Constant factor rule in integration and Arroyo High School (San Lorenzo, California): Difference between pages

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{{Infobox School2

The '''constant factor rule in integration''' is a dual of the [[constant factor rule in differentiation]], and is a consequence of the [[linearity of integration]].

| name = Arroyo High School

| image =

| established = 1955

| type = Public

| head_name = Principal

| head = Larry Smith

| city = San Lorenzo

| state = California

| country = USA

| faculty =

| students= 1969

| color = Red, Black, White

| school_colors = Red and Black

| mascot = [[Dons]]

| free_label =

| free =

| website = [http://www.slzusd.org/arroyo/ www.slzusd.org/arroyo]

}}

'''Arroyo High School''' is located in [[San Lorenzo, California]], and is part of the [[San Lorenzo Unified School District]]. Serving as one of two public high school for San Lorenzo and parts of [[San Leandro, California|San Leandro]], its sister high school is [[San Lorenzo High School]].

Start by noticing that, from the definition of [[integral|integration]] as the [[inverse]] process of [[derivative|differentiation]]:

==Small Learning Communities==

:<math>y = \int \frac{dy}{dx} dx.</math>

Includes four academies in partnership with the California Partnership Academies program.

*FUTURE Academy

*[[Tech-Links Academy]]

*Health and Medicine Academy

*Arts Academy

==Class Schedule==

Now [[product (mathematics)|multiply]] both sides by a [[constant]] ''k''. Since ''k'' is a constant it is [[not dependent on]] ''x'':

{| class="wikitable"

|-

:<math>ky = k \int \frac{dy}{dx} dx. \quad \mbox{(1)}</math>

! '''Schedule'''

Take the [[constant factor rule in differentiation]]:

! '''Block I'''

:<math>\frac{d\left(ky\right)}{dx} = k \frac{dy}{dx}.</math>

! '''Block II'''

[[integral|Integrate]] with respect to ''x'':

! '''Block III'''

:<math>ky = \int k \frac{dy}{dx} dx. \quad \mbox{(2)}</math>

! '''Block IV'''

Now from (1) and (2) we have:

|-

:<math>ky = k \int \frac{dy}{dx} dx</math>

:<math>ky = \int k \frac{dy}{dx} dx.</math>

| Regular

Therefore:

| 8:00 AM - 9:30 AM

:<math>\int k \frac{dy}{dx} dx = k \int \frac{dy}{dx} dx. \quad \mbox{(3)}</math>

| 9:45 AM - 11:20 AM

Now make a new differentiable [[function (mathematics)|function]]:

| 12:00 PM - 1:30 PM

:<math>u = \frac{dy}{dx}.</math>

| 1:40 PM - 3:10 PM

[[Substitution property of equality|Substitute]] in (3):

|-

:<math>\int ku dx = k \int u dx.</math>

| CPT

Now we can re-[[substitute]] ''y'' for something different from what it was originally:

| 8:00 AM - 9:10 AM

:<math>y = u. \,</math>

| 9:25 AM - 10:40 AM

So:

| 10:45 AM - 11:55 AM

:<math>\int ky dx = k \int y dx.</math>

| 12:35 PM - 1:45 PM

This is the constant factor rule in integration.

|-

A [[special case]] of this, with ''k''=-1, yields:

| Late Start

:<math>\int -y dx = -\int y dx.</math>

| 9:00 AM - 10:15 AM

{{math-stub}}

[[Category:Integral calculus]]

| 10:30 AM - 11:45 AM

[[eo:Konstanta faktora regulo en integralado]]

| 12:25 PM - 1:40 PM

| 1:55 PM - 3:10 PM

|-

| First Final Exam

| 8:30 AM - 10:30 AM

| 11:10 AM - 1:10 PM

|

|

|-

| Second Final Exam

|

|

| 8:30 AM - 10:30 AM

| 11:10 AM - 1:10 PM

|}

== Demographics ==

According to [http://www.slzusd.org/Site/upload_files/SARs/06-Arroyo.pdf Arroyo High's 2005-2006 School Accountability Report Card], there are about 1,969 students. 31% (610) are White, 33.6% (662) are Latino, 18.6% (366) are Asian, 8.4% (165) are Filipino, 6.5% (128) are African-American, 1.1% are Pacific Islander, less than 1% are American Indian or Alaskian Native, and less than 1% are Other.

==See also==

*[[San Lorenzo Unified School District]]

*[[San Lorenzo, California]]

== External link ==

* [http://www.slzusd.org/AHS/ Arroyo High School Homepage]

{{California-school-stub}}

[[Category:High schools in California]]

[[Category:Alameda County, California]]