1 / 14

Chapter 2 Limits and the Derivative

Chapter 2 Limits and the Derivative. Section 6 Differentials. Increments. The derivative of f at x is the limit of the difference quotient:. Increment notation allows interpreting the numerator and the denominator of the difference quotient separately. Example Concept of Increment.

melissat
Télécharger la présentation

Chapter 2 Limits and the Derivative

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 2Limitsand theDerivative Section 6 Differentials

  2. Increments The derivative of f at x is the limit of the difference quotient: Increment notation allows interpreting the numerator and the denominator of the difference quotient separately.

  3. Example Concept of Increment For y = f (x) = x3, a change in x from 2 to 2.1 corresponds to a change in y from y = f (2) = 8 to y = f (2.1) = 9.261. Increment Notation Change in x (the increment in x) is denoted by ∆x. The Greek letter delta, in mathematics stands for a difference or change. Change in y (the increment in y) is denoted by ∆y. In the example, ∆x = 2.1 – 2 = 0.1 ∆y = f (2.1) – f (2) = 9.261 – 8 = 1.261.

  4. Increments Interpreted Graphically For y = f (x), ∆x = x2 – x1, so x2 = x1 + ∆x, and ∆y = y2 – y1 = f (x2) – f (x1) = f (x1 + ∆x) – f (x1) ∆x can be either positive or negative. ∆y represents the change in y corresponding to a ∆x change in x.

  5. Example Increments Solution: (A) Δx = x2 – x1 = 2 – 1 = 1

  6. Example Increments

  7. Differentials

  8. Definition Differentials

  9. Differentials The tangent line has slope f´(x) with horizontal change dx. The vertical change is given by dy= f´(x) dx.

  10. Interpretation of Differentials ∆x and dx both represent change in x. The increment ∆ystands for the actual change in y corresponding to the change in x. The differential dy stands for the approximate change in y, estimated by using derivatives. In applications, we use dy to estimate ∆y.

  11. Example Differentials Find dy for f (x) = x2 + 3x and evaluate dy for x = 2 and dx = 0.1. Solution: dy = f´(x) dx = (2x + 3) dx For x = 2 and dx = 0.1, dy = (2(2) + 3) 0.1 = 0.7

  12. Example Comparing Increments and Differentials (A) Find ∆y and dy when x = 2.

  13. Example Comparing Increments and Differentials (B) Find ∆y and dy from part (A) for Δx = 0.1, 0.2, and 0.3

  14. Example Cost-Revenue A company manufactures and sells x transistor radios per week. If the weekly cost and revenue equations are use differentials to approximate changes in revenue and profit if production is increased from 2,000 to 2,010 units/week. Change in revenue dR Change in profit dP

More Related