Section 14.3 Local Linearity and the Differential

Local Linearity Let y = f(x) where f is differentiable at x = a Recall that when we “zoomed in” on a differentiable graph, it became almost linear, no matter how much curve there was in the original graph Therefore a tangent line at x = a can be a good approximation for a function near a Let’s take a look at the function and its tangent line at x = 0

Between -.1 and.1 both graphs look almost identical Now, if we are given an initial height of b and that the line changes by the amount f’(0) per unit change in x, then our line is Now if we are at x = 3 instead of the intercept, we can adjust our line to get In general an estimate of f(x) near a is given by

Local Linearity in 3 Space Recall the general equation of a plane So we are starting at a height c above the origin and move with slope m in the positive x direction and slope n in the positive y direction Now on the z-axis So the tangent plane approximation to a function z = f(x,y) for (x,y) near (a,b) is

Differentials The differential of f at x = a is defined as The differential is essentially the distance between two values as we increase (or decrease) x Let’s use a differential to approximate Now let’s talk about 3 space

Differentials in 3 space If z = f(x,y) is a function in 3 space and is differentiable at (a,b), the its differential is defined as For all x near (a,b) and

Why do we care about differentials? In two dimensions a differential gives us an expected change in our y-value based on a change in our x-value (and the derivative at that point) In 3 dimensions we get an expected change in our z-value based on a change in our x and y values (and the partial derivatives at that point) –As these are estimates, the larger the change in the inputs, the less accurate our approximation will be