Higher Order Derivatives. Objectives Students will be able to Calculate higher order derivatives Apply higher order derivatives in application problems.

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Presentation transcript:

Higher Order Derivatives

Objectives Students will be able to Calculate higher order derivatives Apply higher order derivatives in application problems

Symbol Representations First Derivati ve Second Derivati ve

Symbol Representations Third Derivati ve Fourth Derivati ve

Symbol Representations nth Derivati ve

Example 1 Calculate the second derivative of the function

Example 2 For the function find

Example 3 Calculate the second derivative of the function

Example 4 For the function find

Example 5 Calculate the second derivative of the function

Example 6 For the function find

Example 7 Calculate the second derivative of the function

Example 8 For the function find

Example 9 Calculate the third and fourth derivative of the function

Example 10 Find the open interval(s) where the function is concave up or concave down. Find any points of inflection.

Example 11 Find the open interval(s) where the function is concave up or concave down. Find any points of inflection.

Example 12-1 For an original function f(x) being a distance function with respect to time, the first derivative of f(x) is the velocity (instantaneous rate of change of distance) and the second derivative of f(x) is called acceleration (instantaneous rate of change of velocity). In terms of the demand

Example 12-2 A car rolls down a hill. Its distance (in feet) from its starting point is given by where t is in seconds. a.How far will the car move in 10 seconds? b.What is the velocity at 5 seconds? At 10 seconds? c.How can you tell from v(t) that the car will not stop?

Example 12-3 A car rolls down a hill. Its distance (in feet) from its starting point is given by where t is in seconds. d.What is the acceleration at 5 seconds? At 10 seconds? e.What is happening to the velocity and the acceleration as t increases?