Section 3.8 Higher Derivatives AP Calculus October 7, 2009 Berkley High School, D2B2

Calculus, Section 3.82 We can take the derivative of a derivative

Calculus, Section 3.83 What First Derivatives Tell Us Instantaneous slope of the function Where the first derivative is zero, the function has a horizontal tangent When the original curve is increasing, the first derivative is positive, and when the original curve is decreasing, the derivative is negative

Calculus, Section 3.84 What Second Derivatives Tell Us Instantaneous slope of the first derivative F(x):F’(x)::F’(x):F’’(x)

Calculus, Section 3.85 What Second Derivatives Tell Us Where the second derivative is positive, the original function is concave up. (Tangent line is below the function.) Where the second derivative is negative, the original curve is concave down. (Tangent line is above the function.) The transition point between concavities is called an inflection point  This can happen only when the second derivative is zero, although it does not happen at all zeros.

Calculus, Section 3.86 Example

Calculus, Section 3.87 Example

Calculus, Section 3.88 Example

Calculus, Section 3.89 Implicit Differentiation

Calculus, Section Implicit Differentiation

Calculus, Section Seeing trends

Calculus, Section Assignment Section 3.8, 1-13 odd, 29-39, odd