Khanh at 05:17, 14 November 2021

2021-11-14T05:17:42Z

Lila at 19:02, 18 October 2021

2021-10-18T19:02:52Z

Lila: Created page with "== The First Derivative Test ==
The definition of a derivatives tells us that a derivative is the slope of the tangent line at a point on the function.

Derivative..."

2021-10-01T16:41:22Z

Created page with "== The First Derivative Test == The definition of a derivatives tells us that a derivative is the slope of the tangent line at a point on the function. Derivative..."

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== The First Derivative Test ==
 The definition of a derivatives tells us that a derivative is the slope of the tangent line at a point on the function. 
 Derivatives can also tell us if a function is decreasing or increasing at a point. 
 A function <math> f(x) </math> is increasing on an interval, if for two numbers <math> x_1 </math> and <math> x_2 </math> in the interval <math> x_1 < x_2, </math> that <math> f(x_1) < f(x_2) </math> is true.
 A function <math> f(x) </math> is decreasing on an interval, if for two numbers <math> x_1 </math> and <math> x_2 </math> in the interval <math> x_1 < x_2, </math> that <math> f(x_1) > f(x_2) </math> is true.
 
 If a function <math> f(x) </math> is continuous on a closed interval <math> [a,b], </math> and differentiable on an open interval <math> (a,b), </math> then the following applies: 
 1. If <math> f'(x) > 0 </math> for all <math> x </math> in <math> (a,b), </math> then <math> f(x) </math> is increasing on <math> [a,b]. </math>
 2. If <math> f'(x) < 0 </math> for all <math> x </math> in <math> (a,b), </math> then <math> f(x) </math> is decreasing on <math> [a,b]. </math>
 3. If <math> f'(x) = 0 </math> for all <math> x </math> in <math> (a,b), </math> then <math> f(x) </math> is constant on <math> [a,b]. </math>
 
 In the last section, we learned about absolute minimums/maximums. Inside a function, other extrema, known as relative extrema, can exist. 
 The relative extrema of a function are points on a function that are lower or higher than all of the points near them. Such points create "hills" or "valleys" within a given function. 
 Relative extrema occur at points on a function where the derivative at that point changes from increasing to decreasing, or decreasing to increasing. 
 If the derivative changes from increasing to decreasing, that point is known as a relative maximum. 
 If the derivative changes from decreasing to increasing, that point is known as a relative minimum. 
 By finding the relative extrema of a function, you can then calculate whether or not those extrema are relative minima or maxima using the derivative of the function at those points. 
Relative extrema are always critical points of a function. 
 
===Example===
 Find the relative extrema of <math> f(x) = x^3 - \frac {3}{2}x^2. </math>
 First, check if the function is continuous for all <math> x. </math>
 We can see the function exists for all <math> x </math> therefore, it is continuous. 

 Second, find the critical numbers of <math> f(x) </math> by using the derivative of the function. 
 Find the critical numbers by setting <math> f'(x) = 0. </math>
<math> f'(x) = 3x^2 - 3x </math>
<math> 3x^2 - 3x = 0 </math>
<math> x(3x - 3) = 0 </math>
<math> x = 0,1. </math>

 Third, create intervals with your critical numbers. 
 Since we have two critical numbers, we will have three intervals. They are: 
<math> -\infty < x < 0, 0 < x < 1, 1 < x < \infty. </math>

 Fourth, determine if <math> f'(x) </math> is increasing or decreasing over each interval. Do this by evaluating a test number within each interval. 
 In most cases, it is beneficial to create a table to arrange the present data. 

<table border="1"><tr><td>Interval</td><td><math>-\infty < x < 0</math></td><td><math> 0 < x < 1</math></td><td> <math>1 < x < \infty</math></td></tr>
<tr><td> Test Value </td><td><math> x = -1 </math></td><td><math> x = \frac {1}{2}</math> </td><td><math> x = 2 </math></td></tr>
<tr><td> Sign of <math> f'(x) </math></td><td><math> f'(-1) = 6 </math></td><td><math> f'(\frac {1}{2}) = \frac {-3}{4} </math> </td><td><math> f'(2) = 6 </math></td></tr>
<tr><td>Increasing/Decreasing </td><td> Increasing </td><td> Decreasing </td><td> Increasing </td></tr></table>

 Lastly, determine if any relative maximums or minimums are present. 
Since <math> f'(x) </math> changes from increasing to decreasing to increasing, we can conclude that there is a relative maximum at <math> x = 0, </math> and a relative minimum at <math> x = 1. </math>
====Practice Problems====
 Find the relative extrema of the given functions. 
<math> 1. f(x) = x^2 - 6x </math>
<math> 2. f(x) = x^4 - 32x + 4 </math>
<math> 3. f(x) = x + \frac {1}{x} </math>

@@ Line 50: / Line 50: @@
 <p><math> 3. f(x) = x + \frac {1}{x} </math></p>
-==Resources==
+== Licensing ==
-* [https://en.wikibooks.org/wiki/High_School_Calculus/The_First_Derivative_Test The First Derivative Test], High School Calculus
+Content obtained and/or adapted from:

← Older revision		Revision as of 19:02, 18 October 2021
Line 49:		Line 49:
	<p><math> 2. f(x) = x^4 - 32x + 4 </math></p>		<p><math> 2. f(x) = x^4 - 32x + 4 </math></p>
	<p><math> 3. f(x) = x + \frac {1}{x} </math></p>		<p><math> 3. f(x) = x + \frac {1}{x} </math></p>
		+
		+	==Resources==
		+	* [https://en.wikibooks.org/wiki/High_School_Calculus/The_First_Derivative_Test The First Derivative Test], High School Calculus

The First Derivative Test - Revision history

Khanh at 05:17, 14 November 2021

Lila at 19:02, 18 October 2021

Lila: Created page with "== The First Derivative Test == The definition of a derivatives tells us that a derivative is the slope of the tangent line at a point on the function. Derivative..."

Lila: Created page with "== The First Derivative Test ==
The definition of a derivatives tells us that a derivative is the slope of the tangent line at a point on the function.

Derivative..."