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7/31/2019 h31 Higher Order Derivatives Velocity and Acceleration
1/2
Calculus and Vectors How to get an A+
3.1 Higher Order Derivatives. Velocity and Acceleration2010 Iulia & Teodoru Gugoiu - Page 1 of 2
3.1 Higher Order Derivatives. Velocity and Acceleration
A Higher Order DerivativesLet consider the function )(xfy = .
The firstderivative of for fprime is:
dx
dyyxf == ')('
The secondderivative of f or fdouble prime is:
2
2
)''(''))'('()(''dx
yd
dx
dy
dx
dyyxfxf =
====
The thirdderivative of f or f triple prime is:
3
3
2
2
)'''('''))'(''()('''dx
yd
dx
yd
dx
dyyxfxf =
====
The fourth derivative of f or super 4 is:
4
4
3
3
)4()4( )''''())'('''()(dx
yddxyd
dxdyyxfxf =
====
The n-th derivative of f or super n is:
n
n
n
nnnnn
dx
yd
dx
yd
dx
dyyxfxf =
====
1
1)1()()1()()'())'(()(
Notes.1. f is differentiable at x if )(' xf exists.
2. f is double differentiable at x if )('' xf exists.
3. fis n differentiable at x if )()( xf n exists.
Ex 1. Find 'f , ''f , '''f , )4(f , and )5(f for
13)( 24 ++= xxxxf .
Ex 2. Find 'f , ''f , and '''f for1
)(2
3
+=x
xxf .
Ex 3. Find )(nf for1
)(+
=x
xxf . Ex 4. Show that 13
3 ++= xxy satisfies 0'2''''' =+ yxyy .
7/31/2019 h31 Higher Order Derivatives Velocity and Acceleration
2/2
Calculus and Vectors How to get an A+
3.1 Higher Order Derivatives. Velocity and Acceleration2010 Iulia & Teodoru Gugoiu - Page 2 of 2
B Velocity and AccelerationLet )(ts or )(th (height or altitude) be theposition
function of a particle.
mhsISIS
== (meter)
Displacementis the change in position over a time
interval ],[ 21 tt :
)()()()( 1212 ththhtstss ==
Average Velocityis defined by:
12
12 )()(
tt
tsts
t
sAV
=
= )/( sm
Velocityfunction is the derivative of the positionfunction:
)(')(')( thtstv ==
Speedfunction is the absolute value of velocity |)(| tv .
Acceleration function is the derivative of velocity:
)('')(')( tstvta == )/( 2sm
Jerkfunction is the derivative of acceleration:)(''')('')(')( tstvtatj === )/( 3sm
Notes:
0)( =ts (particle is in the origin)
0)( =th (particle is on the ground)
0)( >ts (particle is on the positive side of the origin)
0)( th (particle is above the ground)
0)( tv (particle moves in the positive direction (right
or up))0)( tvts .
If || s is decreasing then the particle is moving towards
the origin (MTO) and 0)()( tatv .
If || v is decreasing then the particle is slowing down
(SD) and 0)()(