3.9 Differentials

Objectives Understand the concept of a tangent line approximation. Compare the value of a differential, dy, with the actual change in y, Δy. Estimate a propagated error using a differential. Find the differential of a function using differentiation formulas.

In Newton’s Method, we used the tangent line to approximate zeros. Now we’ll use the tangent line to approximate the graph of a function in other situations.

For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.

Find the tangent line approximation of at the point (0,1). (In other words, find the tangent line at (0,1) and then use the tangent line to approximate values of f(x).)

Find the tangent line approximation of at the point (0,1). (In other words, find the tangent line at (0,1) and then use the tangent line to approximate values of f(x).) x f(x) y=1+x For values not close to x=0, tangent line would not be a good approximation.

Differentials: c (c,f(c)) ΔxΔx (c+ Δx,f(c+ Δx)) ΔyΔy dy

Example: Let y=x 2. Find dy when x=1 and dx=0.01. Compare this value with Δy for x=1 and Δx=0.01. Should be close for small dx and Δx values

Error Propagation Exact valueMeasured value

The radius of a ball bearing is measure to be 0.7”. If the measurement is correct to within 0.01”, estimate the propagated error in the volume of the ball bearing. very small change in V very small change in r (approximate change in volume)

Relative error: Percent error:

Find the differential.

Homework 3.9 (page 240) # 1 – 21 odd, 25 – 33 odd