Math 22 Higher-Order Derivative - Revision history

Tphan046: /* Acceleration */

2020-07-25T16:48:50Z

Acceleration

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2020-07-25T16:48:25Z

Acceleration

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2020-07-25T16:48:13Z

Acceleration

Tphan046 at 16:47, 25 July 2020

2020-07-25T16:47:40Z

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2020-07-25T16:39:05Z

Acceleration

Tphan046: /* Notes */

2020-07-25T16:37:33Z

Notes

Tphan046: /* Higher-Order Derivatives */

2020-07-25T16:36:39Z

Higher-Order Derivatives

Tphan046: /* Higher-Order Derivatives */

2020-07-25T16:35:52Z

Higher-Order Derivatives

Tphan046: Created page with "==Higher-Order Derivatives== The "standard" derivative $f'(x)$ is called the first derivative of $f(x)$ . The derivative of $f'(x)$ is the se..."

2020-07-25T16:30:16Z

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..."

New page

==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 second derivative of<math>f(x)</math>, denoted by <math>f''(x).</math>
By continuing this process, we obtain '''higher-order derivative''' of <math>f(x)</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>

==Notes==
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.

[[Math_22| '''Return to Topics Page''']]

'''This page were made by [[Contributors|Tri Phan]]'''

@@ Line 37: / Line 37: @@
 ==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.
 '''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>

@@ Line 37: / Line 37: @@
 ==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.
 '''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>

@@ Line 53: / Line 53: @@
 |<math>\text{Height = }f(4)=-16(4^2)+32(4)+200=72</math>
 |-
-|<math>\text{Velocity = }f'(4)=-32(4)+32=-96</math> (Velocity function)
+|<math>\text{Velocity = }f'(4)=-32(4)+32=-96</math>
 |-
-|<math>\text{Acceleration = }f''(4)=-32</math> (Acceleration function)
+|<math>\text{Acceleration = }f''(4)=-32</math>
 |}
 [[Math_22| '''Return to Topics Page''']]
 '''This page were made by [[Contributors|Tri Phan]]'''

@@ Line 11: / Line 11: @@
 !Solution: &nbsp;
 |-
-|f'(x)=4x^3+15x^2-4x
+|<math>f'(x)=4x^3+15x^2-4x</math>
 |-
-|f''(x)=12x^2+30x-4
+|<math>f''(x)=12x^2+30x-4</math>
 |-
-|f'''(x)=24x+30
+|<math>f'''(x)=24x+30</math>
 |-
-|f^{(4)}(x)=24
+|<math>f^{(4)}(x)=24</math>
 |}

← Older revision		Revision as of 16:35, 25 July 2020
Line 4:		Line 4:

	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:
		+	\|-
		+	\|f'(x)=4x^3+15x^2-4x
		+	\|-
		+	\|f''(x)=12x^2+30x-4
		+	\|-
		+	\|f'''(x)=24x+30
		+	\|-
		+	\|f^{(4)}(x)=24
		+	\|}
		+
		+	'''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>
		+	\|}

	==Notes==		==Notes==

Math 22 Higher-Order Derivative - Revision history

Tphan046: /* Acceleration */

Tphan046: /* Acceleration */

Tphan046: /* Acceleration */

Tphan046 at 16:47, 25 July 2020

Tphan046: /* Acceleration */

Tphan046: /* Notes */

Tphan046: /* Higher-Order Derivatives */

Tphan046: /* Higher-Order Derivatives */

Tphan046: Created page with "==Higher-Order Derivatives== The "standard" derivative f'(x) is called the first derivative of f(x). The derivative of f'(x) is the se..."

Tphan046: Created page with "==Higher-Order Derivatives== The "standard" derivative $f'(x)$ is called the first derivative of $f(x)$ . The derivative of $f'(x)$ is the se..."