1. Differentials, Estimating Change Section 4.5b

2. Recall that we sometimes use the notation dy/dx to represent the derivative of y with respect to x  this notation is not truly a ratio!!! This leads us to the definition of new variables: Differentials Let be a differentiable function. The differential dx is an independent variable. The differential dy is (dy is always a dependent variable that depends on both x and dx)

3. Guided Practice Find if

4. Guided Practice Find and evaluate for the given values of and. With the given data:

5. Differentials can be used to estimate change: Let be differentiable at. The approximate change in the value of when changes from to is

6. Guided Practice The given function changes value when x changes from a to a + dx. the absolute change Find: the estimated change

7. Guided Practice The given function changes value when x changes from a to a + dx. Find: the approximation error

8. Guided Practice The radius r of a circle increases from a = 10 m to 10.1 m. Use dA to estimate the increase in the circle’s area A. Compare this estimate with the true change in A.  Estimated increase is dA: m 2 True change: m 2 dAerror

9. Guided Practice Write a differential formula that estimates the given change in area. The change in the surface area of a sphere when the radius changes from a to a + dr. When r changes from a to a + dr… The change in surface area is approximately

10. Guided Practice Write a differential formula that estimates the given change in area. The change in the surface area of a cube when the edge lengths change from a to a + dx. When x changes from a to a + dx… The change in surface area is approximately

11. Guided Practice The differential equation tells us how sensitive the output of is to a change in input at different values of x. The larger the value of at x, the greater the effect of a given change dx.

12. Guided Practice You want to calculate the depth of a well from the given equation by timing how long it takes a heavy stone you drop to splash into the water below. How sensitive will your calculations be to a 0.1 second error in measuring the time? The size of ds in the equation depends on how big t is. If t = 2 sec, the error caused by dt = 0.1 is only ft Three seconds later at t = 5 sec, the error caused by the same dt: ft

13. Guided Practice The height and radius of a right circular cylinder are equal, so the cylinder’s volume is. The volume is to be calculated with an error of no more than 1% of the true value. Find approx. the greatest error that can be tolerated in the measurement of h, expressed as a percentage of h. We want, which gives The height should be measured with an error of no more than.

14. Guided Practice A manufacturer contracts to mint coins for the federal government. How much variation dr in the radius of the coins can be tolerated if the coins are to weigh within 1/1000 of their ideal weight? Assume the thickness does not vary. We want, which gives The variation of the radius should not exceed 1/2000 of the ideal radius, that is, 0.05% of the ideal radius.