Calc
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- 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)
- f(-4)=-(-4)2+2=-14f(4)=-(4)2+2=-14f(-4)=f(4) f(a)=f(b)
- f(c)=-c2+2f'(c)=-2c0=-2cc=0
- 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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