the differential is dx, an approximation. This is quite useful when it is difficult to calculate delta x. How do we calculate dx? Click "oil slick" to see an example.

For the first year calculus student, the differentials involved in a linearization are a topic for serious study. The word "linear" is part of the word "linearization", so we would expect this to have something to do with a line. In fact the tangent line to the curve at any point can be used to approximate the curve, as long as one is in the neighborhood of that point. Why would this be important? It would be very important to you if the equation of the linearization line were much simpler than some difficult function you were estimating. In the case of the oil slick, suppose you want to do more than just use the differential to estimate the change in the area of the slick. Suppose you wish to predict its future size from some point in time (a particular radius of the slick). Go to that value r=b, and find the tangent line to the curve there. This is the linearization at b. You may use this linearization to estimate the size of the slick at nearby r's. Pretty slick, huh? (sorry) Click "linearize" to see this graphically.

What would be the general equation for a linearization? Click "explore" and see how much you can derive. Then use the equation to ..........

1. Find the linearization of

What is f(1.1)? f(2.1)? Estimate these with the linearization, then calculate them from f. How much error is there?

2. Find the linearization of

To return to the homepage, click " loondog".