3.8 Local Linear Approximations; Differentials (page 226) b We have been interpreting dy/dx as a single entity representing the derivative of y with respect.

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Presentation transcript:

3.8 Local Linear Approximations; Differentials (page 226) b We have been interpreting dy/dx as a single entity representing the derivative of y with respect to x. b We will now give the quantities dy and dx separate meanings that will allow us to treat dy/dx as a ratio. b dy terminology and the concept of differentials will also be used to approximate functions by simpler linear functions.

Differentials (page 228)

Historical Note (See page 4) b Woolsthorpe, England b Not gifted as a youth b Entered Trinity College with a deficiency in geometry. b 1665 to discovered Calculus b Calculus work not published until 1687 b Viewed value of work to be its support of the existence of God. Isaac Newton ( )

Historical Note (See page 5) b Leipzig, Germany b Gifted Genius b Entered University of Altdorf at age 15 b Doctorate by age 20 b Developed Calculus in 1676 b Developed the dy/dx notation we use today Gottfried Wilhelm Leibniz (

Differentials (page 229)

Differential Notation (page 212)

Example / Not in this edition

Example 4 (page 229)

Example 2 (page 229)

Error Calculation

Homework Example #30, #36 page 233

Local Linear Approximation (page 226) b If the graph of a function is magnified at a point “P” that is differentiable, the function is said to be locally linear at “P”. b The tangent line through “P” closely approximates the graph. b A technique called “ local linear approximation” is used to evaluate function at a particular value.

Locally Linear Function at a Differentiable Point “P” (page 226)

Function Not Locally Linear at Point “P” Because Function Not Differentiable at Point “P”

Local Linear Approximation Formulas (page 227)

Example 1 (page 227) / 1

Example 1 (page 227)

Error Propagation in Applications (page 230) b In applications small errors occur in measured quantities. b When these quantities are used in computations, those errors are “propagated” throughout the calculation process. b To estimate the propagated error use the formula below:

Example 6 (page 231) / 6

Homework Example - 20 (page 233)