1. Section 3.9 - Differentials

2. Local Linearity If a function is differentiable at a point, it is at least locally linear. Differentiable

3. Local Linearity If a function is differentiable at a point, it is at least locally linear. Not differentiable. The function is NOT smooth at this point.

4. Local Linearity If a function is differentiable at a point, it is at least locally linear. The function is differentiable at every point. The function is always smooth.

5. How to Approximate a Function A line that best approximates the graph of a function is the tangent line (if the values are near the point of tangency).

6. How to Approximate a Function Point of Tangency ( c, f (c) ) Tangent Line y – f (c) = f '(c) (x – c) A point (x,y) on the tangent line approximates the value of the function (must be near the point of tangency) y = f (c) + f '(c) (x – c)

7. Linear Approximation The linear approximation or tangent line approximation of a function f at x = c : This is also referred to as the linearization of f at x = c. Our Book uses T(x) instead of L(x)

8. Example 1 Find the local linear approximation of at x 0 = 1. Use the linearization formula if c = x 0 = 1.

9. Example 2 Use the linearization of at π/6 to approximate sin 0.5. Use the linearization formula if c = π/6. Evaluate the linearization formula at x = 0.5.

10. Error in Local Linear Approximations As a general rule, the accuracy of the linearization to f (x) at c deteriorates as x gets further away from c. The size of the error of the approximation is simply the vertical gap between the graph and the tangent line: Error Under EstimateOver Estimate

11. Example 2 (Continued) Use the linearization of at π/6 to approximate sin 0.5. Since Our approximation is an over estimate and the error is: The further away from π/6 the worse the approximation: xLinearizationActualError 0.50.4795630.4794260.000137 0.4 0.39296 0.389418 0.003542 0.010.05521.01000 0.045211

12. Example 3 Use linearization to approximate. Is the approximation an overestimate or underestimate. Since the square root of 4 is well known, let c = 4. Evaluate the linearization formula at x = 0.5. Let the function be: Since Our approximation is an over estimate

13. Introduction to Differentials So far we associated the following items to be equivalent: Now we will consider the differentials “ dy ” and “ dx ” to represent two different real numbers. Thus dy/dx becomes a real number ratio. Also, if f is differentiable at x, then becomes

14. Introduction to Differentials Slope of tangent line: Error Solve for the differential dy : Find the change in y : Thus the two vertical differences are almost equal: Thus the horizontal differences are always equal: Error is not the only “differen”ce we can calculate

15. Differentials If y = f (x), where f is a differentiable function, then the differential of x, dx, is an independent variable; that is, dx can be given the value of any real number. The differential of y, dy, is then defined in terms of dx by the equation: So dy is the is a dependent variable; it depends on the values of x and dx. Remember:

16. Example 1 Compare the values of Δy and dy if f (x) = x 3 + x 2 – 2x + 1 and dx = 0.05. Thus, Δy and dy are approximately equal (with Δy being slightly larger)

17. Example 2 Find the differential of Use the differential formula for dy.

18. Example 3 Find the differential dy of Since this is an implicit equation, you can either solve for y or use a method similar to the differentiation process in related rates/implicit differentiation. The following shows the latter: Now solve for d y :

19. Example 4 The radius r of a circle increases from 10 m to 10.1 m. Use differentials to estimate the increase in the circles area A. The function for area is: Use the differential formula to find dA : Evaluate the differential formula at r = 10 The increase in the circles areas is about 2π m 2

20. Three Ways to Describe Change ActualEstimated Actual Error (change) Relative Error (change) Percentage Error (change) As we move from c to a nearby point c + dx, we can describe the change in f in three ways: Other Vocab dx = Δx dy Measurement Error: Propagated Error:

21. Example 1 Inflating a bicycle tire changes its radius from 12 inches to 13 inches. Use differentials to estimate the actual change, the relative change, and the percentage change in the perimeter of the tire. Actual:Relative:Percentage: Calculate necessary information for the formulas: Estimate the changes (Errors): Actual Relative Percentage

22. Example 2 Suppose that the side of a square is measured with a ruler to be 10 inches with a measurement error of at most ± 1/32 in. Estimate the error in the computed area of the square. The function for area is: Use the differential formula to find dA : Evaluate the differential formula at s = 10 Thus, the propagated error in the area is estimated to be within ± 5/8 in 2

23. Example 3 The volume of a sphere is to be computed from a measured value of its radius. Estimate the maximum permissible percentage error in the measurement if the percentage error in the volume must be kept within ± 1.2%. The function for volume is: Thus, the estimated percentage error in the volume is within ± 0.4% Use the Differential Formula to find dV : Use the known information to Find % Error of the Radius : Definition of percentage error