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 3 years ago
Give the number of values of c that verify the mean value theorem for f(x) = sin(x) on the
interval [1,5].
 3 years ago
Give the number of values of c that verify the mean value theorem for f(x) = sin(x) on the interval [1,5].

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David_Ho
 3 years ago
Best ResponseYou've already chosen the best response.0Hiya! So far I have cos(x) = (sin(5)  sin(1))/4 I really have no clue where to go with that.

elecengineer
 3 years ago
Best ResponseYou've already chosen the best response.0Find a value for the RHS

elecengineer
 3 years ago
Best ResponseYou've already chosen the best response.0then you can figure out how many solns for x are in the interval

JamesJ
 3 years ago
Best ResponseYou've already chosen the best response.3What is C here? (And btw, the derivative isn't a feature of the MVT.)

JamesJ
 3 years ago
Best ResponseYou've already chosen the best response.3I think I see. Look at f(1) and f(5). Then the MVP says for every y such that f(1) < y < f(5) there is a c such that f(c) = y. Is that what you're asking about?

David_Ho
 3 years ago
Best ResponseYou've already chosen the best response.0Well, what I've recently learned is that MVT is (f(a)  f(b))/b a = f' (c)

David_Ho
 3 years ago
Best ResponseYou've already chosen the best response.0I had this problem earlier where I had to find the c value.

JamesJ
 3 years ago
Best ResponseYou've already chosen the best response.3Ok, that's another MVT, sometimes called Rolle's theorem. But be that as it may, we'll use the one you're talking about. Here a = 1, b = 5. What is the LHS that expression?

elecengineer
 3 years ago
Best ResponseYou've already chosen the best response.0Rolle's theorm is a specific case of MTV

David_Ho
 3 years ago
Best ResponseYou've already chosen the best response.0oh ok, it would be the derivative of f(x) or cos(x)

David_Ho
 3 years ago
Best ResponseYou've already chosen the best response.0cos(x) = (sin(5)  sin(1))/4

JamesJ
 3 years ago
Best ResponseYou've already chosen the best response.3Right. so solutions to this equation are teh values c you're looking for. Remember also that c must lie between 1 and 5 So how many such solutions are there?

JamesJ
 3 years ago
Best ResponseYou've already chosen the best response.3Actually no  check your equation carefully . It is NOT divided by 4.

David_Ho
 3 years ago
Best ResponseYou've already chosen the best response.0There would only be one value, right? Since there are only constants on the right side the value of c can only be one value. Is this logic sound?

JamesJ
 3 years ago
Best ResponseYou've already chosen the best response.3No, it is not sound, because the function cos x isn't 1:1; it has periodicity; it goes up and down with period 2pi, which is almost exactly equal to the length of the interval [1.5]. So there MIGHT be one c, but there could easily be two values of c. So you are going to have to calculate explicitly and check.

David_Ho
 3 years ago
Best ResponseYou've already chosen the best response.0oohh, alright. I see what you mean now. I have to find all of the points on cos x that have a tangent line = to (sin(5)sin(1))/6.

JamesJ
 3 years ago
Best ResponseYou've already chosen the best response.3No you need to find x such that cos x = (sin(5)sin(1))/6

JamesJ
 3 years ago
Best ResponseYou've already chosen the best response.3cos x is the formula for the slope of the tangents to sin x. That's what we're looking for.
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