This is not the document you are looking for? Use the search form below to find more!

Report

# Lesson 12: Linear Approximation and Differentials

Document Description
Lesson 12: Linear Approximation and Differentials
File Details
Submitter
• Name: imogen
Embed Code:

#### Add New Comment

Related Documents

## Lesson 13: Linear Approximation

by: ishaan, 53 pages

Lesson 13: Linear Approximation

## Linear Algebra and its Applications Lay 3rd Edition Solutions Manual

by: georgesheslers, 48 pages

Linear Algebra and its Applications Lay 3rd Edition Solutions Manual

## Linear Approximation Examples

by: ramsingh11, 5 pages

By now we have seen many examples in which we determined the tangent line to the graph of a function f(x) at a point x = a. A linear approximation (or tangent line approximation) is the simple idea ...

## Lesson 6: Polar, Cylindrical, and Spherical coordinates

by: morela, 20 pages

Lesson 6: Polar, Cylindrical, and Spherical coordinates

## Lesson 15: Exponential Growth and Decay (Section 041 handout)

by: jansen, 13 pages

Lesson 15: Exponential Growth and Decay (Section 041 handout)

## Lesson 9: The Product and Quotient Rule

by: jayden, 39 pages

Lesson 9: The Product and Quotient Rule

## Lesson 15: Inverse Functions And Logarithms

by: madison, 31 pages

Lesson 15: Inverse Functions And Logarithms

## Linear Approximation Examples

by: ramsingh11, 5 pages

Through the examples we looked at for the area under graphs of functions, we were led to an interesting observation: there seems to be a relationship between the process of integration, which is just ...

## Spire Tech Greater Noida,Spire Tech Park Greater Noida,Tech Park Noida , Spire Tech Noida,12% ASSURED RETURN and Best Deal to Call : 011-64-666777 on Sabakhub.com

by: sabkahub1@gmail.com, 10 pages

Spire Tech Greater Noida - India’s First Mainstream Green Office Complex. SPIRE TECH PARK Greater Noida is being developed as a State-of-Art IT park with an aim to create an ...

## Pattern and differentials of morbidity among under-five children in Bangladesh

by: mortuza ahmmed, 13 pages

The study of infant and child mortality in developing countries is an important issue in public health programs. With the increasing emphasis on planning programs in recent years, it becomes ...

Content Preview
Section 2.8Linear Approximation andDifferentialsV63.0121.027, Calculus IOctober 13, 2009AnnouncementsMidterm Thursday on Sections 1.1–2.4OutlineThe linear approximation of a function near a pointExamplesDifferentialsThe not-so-big ideaUsing differentials to estimate errorMidterm ReviewAdvanced ExamplesAnswerThe tangent line, of course!QuestionWhat is the equation for the line tangent to y = f(x) at (a, f(a))?AnswerL(x) = f(a) + f′(a)(x − a)The Big IdeaQuestionLet f be differentiable at a. What linear function bestapproximates f near a?QuestionWhat is the equation for the line tangent to y = f(x) at (a, f(a))?AnswerL(x) = f(a) + f′(a)(x − a)The Big IdeaQuestionLet f be differentiable at a. What linear function bestapproximates f near a?AnswerThe tangent line, of course!AnswerL(x) = f(a) + f′(a)(x − a)The Big IdeaQuestionLet f be differentiable at a. What linear function bestapproximates f near a?AnswerThe tangent line, of course!QuestionWhat is the equation for the line tangent to y = f(x) at (a, f(a))?The Big IdeaQuestionLet f be differentiable at a. What linear function bestapproximates f near a?AnswerThe tangent line, of course!QuestionWhat is the equation for the line tangent to y = f(x) at (a, f(a))?AnswerL(x) = f(a) + f′(a)(x − a)√3 1 ()+x − π2230.87475√3212So L(x) =Thus ( )61πsin≈1800.87462.Solution (i)Solution (ii)If f( )(x) = sin x, then f(0) = 0We have f π =andand f′3)(0) = 1.f′ (π.3=So the linear approximationnear 0 isL(x) = 0 + 1 · x = x.Thus()61π61πsin≈≈ 1.06465180180Calculator check: sin(61◦) ≈ExampleExampleEstimate sin(61◦) by using a linear approximation(i) about a = 0(ii) about a = 60◦ = π/3.√3 1 ()+x − π2230.87475√3212So L(x) =Thus ( )61πsin≈1800.87462.Solution (ii) ( )We have f πand3=)f′ (π.3=Calculator check: sin(61◦) ≈ExampleExampleEstimate sin(61◦) by using a linear approximation(i) about a = 0(ii) about a = 60◦ = π/3.Solution (i)If f(x) = sin x, then f(0) = 0and f′(0) = 1.So the linear approximationnear 0 isL(x) = 0 + 1 · x = x.Thus()61π61πsin≈≈ 1.06465180180√3 1 ()+x − π2230.874750.87462.√3212So L(x) =Thus ( )61πsin≈180Calculator check: sin(61◦) ≈ExampleExampleEstimate sin(61◦) by using a linear approximation(i) about a = 0(ii) about a = 60◦ = π/3.Solution (i)Solution (ii)If f( )(x) = sin x, then f(0) = 0We have f π =andand f′3)(0) = 1.f′ (π.3=So the linear approximationnear 0 isL(x) = 0 + 1 · x = x.Thus()61π61πsin≈≈ 1.06465180180√3 1 ()+x − π2230.874750.87462.12So L(x) =Thus ( )61πsin≈180Calculator check: sin(61◦) ≈ExampleExampleEstimate sin(61◦) by using a linear approximation(i) about a = 0(ii) about a = 60◦ = π/3.Solution (i)Solution (ii)√If f( )(x) = sin x, then f(0) = 0We have f π = 3 andand f′32)(0) = 1.f′ (π.3=So the linear approximationnear 0 isL(x) = 0 + 1 · x = x.Thus()61π61πsin≈≈ 1.06465180180Document Outline
• Announcements
• The linear approximation of a function near a point
• Examples
• Differentials
• The not-so-big idea
• Using differentials to estimate error
• Midterm Review

Lesson 12: Linear Approximation and Differentials

If it doesn't, click here to try again.

Share Lesson 12: Linear Approximation and Differentials to:

Insert your wordpress URL:

example:

http://myblog.wordpress.com/
or
http://myblog.com/

Share Lesson 12: Linear Approximation and Differentials as:

From:

To:

Share Lesson 12: Linear Approximation and Differentials.

Enter two words as shown below. If you cannot read the words, click the refresh icon.

Share Lesson 12: Linear Approximation and Differentials as:

Copy html code above and paste to your web page.