AP Calculus BC Wednesday , 30 October 2013

1. AP Calculus BCWednesday, 30 October 2013 • OBJECTIVETSW use differentials to solve calculus problems. • ASSIGNMENTS DUE MONDAY • Sec. 3.9 • 4 problems listed on calendar

2. Sec. 3.9: Differentials

3. Sec. 3.9: Differentials Consider a function f that is differentiable at c. The equation for the tangent line at the point is or This is called the tangent line approximation(or linear approximation)of f at c.

4. Sec. 3.9: Differentials By restricting the values of x to be sufficiently close to c, the values of y can be used as approximations of the values of the function f. In other words, as x→ c, the limit of y is f (c). The quantity x – c is called the change in xand is denoted Δx.

5. Sec. 3.9: Differentials Now consider this, the secant line through The slope is When Δx is small, Δy can be approximated as “Approximate” because we are not taking a limit.

6. Definition The differential of y(denoted by dy) is Sec. 3.9: Differentials DefinitionΔx is called the differential of xand is denoted by dx.

7. Sec. 3.9: Differentials Ex: Find the differential dy of

8. Sec. 3.9: Differentials Ex: Find the differential dy of

9. Sec. 3.9: Differentials Ex: Find the differential dy of

10. Sec. 3.9: Differentials Ex: Find the differential dy of

11. Sec. 3.9: Differentials Ex: Find the differential dy of when x = 4 and dx = Δx = 3. Compare it to Δy. Use descriptive words: "bigger," "smaller," "larger," "greater than," "less than," etc.

12. Sec. 3.9: Differentials Ex: Find the differential dy of when x = 1 and dx = Δx = 0.01. Compare it to Δy.

13. Sec. 3.9: Differentials Ex: Use differentials to approximate Closest perfect square to 197. 14.03566885  Actual

14. Sec. 3.9: Differentials Ex: Use differentials to approximate Closest perfect cube to 6. 1.817120593  Actual