Difference between revisions of "Math 22 Higher-Order Derivative"

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If <math>f(x)</math> is the position function, then <math>f'(x)</math> is the velocity function and <math>f''(x)</math> is the acceleration function.
 
If <math>f(x)</math> is the position function, then <math>f'(x)</math> is the velocity function and <math>f''(x)</math> is the acceleration function.
  
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'''Word-Problem Example''': A ball is thrown upward from the top of a <math>200</math>-foot cliff. The initial velocity of the ball is <math>32</math> feet per second. The position function is <math>f(t)=-16t^2+32t+200</math> where <math>t</math> is measured in seconds. Find the height, velocity, and acceleration of the ball at <math>t=4</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>f(t)=-16t^2+32t+200</math> (Position function)
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|-
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|<math>f'(t)=-32t+32</math> (Velocity function)
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|-
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|<math>f''(x)=-32</math> (Acceleration function)
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|-
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|So, when <math>t=4</math>, from the functions above, we can have:
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|-
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|<math>\text{Height = }f(4)=-16(4^2)+32(4)+200=72</math>
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|-
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|<math>\text{Velocity = }f'(4)=-32(4)+32=-96</math> (Velocity function)
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|-
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|<math>\text{Acceleration = }f''(4)=-32</math> (Acceleration function)
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|}
  
 
[[Math_22| '''Return to Topics Page''']]
 
[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Revision as of 09:47, 25 July 2020

Higher-Order Derivatives

 The "standard" derivative  is called the first derivative of . The derivative of  is the second derivative of, denoted by 
 By continuing this process, we obtain higher-order derivative of .

Note: The 3rd derivative of is . However, we simply denote the derivative as for

Example: Find the first four derivative of

1)

Solution:  

2)

Solution:  
It is better to rewrite
Then,

Acceleration

If is the position function, then is the velocity function and is the acceleration function.

Word-Problem Example: A ball is thrown upward from the top of a -foot cliff. The initial velocity of the ball is feet per second. The position function is where is measured in seconds. Find the height, velocity, and acceleration of the ball at

Solution:  
(Position function)
(Velocity function)
(Acceleration function)
So, when , from the functions above, we can have:
(Velocity function)
(Acceleration function)

Return to Topics Page

This page were made by Tri Phan