David_Ho
Give the number of values of c that verify the mean value theorem for f(x) = sin(x) on the
interval [-1,5].
Delete
Share
This Question is Closed
naf
Best Response
You've already chosen the best response.
1
|dw:1317571240881:dw|
David_Ho
Best Response
You'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.
naf
Best Response
You've already chosen the best response.
1
|dw:1317571301234:dw|
elecengineer
Best Response
You've already chosen the best response.
0
Find a value for the RHS
elecengineer
Best Response
You've already chosen the best response.
0
then you can figure out how many solns for x are in the interval
JamesJ
Best Response
You've already chosen the best response.
3
What is C here? (And btw, the derivative isn't a feature of the MVT.)
naf
Best Response
You've already chosen the best response.
1
|dw:1317571400618:dw|
naf
Best Response
You've already chosen the best response.
1
|dw:1317571475430:dw|
JamesJ
Best Response
You'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?
elecengineer
Best Response
You've already chosen the best response.
0
c is a value of x
David_Ho
Best Response
You'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)
David_Ho
Best Response
You've already chosen the best response.
0
I had this problem earlier where I had to find the c value.
JamesJ
Best Response
You'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?
David_Ho
Best Response
You've already chosen the best response.
0
What is LHS? sorry
elecengineer
Best Response
You've already chosen the best response.
0
Rolle's theorm is a specific case of MTV
David_Ho
Best Response
You've already chosen the best response.
0
oh ok, it would be the derivative of f(x) or cos(x)
JamesJ
Best Response
You've already chosen the best response.
3
the left hand side LHS
David_Ho
Best Response
You've already chosen the best response.
0
cos(x) = (sin(5) - sin(-1))/4
David_Ho
Best Response
You've already chosen the best response.
0
replace x for c
JamesJ
Best Response
You'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?
JamesJ
Best Response
You've already chosen the best response.
3
Actually no -- check your equation carefully . It is NOT divided by 4.
naf
Best Response
You've already chosen the best response.
1
|dw:1317572230820:dw|
David_Ho
Best Response
You've already chosen the best response.
0
ohh, divide by 6.
David_Ho
Best Response
You'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?
JamesJ
Best Response
You'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.
David_Ho
Best Response
You'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.
JamesJ
Best Response
You've already chosen the best response.
3
No you need to find x such that
cos x = (sin(5)-sin(-1))/6
JamesJ
Best Response
You'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.