Linear approximation: Difference between revisions

Revision as of 19:32, 28 July 2020

Revision as of 07:48, 16 December 2020

Line 14:

This is a good approximation when <math>x</math> is close enough to <math>a</math>; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the [[tangent line]] to the graph of <math>f</math> at <math>(a,f(a))</math>. For this reason, this process is also called the '''tangent line approximation'''.

If <math>f</math> is [[concave down]] in the interval between <math>x</math> and <math>a</math>, the approximation will be an overestimate (since the derivative is decreasing in that interval). If <math>f</math> is [[concave up]], the approximation will be an underestimate.<ref>{{cite web|title=12.1 Estimating a Function Value Using the Linear Approximation|url=http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html|accessdateaccess-date=3 June 2012}}</ref>

Linear approximations for [[vector (geometric)|vector]] functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the [[Jacobian matrix and determinant|Jacobian]] matrix. For example, given a differentiable function <math>f(x, y)</math> with real values, one can approximate <math>f(x, y)</math> for <math>(x, y)</math> close to <math>(a, b)</math> by the formula

Line 38:

| last = Nelson

| first = Robert

| authorlink =

|author2=M. G. Olsson

| title = The pendulum – Rich physics from a simple system

Line 45 ⟶ 44:

| issue = 2

| pages = 112–121

| publisher =

| ___location =

| date = February 1987

| url = http://fy.chalmers.se/~f7xiz/TIF080/pendulum.pdf

| doi = 10.1119/1.14703

| accessdateaccess-date = 2008-10-29|bibcode = 1986AmJPh..54..112N }}</ref><ref>{{cite encyclopedia

| id =

| accessdate = 2008-10-29|bibcode = 1986AmJPh..54..112N }}</ref><ref>{{cite encyclopedia

| title = Clock

| encyclopedia = Encyclopædia Britannica, 11th Ed.

Line 58 ⟶ 54:

| publisher = The Encyclopædia Britannica Publishing Co.

| year = 1910

| id =

| url= https://books.google.com/books?id=cLsUAAAAYAAJ&pg=PA538

| accessdateaccess-date = 2009-03-04}} includes a derivation</ref>

:<math>

Line 70 ⟶ 65:

However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,<ref group = Note>A "small" swing is one in which the angle θ is small enough that sin(θ) can be approximated by θ when θ is measured in radians</ref> ) the [[Frequency|period]] is:<ref>{{cite book |last = Halliday

|first = David

|authorlink =

|author2 = Robert Resnick

|author3 = Jearl Walker

Line 80 ⟶ 74:

|url = https://archive.org/details/fundamentalsofp000davi/page/381

|url-access = registration

|doi =

|id =

|isbn = 0-471-14854-7

}}</ref>

Line 96 ⟶ 88:

| page = 162

| url = https://www.google.com/books?id=t7OoPLzXwiQC&pg=PA162

| doi =

| id =

| isbn = 1-4067-6879-0}}</ref> Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

Line 122 ⟶ 112:

==Further reading==

*{{cite book |author1=Weinstein, Alan |author2=Marsden, Jerrold E. |title=Calculus III |publisher=Springer-Verlag |___location=Berlin |year=1984|isbn=0-387-90985-0 |oclc= |doi= |page= 775}}

* {{cite book |author=Strang, Gilbert |title=Calculus |publisher=Wellesley College |___location= |year=1991|isbn=0-9614088-2-0 |oclc= |doi= |page= 94}}

*{{cite book |author1=Bock, David |author2=Hockett, Shirley O. |title=How to Prepare for the AP Calculus |publisher=Barrons Educational Series |___location=Hauppauge, NY |year=2005 |isbn=0-7641-2382-3 |oclc= |doi= |page=[https://archive.org/details/isbn_9780764177668/page/118 118] |url-access=registration |url=https://archive.org/details/isbn_9780764177668/page/118 }}

[[Category:Differential calculus]]