Linearization and Newton's Method Completed 2017.notebook Local Linear Approximations/Linearization Graph the function y = x2

Now Graph in your calculator and  Zoom in 3 separate times around the point (1,1).  

What do you notice as you zoom in around (1, 1)?

What does your graph look like now?

Why does your graph look like this? (What is this?)

Write the equation of the tangent line to y = x2 at (1, 1).   

Graph both the function and the tangent Line by hand. Now graph both of the functions in your calculator and go back to  your standard window. Zoom in 3 separate times around the point (1,1).  

What do you notice as you zoom in around (1, 1)?

Why does this matter?

If a function is locally linear (looks like a line as you continue to  zoom in) what does that tell you about the function?

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Finding a linear approximation @ x = a 1.  Take the derivative

2.  Find the equation of the tangent line           If f(x) is differentiable @ x = a, then the approximating function 

is called the linearization of f(x) @ x = a.

L(x) is the standard linear approximation of f(x) @ x (center of  approximation)

Remember that

Error of Approximation

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

In general, the closer x is to a, the better the approximation is to f(x).

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Example of Linearization: Find the tangent line approximation of 

@ x = 0.

Linearization and Newton's Method Completed 2017.notebook

x

­0.5

­0.1

­0.01

0

Find the error of the approximation @ x = 0.01

0.01

March 20, 2017

0.1

0.5

Linearization and Newton's Method Completed 2017.notebook

x

­0.5

­0.1

­0.01

0

Find the error of the approximation @ x = 0.01

0.01

March 20, 2017

0.1

0.5

Linearization and Newton's Method Completed 2017.notebook

Let 

March 20, 2017

and use the Linear approximation of the function @  x = 0 to approximate 

find the error approximation @  x = 0.275

Linearization and Newton's Method Completed 2017.notebook

Let 

March 20, 2017

and use the Linear approximation of the function @  x = 0 to approximate 

find the error approximation @  x = 0.275

Linearization and Newton's Method Completed 2017.notebook

find 

without a calculator.

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

find 

without a calculator.

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Find the linearization of                       at x = 2 to  approximate g(2.5).

Find the exact value of g(2.5) and the error of your  approxiimation.

Linearization and Newton's Method Completed 2017.notebook

Suppose that after you stuff a turkey its temperature is 

March 20, 2017

and you then put it in a 

oven.  After an hour the meat thermometer indicates that the temperature of the turkey is  and after two hours it indicates 

Predict the temperature of the turkey after three hours. 

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

Approximate the zero of 

March 20, 2017

within 0.0001

Linearization and Newton's Method Completed 2017.notebook

Compare the values of 

and dy if 

and x changes from 2 to 2.05 and from 2 to 2.01

March 20, 2017

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

The radius of a sphere was measured and found to be 21cm with a  possible error in measurement of at most 0.05cm.  What is the  maximum error in using this value of the radius to compute the  volume of the sphere?  What is the relative error?  What is the  percentage error?

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

The radius of a ball bearing is measured to be 0.7in.  If the  measurement is correct to within 0.01in, estimate the propagated  error in the volume V of the ball bearing.  Then find the relative and  percentage error.

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

The circumference of a sphere was measured to be 84 cm  with a possible error of 0.5 cm.  Estimate the maximum  error in the calculated surface error.  What is the relative  error? b)  Estimate the maximum error in the calculated volume.   What is the relative error?

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Estimate the amount of paint needed to apply a coat of paint  0.05 cm thick to a hemispherical dome with diameter 50 m.

How many gallons is that if  1 gallon = 0.00378541178 Cubic Meters ?

Linearization and Newton's Method Completed 2017.notebook

March 20, 2017

Linearization and Newton's Method Completed 2017.pdf ...

There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. Linearization ...

1MB Sizes 3 Downloads 171 Views

Recommend Documents

LOCAL LINEARIZATION METHOD FOR NUMERICAL INTEGRATION ...
INTEGRATION OF DELAY DIFFERENTIAL EQUATIONS∗. J.C. JIMENEZ ..... Bi n(˜yi tn (u − τi) − ˜yi tn (−τi)) + dn)du. + hn. ∫. 0 u. ∫. 0. eAn(hn−u)cndrdu. (2.16) ...

Newton's Method Completed 2017.pdf
There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. Newton's ...

The Local Linearization method for numerical integration of random ...
A Local Linearization (LL) method for the numerical integration of Random ... interest in the study of RDEs has been motivated by the development of the ... However, the application of these averaged methods is not only restricted to the ...

A Higher Order Local Linearization Method for Solving ...
A Higher Order Local Linearization Method for. Solving Ordinary Differential Equations. H. De la Cruz∗, R. J. Biscay†, F. Carbonell†, T. Ozaki‡, J. C. Jimenez†.

Newtons 3rd Intro.pdf
Page 2 of 3. Page 3 of 3. Newtons 3rd Intro.pdf. Newtons 3rd Intro.pdf. Open. Extract. Open with. Sign In. Main menu. Displaying Newtons 3rd Intro.pdf. Page 1 of ...

SPH 4U, T1L3, Newtons Laws and Friction v2
a = acceleration m/s2 (metres per second squared). Unit Analysis: Inertial mass – the m used in the second law is correctly described as the inertial mass.

Newtons Laws (Murray).pdf
Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. Newtons Laws (Murray).pdf. Newtons Laws (Murray).pdf. Open.

Completed Projects.pdf
1000units video brochure for the Nigeria. Custom Service. Product Specification. Product Design & Customization. Unit Testing. Project Management. Triune Built-Tech. Solutions. August 2014 Design and Development of Blubird - a retail. management plat

Newtons Laws Notes Review.pdf
There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. Newtons Laws Notes Review.pdf. Newtons Laws Notes Review.pdf. Open. Extract. Open with. Sign In. Main menu.

9) Linearization & Newton's Method.pdf
There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. 9) Linearization ...

Newtons 3rd Law Reading_20161218_0001_NEW.pdf
Page 1 of 7. Newton's ThirdLaw. 'When you push on the edge o{ etable or desk with your hand, your palm. indents in a way that can only happen if something is ...

Newtons Laws (Murray).pdf
Ex. A force of 49 N causes a 7 m/s2. acceleration. Find the mass of the. object it was pulling. Variables: 49 N = F. 7 m/s2. = a. m = ? Equation: F = ma. Solve: Newton's 2nd Law tells us that when. you accelerate (stomp on the gas) or. decelerate qui

2.3 Completed Notes.pdf
Sign in. Loading… Whoops! There was a problem loading more pages. Retrying... Whoops! There was a problem previewing this document. Retrying.

Rate of convergence of Local Linearization schemes for ...
Feb 17, 2005 - The proposal of this paper is studying the convergence of the LL schemes for ODEs. Specif- ..... [20] T. Barker, R. Bowles and W. Williams, Development and ... [27] R.B. Sidje, EXPOKIT: software package for computing matrix ...

Rate of convergence of local linearization schemes for ...
Linear discretization, the order of convergence of the LL schemes have not been ... In this paper, a main theorem on the convergence rate of the LL schemes for ...

Electrosurgery system and method
Dec 19, 2002 - FOREIGN PATENT DOCUMENTS. (22) Filed: Dec. ... US PATENT DOCUMENTS pulsed r.f. ...... voltage detector by the doctor. 4. A generator ...

Scanning apparatus and method
24 Dec 2009 - 29, 1991 from Mr. Stephen Crampton of 3D Scan ners Ltd. to Mr. Michel Brunet of Vision 3D, Marked as Page Nos. M0083274-M0083275. Vision 3D document labeled “Potential Partners”, addressed to 3D. Scanners Ltd., dated Jan. 10, 1991,

Scanning apparatus and method
Dec 24, 2009 - FOREIGN PATENT DOCUMENTS. DE. 3 938 714 A1. 5/1991. EP. 0159187 A1 10/1985. EP. 0159187. 10/1985. EP. 0 328 443. 8/1989. EP. 0 348 247. 12/1989. EP. 0 550 300. 7/1993. EP. 0 589 750. 3/1994. EP. 0 750 175. 12/1996. EP. 0 750 176. 12/19

11-2 notes (completed).pdf
Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. 11-2 notes (completed).pdf. 11-2 notes (completed).pdf. Open.

Tree Linearization in English: Improving Language ...
NLP applications. ... draw such a comparison, especially on English data, ... list of possible linearizations is too long and has to ... big N, etc.). Next, on the clause level, for every finite verb in the tree we find the order of its dependents us