Difference between revisions of "Derivatives Rates of Change"
| (2 intermediate revisions by the same user not shown) | |||
| Line 2: | Line 2: | ||
Rate of Change is used to describe the ratio of a change in one variable compare to a change of another variable | 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 | + | 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> | :<math>\frac{\Delta y}{\Delta x}=\frac{\Delta f(x)}{\Delta x}</math> | ||
| Line 14: | Line 14: | ||
(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== | ||
| Line 23: | Line 32: | ||
* [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 | ||
Latest revision as of 09:26, 28 October 2021
Contents
Average Rate of Change
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=y} then the rate of change of the function correspond to the rate of change of variable x is
For example, if there are two points and , then
The Rate of Change of a Function at a Point
Consider the formula for average velocity in the direction, , where is the change in over the time interval . This formula gives the average velocity over a period of time, but suppose we want to define the instantaneous velocity. To this end we look at the change in position as the change in time approaches 0. Mathematically this is written as: , which we abbreviate by the symbol . (The idea of this notation is that the letter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} denotes change.) Compare the symbol Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta} . The idea is that both indicate a difference between two numbers, but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta} denotes a finite difference while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} denotes an infinitesimal difference. Please note that the symbols Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dt} have no rigorous meaning on their own, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{\Delta t\to 0}\Delta t=0} , and we can't divide by 0.
(Note that the letter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} is often used to denote distance, which would yield Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ds}{dt}} . The letter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is often avoided in denoting distance due to the potential confusion resulting from the expression Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dd}{dt}} .)
Definition of Instantaneous Rate of Change (Derivative)
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} be a function. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}} wherever this limit exists. In this case we say that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and its derivative at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)} .
Think of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h } as being the distance between the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0, f(x_0)) } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0 + h, f(x_0 + h)) } . Then, the average rate of change between these two points is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}} .
As the distance between the points goes to 0, we approach the instantaneous rate of change (slope) of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } at the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = x_0 } . This is how we derive (no pun intended) the definition of the derivative.
Resources
- Derivatives as Rates of Change PowerPoint file created by Dr. Sara Shirinkam, UTSA.
- Average and Instantaneous Rate of Change of a function over an interval & a point - Calculus by The Organic Chemistry Tutor
Licensing
Content obtained and/or adapted from:
- Differentiation Defined, Wikibooks: Calculus/Differentiation under a CC BY-SA license
