1. Section 14.1 The Partial Derivative

2. The following contour plot shows the monthly payment as a function of the interest rate and loan amount

3. For a fixed interest rate the spacing between contours is even, why? • Holding rate constant, is the rate of change positive or negative? • Does the constant spacing hold for a fixed loan amount?

4. Old stuff • Let y = f(x), then • This gives us • The slope of the tangent line at x = a • The instantaneous rate of change at x = a

5. New stuff • Let z = f(x, y) • We can find partial derivates with respect to each of the input variables (we hold the other constant) • The partial derivates at the point (a, b) is

6. Interpretations • Numerical • It gives us the instantaneous rate of change of f(x,y) at (a,b) per unit increase in x with y held fixed at b (or in y with x held fixed at a) • Graphical • It gives us the slope of the tangent line at (a,b) in the x direction (or y direction)

7. Estimating partial derivatives • If

8. Suppose z = f(x,y) is given by the table • Estimate the following

9. Notations for the partial derivatives • If z = f(x,y) we can write

10. C A B Determine the sign of the partials at the given points D