Rolle's theorem is just the mean value theorem, where f(x) = 0 at both endpoints.
What did you get to parts a and b?
Verify the hypothesis of the mean value theorem for each function below defined on the indicated interval. Then find the value “C” referred to by the theorem.
Q1a) h(x)=√(x+1 ) [3,8]
Q1b) K(x)=(x-1)/(x=1) [0,4]
Q1c) Explain the difference between the Mean Value Theorem and Rollo’s Theorem.
5 answers
I get stuck at this and don't know how to from here.... can you help...
Q1a) h(x)=√(x+1 ) [3,8]
MVT=[h(b)-h(a)]/(b-a)=h'(c)
To find h(b) and h(a), we just plug endpoints into original function
h(b)=h(8)=√(x+1 )
h(b)=h(8)=√(8+1 ) = 3
h(a)=h(3)= √(3+1 ) = 2
MVT=[3-2]/[8-3] =f^' (c)
MVT=1/2=f^' (c)
Next, we find the derivative for h(x)
h'(x)=√(x+1 )
h'(x)=(d/dx(x+1))/(2√(x+1))
h'(x)=(1+0)/(2√(x+1))
h'(x)=1/(2√(x+1))
h'(c)=h'(x)
1/2=(1+0)/(2√(x+1))
Q1a) h(x)=√(x+1 ) [3,8]
MVT=[h(b)-h(a)]/(b-a)=h'(c)
To find h(b) and h(a), we just plug endpoints into original function
h(b)=h(8)=√(x+1 )
h(b)=h(8)=√(8+1 ) = 3
h(a)=h(3)= √(3+1 ) = 2
MVT=[3-2]/[8-3] =f^' (c)
MVT=1/2=f^' (c)
Next, we find the derivative for h(x)
h'(x)=√(x+1 )
h'(x)=(d/dx(x+1))/(2√(x+1))
h'(x)=(1+0)/(2√(x+1))
h'(x)=1/(2√(x+1))
h'(c)=h'(x)
1/2=(1+0)/(2√(x+1))
if h(x) = √(x+1)
h'(x) = 1/(2√(x+1))
so, we want c where
h'(c) = (3-2)/(8-3) = 1/5
1/2√(x+1) = 1/5
5 = 2√(x+1)
√(x+1) = 5/2
x+1 = 25/4
x = 21/4
and 3 < 21/4 < 8
-----------------------------
if k(x) = (x-1)/(x+1)
k'(x) = 2/(x+1)^2
k(0) = 0
k(4) = 3/5
so, we want c where k'(c) = (3/5)/4 = 3/20
3/20 = 2/(x+1)^2
3(x+1)^2 = 40
(x+1)^2 = 40/3
x = -1 + 2√(10/3) = 2.65
0 < 2.65 < 4, so we're ok.
h'(x) = 1/(2√(x+1))
so, we want c where
h'(c) = (3-2)/(8-3) = 1/5
1/2√(x+1) = 1/5
5 = 2√(x+1)
√(x+1) = 5/2
x+1 = 25/4
x = 21/4
and 3 < 21/4 < 8
-----------------------------
if k(x) = (x-1)/(x+1)
k'(x) = 2/(x+1)^2
k(0) = 0
k(4) = 3/5
so, we want c where k'(c) = (3/5)/4 = 3/20
3/20 = 2/(x+1)^2
3(x+1)^2 = 40
(x+1)^2 = 40/3
x = -1 + 2√(10/3) = 2.65
0 < 2.65 < 4, so we're ok.
actually, I think Rolle's Theorem is the MVT where f(a) = f(b), so that f'(c) = 0.
oops. k(0) = -1
adjust the calculation accordingly.
adjust the calculation accordingly.