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Give the number of values of c that verify the mean value theorem for f(x) = sin(x) on the
interval [1,5].
 2 years ago
 2 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].
 2 years ago
 2 years ago

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

elecengineerBest ResponseYou've already chosen the best response.0
Find a value for the RHS
 2 years ago

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

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

JamesJBest ResponseYou've already chosen the best response.3
I 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?
 2 years ago

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

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

JamesJBest ResponseYou've already chosen the best response.3
Ok, 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?
 2 years ago

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

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

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

JamesJBest ResponseYou've already chosen the best response.3
Right. 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?
 2 years ago

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

David_HoBest ResponseYou've already chosen the best response.0
There 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?
 2 years ago

JamesJBest ResponseYou've already chosen the best response.3
No, 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.
 2 years ago

David_HoBest ResponseYou've already chosen the best response.0
oohh, 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.
 2 years ago

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

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