5.3:higher order derivatives, concavity and the 2 nd derivative test objectives: to find higher...
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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
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)
)(,),('' 2
2
2
xfDdx
ydxf x
3
3
),('''dx
ydxf
1. Find f(4)(x). 10764)( 234 xxxxxf
2. Find f’’(0). xxxxf 23 125)(
Asking to find the 4th derivative of f(x):
24)(
2424)('''
122412)(''
712124)('
)4(
2
23
xf
xxf
xxxf
xxxxf
Asking to find the 2nd derivative and evaluate for x = 0:
24)0(''
2430)(''
2415)(' 2
f
xxf
xxxf
Find f’’(x).
1. 22 7)( xxf 2. x
xxf
1
)(
2812
2848
4)7()2(4)(''
)7(4)2)(7(2)('
:
2
22
2
22
x
xx
xxxxf
xxxxxf
Answer
2
2
1
1
1
)1(1)1()('
x
x
xxxf
For f ‘’(x), easiest to bring up with a negative exponent:
3
3
2
1
2)(''
12)(''
11)('
xxf
xxf
xxf
Find f’’’(x).
2
3)(
x
xxf
44
3
2
22
2
2
36)2(36)('''
)2(12)(''
)2(6)('
2
6
2
363
2
1)3(3)2()('
xxxf
xxf
xxf
xx
xx
x
xxxf
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
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
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
Find the open intervals where the functions are concave up or concave down. Find any inflection points.
1. 34 4)( xxxf
3
6)(
2 x
xf
3
5
3
8
4)( xxxf
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
Find all relative extrema using the
2nd Derivative Test.1. 2. 133)( 23 xxxf 3
5
3
8
)( xxxf
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
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?
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