5.3 :higher order derivatives, concavity and the 2 nd derivative test
DESCRIPTION
5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test. Objectives: To find Higher Order Derivatives To use the second derivative to test for concavity To use the 2 nd Derivative Test to find relative extrema. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/1.jpg)
5.3:Higher Order Derivatives, Concavity and the 2nd Derivative Test
Objectives:•To find Higher Order Derivatives•To use the second derivative to test for concavity•To use the 2nd Derivative Test to find relative extrema
![Page 2: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/2.jpg)
If a function’s derivative is f’, the derivative of f’, if it exists, is the second derivative, f’’. You can take 3rd, 4th,5th, etc. derivative
NotationsSecond Derivative:
Third Derivative:
For n> 4, the nth derivative is written f(n)(x)
)(,),('' 22
2
xfDdxydxf x
3
3
),('''dxydxf
![Page 3: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/3.jpg)
1. Find f(4)(x). 10764)( 234 xxxxxf
2. Find f’’(0). xxxxf 23 125)(
![Page 4: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/4.jpg)
Find f’’(x).
1. 22 7)( xxf 2. xxxf
1
)(
![Page 5: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/5.jpg)
Find f’’’(x).
23)(
xxxf
![Page 6: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/6.jpg)
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 up
If v(t) and a(t) are opposite signs, object is slowing down
![Page 7: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/7.jpg)
Suppose 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+9
a.) Find where the car is moving forwards and backwards.
b.) When is the car speeding up and slowing down?
![Page 8: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/8.jpg)
Concavity of a Graph How 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 bowl Concave Down: y’ is decreasing, graph is “frowning”, arch Inflection point: where a function changes concavity
f’’ = 0 or f’’ does not exist here
![Page 9: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/9.jpg)
Precise Definition of Concave Up and Down
A 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 line
![Page 10: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/10.jpg)
Test for Concavity
• f’ 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 < 0
![Page 11: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/11.jpg)
Find the open intervals where the functions are concave up or concave down. Find any inflection points.
1. 34 4)( xxxf
![Page 12: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/12.jpg)
36)( 2
x
xf
![Page 13: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/13.jpg)
35
38
4)( xxxf
![Page 14: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/14.jpg)
Second Derivative Test for Relative Extrema
Let f’’(x) exist on some open interval containing c, and let f’(c) = 0.
1. If f’’(c) > 0, then f(c) is a relative minimum2. If f’’(c) < 0, then f(c) is a relative maximum3. If f’’(c) = 0 or f’’(c) does not exist, use 1st
derivative test
![Page 15: 5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test](https://reader034.vdocument.in/reader034/viewer/2022050802/56815f01550346895dcdbe6c/html5/thumbnails/15.jpg)
Find all relative extrema using the
2nd Derivative Test.1. 2. 133)( 23 xxxf 3
538
)( xxxf