5.3:Higher Order Derivatives, Concavity and the 2nd Derivative Test
5.3:Higher Order Derivatives, Concavity and the 2nd Derivative TestObjectives:To find Higher Order DerivativesTo use the second derivative to test for concavityTo use the 2nd Derivative Test to find relative extrema
If a functions derivative is f, the derivative of f, if it exists, is the second derivative, f. You can take 3rd, 4th,5th, etc. derivativeNotationsSecond Derivative:
For n> 4, the nth derivative is written f(n)(x)
1. Find f(4)(x).
2. Find f(0).
Asking to find the 4th derivative of f(x):
Asking to find the 2nd derivative and evaluate for x = 0:
For f (x), easiest to bring up with a negative exponent:
Concavity of a GraphHow the curve is turning, shape of the graph
Determined by finding the 2nd derivative
Rate of change of the first derivative
Concave Up: y is increasing, graph is smiling, cup or bowlConcave Down: y is decreasing, graph is frowning, archInflection point: where a function changes concavityf = 0 or f does not exist herePrecise Definition of Concave Up and DownA graph is Concave Up on an interval (a,b) if the graph lies above its tangent line at each point in (a,b)
A graph is Concave Down on an interval (a,b) if graph lies below its tangent line at each point in (a,b)
At inflection points, the graph crosses the tangent lineTest for Concavityf and f need to exist at all point in an interval (a,b)Graph is concave up where f(x) > 0 for all points in (a,b)Graph is concave down where f(x) < 0 for all points in (a,b)
Find inflection points and test on a number line. Pick x-values on either side of inflection points to tell whether f is > 0 or < 0Find the open intervals where the functions are concave up or concave down. Find any inflection points.1.
Second Derivative Test for Relative ExtremaLet f(x) exist on some open interval containing c, and let f(c) = 0.
If f(c) > 0, then f(c) is a relative minimumIf f(c) < 0, then f(c) is a relative maximumIf f(c) = 0 or f(c) does not exist, use 1st derivative testFind all relative extrema using the 2nd Derivative Test.1. 2.
If a function describes the position of an object along a straight line at time t:s(t) = positions(t) = v(t) = velocity (can be + or - )s(t) = v(t) = a(t) = acceleration
If v(t) and a(t) are the same sign, object is speeding upIf v(t) and a(t) are opposite signs, object is slowing downSuppose a car is moving in a straight line, with its position from a starting point (in ft) at time t (in sec) is given by s(t)=t3-2t2-7t+9a.) Find where the car is moving forwards and backwards. b.) When is the car speeding up and slowing down?