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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...