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

Jump to navigation
Jump to search

(Created page with "==Higher-Order Derivatives== The "standard" derivative <math>f'(x)</math> is called the first derivative of <math>f(x)</math>. The derivative of <math>f'(x)</math> is the se...") |
|||

(7 intermediate revisions by the same user not shown) | |||

Line 5: | Line 5: | ||

Note: The 3rd derivative of <math>f(x)</math> is <math>f'''(x)</math>. However, we simply denote the <math>n^{th}</math> derivative as <math>f^{(n)}(x)</math> for <math>n\ge 4</math> | Note: The 3rd derivative of <math>f(x)</math> is <math>f'''(x)</math>. However, we simply denote the <math>n^{th}</math> derivative as <math>f^{(n)}(x)</math> for <math>n\ge 4</math> | ||

− | == | + | '''Example''': Find the first four derivative of |

+ | |||

+ | '''1)''' <math>f(x)=x^4+5x^3-2x^2+6</math> | ||

+ | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||

+ | !Solution: | ||

+ | |- | ||

+ | |<math>f'(x)=4x^3+15x^2-4x</math> | ||

+ | |- | ||

+ | |<math>f''(x)=12x^2+30x-4</math> | ||

+ | |- | ||

+ | |<math>f'''(x)=24x+30</math> | ||

+ | |- | ||

+ | |<math>f^{(4)}(x)=24</math> | ||

+ | |} | ||

+ | |||

+ | '''2)''' <math>f(x)=(x^3+1)(x^2+3)</math> | ||

+ | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||

+ | !Solution: | ||

+ | |- | ||

+ | |It is better to rewrite <math>f(x)=(x^3+1)(x^2+3)=x^5+3x^3+x^2+3</math> | ||

+ | |- | ||

+ | |Then, <math>f'(x)=5x^4+9x^3+2x</math> | ||

+ | |- | ||

+ | |<math>f''(x)=20x^3+27x^2+2</math> | ||

+ | |- | ||

+ | |<math>f'''(x)=60x^2+54x</math> | ||

+ | |- | ||

+ | |<math>f^{(4)}(x)=120x+54</math> | ||

+ | |} | ||

+ | |||

+ | ==Acceleration== | ||

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. | ||

+ | |||

+ | '''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> | ||

+ | |||

+ | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||

+ | !Solution: | ||

+ | |- | ||

+ | |<math>f(t)=-16t^2+32t+200</math> (Position function) | ||

+ | |- | ||

+ | |<math>f'(t)=-32t+32</math> (Velocity function) | ||

+ | |- | ||

+ | |<math>f''(x)=-32</math> (Acceleration function) | ||

+ | |- | ||

+ | |So, when <math>t=4</math>, from the functions above, we can have: | ||

+ | |- | ||

+ | |<math>\text{Height = }f(4)=-16(4^2)+32(4)+200=72</math> | ||

+ | |- | ||

+ | |<math>\text{Velocity = }f'(4)=-32(4)+32=-96</math> | ||

+ | |- | ||

+ | |<math>\text{Acceleration = }f''(4)=-32</math> | ||

+ | |} | ||

+ | |||

[[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]]''' |

## Latest revision as of 08:48, 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 obtainhigher-order derivativeof .

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: |

**This page were made by Tri Phan**