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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.

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Chapter 2 Limits and the Derivative

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  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

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