why is the mean value theorem so important? ›

The main use of the mean value theorem is in justifying statements that many people wrongly take to be too obvious to need justification. One example of such a statement is the following. (*) If the derivative of a function f is everywhere strictly positive, then f is a strictly increasing function.

What is the mean value theorem in real life? ›

In a real-world application, the Mean Value Theorem says that if you drive 40 miles in one hour, then at some point within that hour, your speed will be exactly 40 miles per hour.

How to determine if MVT applies? ›

We can see there are two conditions on our function 𝑓 to use the mean value theorem. First, our function 𝑓 must be continuous on the closed interval from 𝑎 to 𝑏. Second, 𝑓 must be differentiable on the open interval from 𝑎 to 𝑏.

What is the hypothesis of the Mean Value Theorem? ›

The hypothesis of the Mean Value Theorem requires that the function be continuous on some closed interval [a, b] and differentiable on the open interval (a, b). Hence MVT is satisfied.

What is the geometrical interpretation of the Mean Value Theorem? ›

Geometrically, Lagrange's Mean Value Theorem states that If the function is continuous and smooth in some interval then there must be a point (which is mention as c in the theorem) at which the slope of the tangent of the function will be equal to the slope of the secant through its endpoints.

What is the difference between Rolle's theorem and MVT? ›

In layman's terms, the Mean Value Theorem states that a continuous, differentiable function on an interval has a point where the slope is equal to the average slope over the interval. Rolle's theorem simply makes the average slope zero which means the function has a point of zero slope within an interval.

When can you not use the Mean Value Theorem? ›

The Mean Value Theorem does not work if our function is not continuous or differentiable over our interval.

Does MVT require continuity? ›

The MVT is a consequence of Rolle's Theorem. you need continuity at [a,b] to be sure that the function is bounded.

What are the rules of MVT? ›

The Mean Value Theorem states that if f is continuous over the closed interval [a,b] and differentiable over the open interval (a,b), then there exists a point c∈(a,b) such that the tangent line to the graph of f at c is parallel to the secant line connecting (a,f(a)) and (b,f(b)).

What is the result of the Mean Value Theorem? ›

Mean Value Theorem states that for any function f(x) passing through two given points [a, f(a)], [b, f(b)], there exists at least one point [c, f(c)] on the curve such that the tangent through that point is parallel to the secant passing through the other two points.

What is the Mean Value Theorem expression? ›

f'(c) = [f(b) – f(a)]/(b-a) This theorem is also known as the first mean value theorem or Lagrange's mean value theorem.

What does the extreme value theorem say? ›

The Extreme value theorem states that if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interval.

What is the Mean Value Theorem argument? ›

In mathematics, the mean value theorem (or Lagrange 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.

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