calc u5

Rolle’s theorem

If f(a)=f(b) then there must be at least one number c in (a,b) such that f’(c)=0

Continuous interval

Closed []

Diff interval

Open interval ()

Steps for Rolle’s theorem

1. State if function is continuous over closed interval

2. State is function is differentiable over open interval

3. Show if f(a)=(fb)

4. State that the theorem applies

5. Take the derivative of ur function, set to zero, and solve for x

6. Make sure ur x is between (a,b)

7. The x value(s) you get are now guaranteed by the theorem

Mean value theorem steps

1. Determine if f(x) is continuous over closed interval

2. Determine if f(x) is differentiable over open interval

3. Differentiate f(x)

4. Find slope of intervals given (y2-y1/x2-x1)

5. Set f’(x) equal to the slope of the intervals you found and solve for x

6. Make sure your x(s) lie within the interval given

Mean value theorem

If f is continuous over the closed interval [a,b] and differentiable over the open interval (a,b), then there exists a number c in (a,b) such that f’(c) =( f(b)-f(a) ) / (b-a)

Basically it’s saying derivative = slope

Extreme value theorem

If f(x) is continuous over a closed interval, then f(x) has both a absolute maximum and a minimum on the interval

When is there Critical numbers

If f(x) is defined at c, and f’(c)=0 or f(x) is not differentiable at c (aka f’(c)=DNE)

Relative extrema

Occur at horizontal tangent lines and only occur at critical numbers

Relative minimum f(x)

f’(x) changes from - to +

Relative maximum of f(x)

f’x) changes from + to -

differentiable

continuous

continuous

not always differentiable

Continuity breaks when there is

• Hole (removable discontinuity)

• Jump discontinuity

• Vertical asymptote

• Any break in the graph

A function is not differentiable when

• The function is not continuous

• Corner or cusp

• Vertical tangent line

• Discontinuity (hole, jump, asymptote)

• Fast Oscillation / wiggle / “infinite bouncing”

NOT continuous

NOT differentiable