After Mission Plutonius fails, stranding a crew of astronauts on a far away planet, the astronauts must find a way to make it back to Earth
How are we going to get back to Earth?
To get back it is imperative we find the critical points in our trajectory to return home safely
To find the critical points either Rolle's or Mean Value Theorem can be applied depending on the circumstances
Well how are we going to do that?
3.2: Rolle's Theorem and the Mean Value Theorem
To apply either of theses theorems you must know that f(x) is continuous on the closed interval [a,b] and that f(x) is differentiable on the open interval (a,b)
First let's apply Rolle's Theorem
Let f(x)=-x2+2. Find all values of c in the interval (-4,4) such that f'(c)=0.f(x) is continuous over [-4,4]f(x) is differentiable over (-4,4)
If the function is continuous and differentiable over the interval and f(a)≠f(b) then the Mean Value Theorem can be applied
What if f(a) does not equal f(b)
Now let's apply the Mean Value Theorem
Determine whether the Mean Value Theorem can be applied to f on the closed interval [a,b]. If the Mean Value Theorem can be applied, find all values of c in the interval (a,b) such that f'(c)=(f(b)-f(a))/(b-a)f(x)=x3+6f(x) is continuous over [1,3]f(x) is differentiable over (1,3)
(f(b)-f(a))/(b-a)=(f(3)-f(1))/(3-1)=(33-7)/2=13
f(c)=c3+6f'(c)=3c213=3c2√(13/3) or -√(13/3)=c
Now that the trajectory back to earth has been calculated by using Rolle's and the Mean Value Theorem we are on course to make it back to Earth safely
After years on Plutonius thecrew of astronauts finally made it back to Earth
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