1 / 14

Differentials, Estimating Change

Differentials, Estimating Change. Section 4.5b. 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.

tyler
Download Presentation

Differentials, Estimating Change

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. 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 dyis (dy is always a dependent variable that depends on both x and dx)

  3. Guided Practice Find if 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 xchanges from ato a+ dx. the absolute change Find: the estimated change

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

  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 areaA. Compare this estimate with the true change in A.  Estimated increase is dA: 2 m True change: 2 m dA error

  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.

More Related