Derivatives Rates of Change - Revision history

Lila: /* Average Rate of Change */

2021-10-28T15:26:10Z

Average Rate of Change

Lila: /* Resources */

2021-10-27T21:01:07Z

Resources

Lila at 20:03, 30 September 2021

2021-09-30T20:03:07Z

Lila at 19:55, 30 September 2021

2021-09-30T19:55:25Z

James.kercheville: Added video links

2020-09-03T20:58:58Z

Added video links

James.kercheville: Added links for PowerPoint and worksheet

2020-08-13T22:58:32Z

Added links for PowerPoint and worksheet

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20as%20Rates%20of%20Change/MAT1214-3.4TheDerivativeAsARateOfChangePwPt.pptx Derivatives as Rates of Change ] PowerPoint file created by Dr. Sara Shirinkam, UTSA.

* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20as%20Rates%20of%20Change/MAT1214-3.4TheDerivativeAsARateOfChangeWS1.pdf Derivatives as Rates of Change Worksheet]

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 Rate of Change is used to describe the ratio of a change in one variable compare to a change of another variable
-If there is a function <math>f(x)=y</math> then the rate of change of the function <math>f(x)</math> correspond to the rate of change of variable z is
+If there is a function <math>f(x)=y</math> then the rate of change of the function <math>f(x)</math> correspond to the rate of change of variable x is
 :<math>\frac{\Delta y}{\Delta x}=\frac{\Delta f(x)}{\Delta x}</math>

← Older revision		Revision as of 21:01, 27 October 2021
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	* [https://youtu.be/jlihNi_Mkos Finding Instantaneous Rates of Change Using Def'n of Derivative] by patrickJMT		* [https://youtu.be/jlihNi_Mkos Finding Instantaneous Rates of Change Using Def'n of Derivative] by patrickJMT
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Differentiation/Differentiation_Defined Differentiation Defined, Wikibooks: Calculus/Differentiation] under a CC BY-SA license

← Older revision		Revision as of 20:03, 30 September 2021
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	(Note that the letter <math>s</math> is often used to denote distance, which would yield <math>\frac{ds}{dt}</math> . The letter <math>d</math> is often avoided in denoting distance due to the potential confusion resulting from the expression <math>\frac{dd}{dt}</math>.)		(Note that the letter <math>s</math> is often used to denote distance, which would yield <math>\frac{ds}{dt}</math> . The letter <math>d</math> is often avoided in denoting distance due to the potential confusion resulting from the expression <math>\frac{dd}{dt}</math>.)

		+	==Definition of Instantaneous Rate of Change (Derivative)==
		+	Let <math>f(x)</math> be a function. Then <math>f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}</math> wherever this limit exists. In this case we say that <math>f</math> is '''differentiable''' at <math>x</math> and its '''derivative''' at <math>x</math> is <math>f'(x)</math>.
		+
		+	[[image:Tangent_as_Secant_Limit.svg\|center]]
		+
		+	Think of <math> h </math> as being the distance between the points <math> (x_0, f(x_0)) </math> and <math> (x_0 + h, f(x_0 + h)) </math>. Then, the average rate of change between these two points is
		+	: <math> \frac{\Delta f(x)}{\Delta x} = \frac{f(x_0+h)-f(x_0)}{(x_0 + h) - x_0} = \frac{f(x_0+h)-f(x_0)}{h}</math>.
		+
		+	As the distance between the points goes to 0, we approach the instantaneous rate of change (slope) of <math> f(x) </math> at the point <math> x = x_0 </math>. This is how we derive (no pun intended) the definition of the derivative.

	==Resources==		==Resources==

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 * [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20as%20Rates%20of%20Change/MAT1214-3.4TheDerivativeAsARateOfChangePwPt.pptx Derivatives as Rates of Change ] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20as%20Rates%20of%20Change/MAT1214-3.4TheDerivativeAsARateOfChangeWS1.pdf Derivatives as Rates of Change Worksheet]
 * [https://youtu.be/4Up5gsDeluw Average and Instantaneous Rate of Change of a function over an interval & a point - Calculus] by The Organic Chemistry Tutor
 * [https://youtu.be/jlihNi_Mkos Finding Instantaneous Rates of Change Using Def'n of Derivative] by patrickJMT

← Older revision		Revision as of 20:58, 3 September 2020
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	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20as%20Rates%20of%20Change/MAT1214-3.4TheDerivativeAsARateOfChangeWS1.pdf Derivatives as Rates of Change Worksheet]		* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Derivatives%20as%20Rates%20of%20Change/MAT1214-3.4TheDerivativeAsARateOfChangeWS1.pdf Derivatives as Rates of Change Worksheet]
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		+
		+
		+	* [https://youtu.be/4Up5gsDeluw Average and Instantaneous Rate of Change of a function over an interval & a point - Calculus] by The Organic Chemistry Tutor
		+
		+	* [https://youtu.be/jlihNi_Mkos Finding Instantaneous Rates of Change Using Def'n of Derivative] by patrickJMT