AP Calc Unit 5: Analytical Applications of Differentiation

Mean Value Theorem

If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists a value c, a<c<b, such that f’(c)= f(b)-f(a)/b-a

• Basically saying that if a function is continuous and differentiable, then there has to be a point where the derivative (IROC) equals the AROC

Extreme Value Theorem

If f(x) is continuous on [a,b], then f(x) must obtain a maximum and minimum on value on [a,b].

• Basically saying that if a function is continuous, then there has to be a maximum and minimum on that interval.

What does the first derivative tell us about f(x)?

• f’(x)>0, f(x) is increasing

• f’(x)<0, f(x) is decreasing

• f’(x) inc, f(x) concave up

• f’(x) dec, f(x) concave down

First Derivative Test

• Allows us to find local/relative extrema

  1. Find critical points

  2. Make a sign chart w/ those critical points

  3. Plug them into f’ to see where they are pos/neg

• Positive to Negative: Relative MAX

• Negative to Positive: Relative MIN

Candidates Test

• Allows us to find absolute extrema

  1. Find critical points

  2. Make a table with critical points & end points

  3. Plug into original function f(x)

What does the second derivative tell us about f(x)?

• f’’(x)>0, f(x) is concave up

• f’’(x)<0, f(x) is concave down

• If f’’(x)=0/undefined and changes sign, there is a point of inflection

Second Derivative Test

• Uses CONCAVITY to find local/relative extrema

  1. Find critical points using f’

  2. Find f’’ and plug in critical points

• f’’(x)<0: relative max

• f’’(x)>0: relative min

Critical Points

where f’(x)=0 or is undefined