Linear, Time-varying Approximations to Nonlinear Dynamical ...
Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations...
Transcript of Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations...
![Page 1: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/1.jpg)
Linear Approximations
ACADEMIC RESOURCE CENTER
![Page 2: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/2.jpg)
Table of Contents
Linear Function
Linear Function or Not
Real World Uses for Linear Equations
Why Do We Use Linear Equations?
Estimation with Linear Approximations
References
![Page 3: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/3.jpg)
Table of Contents
Linear Function
Linear Function or Not
Real World Uses for Linear Equations
Why Do We Use Linear Equations?
Estimation with Linear Approximations
References
![Page 4: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/4.jpg)
Linear Function
Definition: A mathematical equation in which no independent-variable”x” is raised to a power greater than one. A simple linear functionwith only one independent variable ”y” (y = ax + b) traces a straightline when plotted on a graph. Also known as a linear equation.
Famous Forms:
Y-axis form y = mx + b
Point-slope form (y − y1) = m(x − x1)
Intercept form(xc
)+(yb
)= 1
![Page 5: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/5.jpg)
Table of Contents
Linear Function
Linear Function or Not
Real World Uses for Linear Equations
Why Do We Use Linear Equations?
Estimation with Linear Approximations
References
![Page 6: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/6.jpg)
Linear Function or Not
I 4y = 3x + 2
I xy = 3
I 2x = 4y + 2
I x2 + 3y = 2
I x + 3 = y3
I√x + 3 = y
I x + y = 3x + 2
I x(3 + x) = y
I y = 3x
I 3(xy + y2) = 4y
I x2 + y
4 = 1
I 4a + 3b = 6
![Page 7: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/7.jpg)
Answers
I 4y = 3x + 2 Linear Function
I xy = 3 Not
I 2x = 4y + 2 Linear Function
I x2 + 3y = 2 Not
I x + 3 = y3 Not
I√x + 3 = y Not
I x + y = 3x + 2 Linear Function
I x(3 + x) = y Not
I y = 3x Linear Function
I 3(xy + y2) = 4y Linear Function
I x2 + y
4 = 1 Linear Function
I 4a + 3b = 6 Linear Function
![Page 8: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/8.jpg)
Reasoning for the Nonlinear Functions
I xy = 3 Not: Because the independent varialbe is multiplied tothe dependent variable.
I x2 + 3y = 2 Not: Because the independent variable is raised to apower other than 1.
I x + 3 = y3 Not: Because the dependent variable is raised to apower other than 1.
I√x + 3 = y Not: Because the independent variable is raised to a
power other than 1. (i.e.√x = x
12 )
I x(3 + x) = y Not: Because after distribution, the indenpendentvariable is raised to a power other than 1.
![Page 9: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/9.jpg)
Table of Contents
Linear Function
Linear Function or Not
Real World Uses for Linear Equations
Why Do We Use Linear Equations?
Estimation with Linear Approximations
References
![Page 10: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/10.jpg)
Real World Uses for Linear Equations
Popular Uses
I Demand Curves (economic analysis)
I Interest Rates and Investments (finance industry)
I Foreign Currency
Jobs
I Managers
I Financial Occupations
I Computer Programmers
I Scientists
I Engineers
I Administrators
I Construction
I Health Care
![Page 11: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/11.jpg)
Table of Contents
Linear Function
Linear Function or Not
Real World Uses for Linear Equations
Why Do We Use Linear Equations?
Estimation with Linear Approximations
References
![Page 12: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/12.jpg)
Why Do We Use Linear Equations?
Linear Equations are used in everyday life.
I Calculating travel times
I Converting hours to minutes
I Weights and measures (Doubling a recipe)
I Estimation
![Page 13: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/13.jpg)
Table of Contents
Linear Function
Linear Function or Not
Real World Uses for Linear Equations
Why Do We Use Linear Equations?
Estimation with Linear Approximations
References
![Page 14: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/14.jpg)
Estimation with Linear Approximations
Suppose we wanted to approximate√
99. We could say that√99 ≈
√100 = 10. However, using linear approximations, we can
obtain a better approximation than 10. Let us take a look at thenon-linear function f (x) =
√x . This function represents all of the
square roots. i.e. f (3) =√
3.
Now using Mathematica to visualize.
![Page 15: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/15.jpg)
Estimation with Linear Approximations
![Page 16: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/16.jpg)
Estimation with Linear Approximations
Now that we have motivation, we should find a linear approximationaround the point x = 100. Our reasoning is simply because we knowthe function value at that point and it is near 99. i.e. f (100) = 10.So we wish to find a line that passes through the function
√x at the
point x = 100, then we will use that line to approximate the pointx = 99. To start, let us take the form
y = mx + b
, where m is the slope and b is the y -intercept.
![Page 17: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/17.jpg)
Estimation with Linear Approximations
In order to determine the linear equation, we must determine whatthe slope of the line is. Since m = f ′(x),
m = f ′(x) =1
2√x
And we wish to know the slope of a line at the point x = 100, so theslope must be f ′(100) = 1
20 .
Now our equation is:
y =1
20x + b
![Page 18: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/18.jpg)
Estimation with Linear Approximations
Next we must determine b. We can use the point at which we aremaking this linear approximation, x = 100. By plugging in 10 for yand 100 for x , we get:
y =1
20x + b
10 =1
20(100) + b
10 = 5 + b
5 = b
Now we have our linear approximation of f (x) =√x about x = 100
in and will use it to approximate f (99).
y =1
20x + 5
![Page 19: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/19.jpg)
Estimation with Linear ApproximationsUsing Mathematica, we can plot the function and the linearapproximation together.
![Page 20: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/20.jpg)
Estimation with Linear ApproximationsZooming in near the point x = 100 we have:
![Page 21: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/21.jpg)
Estimation with Linear ApproximationsPlotting the error of the two functions, we can clearly see that thelinear approximation will be a good approximation for f (99).
![Page 22: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/22.jpg)
Estimation with Linear Approximations
We can see that the error for the linear approximation at x = 99 willbe small. So then we can obtain the estimation for the
√99. The
actual value is given by Mathematica.
This concludes the example for linear approximations. Hopefully youfind more uses in everyday life for linear approximations.
![Page 23: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/23.jpg)
Table of Contents
Linear Function
Linear Function or Not
Real World Uses for Linear Equations
Why Do We Use Linear Equations?
Estimation with Linear Approximations
References
![Page 24: Linear Approximations - Illinois Institute of TechnologyEstimation with Linear Approximations References. Table of Contents Linear Function Linear Function or Not ... I Managers I](https://reader033.fdocuments.in/reader033/viewer/2022041916/5e69f6c8b0359e77931f010b/html5/thumbnails/24.jpg)
References
I http://en.wikipedia.org/wiki/Linear_approximation
I http://www.ehow.com/facts_6027891_
examples-equations-used-real-life.html