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Rolle's Theorem
If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists a point c∈(a,b), where f '(c) = 0.
The Mean Value Theorem
Let f be a function that satisfies the following hypotheses:
Theorem
the surface area of a sphere is 4pi times the sqare of the radius: S=4piRsquared
Corollary
The acute angles of a right triangle are complementary.
First hypothesis of Rolle's Theorem
f is continuous on closed interval [a, b]
Second hypothesis of Rolle's Theorem
f is differentiable on open interval (a, b)
f(a)= f(b)
Third hypothesis of Rolle's Theorem
Conclusion of Rolle's Theorem
Then there exists a number c in (a, b) such that f'(c)=0
What to check after solving using Rolle's Theorem
Check that x is in the open interval and not an endpoint
First hypothesis of Mean Value Theorem
f is continuous on closed interval [a, b]
Second hypothesis of Mean Value Theorem
f is differentiable on open interval (a, b)
Conclusion of Mean Value Theorem
Then there exists a number c in (a,b) such that f'(c)= (f(b)-f(a))/(b-a) (Avg. value in the interval)
the conclusion is known to be true
If the conditions (hypotheses) of a theorem are satisfied,_
It is guarantee to be true
The conclusion is known to be true or_
constant on (a,b)
If f'(x)=0 for all x in an interval (a,b), then f is_
horizontal line
r = k csc θ (or rsinθ = k or y = k or r = k/sinθ)
differ only by a constant
If the derivative of the functions are the same, the difference of the functions will_
Then f is constant on (a,b)
Theorem: If f'(x)=0 for all x in an interval (a,b), then...
Then f - g is constant on (a,b); that is,
Corollary: If f'(x) =g'(x) for all x in an interval (a,b), then...