1. Section 14.7 Second-Order Partial Derivatives

2. Old Stuff Let y = f(x), then Now the first derivative (at a point) gives us the slope of the tangent, the instantaneous rate of change, and whether or not a function is increasing What does the second derivative give us? If f’’(x) > 0 on an interval, f is concave up on that interval If f’’(x) < 0 on an interval, f is concave down on that interval

3. New Stuff Suppose z = f(x,y) Now we’ve looked at and –What do they give us? For z we have 4 second-order partial derivatives

4. Let’s practice a little Find the first and second order partial derivatives of the following functions What do you notice about the mixed partials?

5. Interpretations of second order partial derivatives If then f is concave up in the x direction or the rate of change is increasing at an increasing rate in the x direction (similar for ) When we are looking at the mixed partials, we are looking at how a partial in one variable is changing in the direction of the other For example, tells us how the rate of change of f in the x direction is changing as we move in the y direction

6. x y 40302010 50 60 P●P● Use the following contour plot to determine the sign of the following partial derivatives at the point, P.

7. We can use these in Taylor approximations Recall we can approximate a function using a 1st degree polynomial This polynomial is the tangent line approximation The tangent line and the curve we are approximating have the same slope at x = a The tangent line approximation is generally more accurate at (or around) x = a

8. To get a more accurate approximation we use a quadratic function instead of a linear function –In order to make this more accurate, we require that our approximation have the same value, same slope, and same second derivative at a The Taylor Polynomial of Degree 2 approximating f(x) for x near a is

9. Now let z = f(x,y) have continuous first and second partial derivatives at (a,b) Now we know from 14.3 that Note that our approximation has the same function value and the partial derivatives have the same value as f at (a,b) Therefore our quadratic Taylor approximation is

10. From we can see that our approximation matches the function value at (a,b) as well as all the partials at (a,b) Let’s take a look at this with Maple