Unit 5 - Analytical Applications of Differentiation

Mean Value Theorem

If a function f is:

• continuous on the closed interval

• differentiable on the open interval

there is at least one point where the instantaneous rate of change is equal to the average rate of change

Extreme Value Theorem

If a function f is

• continuous on the closed interval

it has an absolute maximum and an absolute minimum on the interval

• These happen at critical points or endpoints

Critical Point

When the derivative is 0 or doesn’t exist

• if the sign changes from positive to negative it’s a local maximum

• if the sign changes from negative to positive it’s a local minimum

First Derivative Test

When you plug in intermediate values between the critical points into the derivate to test for a sign change

Concave Up

When the rate of change over an interval is increasing (getting more steep)

Concave Down

When the rate of change over an interval is decreasing (getting less steep)

Second Derivative Test

Test for extrema

• If it is positive, the point is concave up and is a minimum

• If it is negative, the point is concave down and is a maximum

Inflection Point

Location where the function switches concavity

* Second derivative of this point will be 0